Martin Van Horn

Hans Jørgen

Aagaard Jensen

Patrick Norman

Ruud Kenneth

Trond Saue

Maen Salman

Jan Brandejs

Johann Pototschnig

Gabriele Riva

Raul Quintero

Miguel Escobar Azor

come

The validity of the dipole approximation in the simulation of X-ray absorption spectroscopy

The first article in this list is marked with an asterisk, because it is based on work from my Master's internship. I have included it to this list regardless, due to its high relevance to current work

As can be inferred from the above list, several authors have contributed to these articles, so in the following, I will explicitly state my own contributions: I provided the analysis to conclude that schemes bases on truncated light-matter interactions converge extremely slowly in the hard X-ray regime; furthermore, I performed the calculations and analyses related to the basis set studies; I derived the equations-and implemented all the code related to electronic circular dichroism and lastly I implemented the calculation of oscillator strengths in the generalized velocity representation with explicit separation between electricand magnetic components.

Introduction

Over the centuries, the evolution of chemistry is characterized by the emergence of distinct sub-disciplines whose boundaries gradually eroded over time. One notable example is the rise of organic chemistry during the nineteenth century. [START_REF] Levere | Transforming matter: a history of chemistry from alchemy to the buckyball[END_REF]Chapter 8] In its early days, this field was mainly preoccupied with the chemistry of living matter, whereas its more developed and older cousin, inorganic chemistry, was rather centered around inorganic salts and mineral acids. For that reason, a dichotomous division of chemistry was prevalent, according to which organic chemistry is based on fundamentally different principles than inorganic chemistry. However, one key discovery that challenged this paradigm is the synthesis of urea from inorganic matter by Wöhler in 1828. [START_REF] Wöhler | Ueber künstliche Bildung des Harnstoffs[END_REF] Combined with the further improvements in the characterization techniques of organic compounds, a unification of both fields was inevitable. The chemists at the heart of these developments can be likened to detectives, meticulously deducing structural features using minimalistic resources, such as decomposition analysis and chemical reactivity. [START_REF] Levere | Transforming matter: a history of chemistry from alchemy to the buckyball[END_REF]Chapter 9] In modern-day chemistry, the resolution of molecular structure has become a routine task due to the advent of spectroscopy. Spectroscopy generally measures light-matter interactions in terms of absorption, emission or scattering as a function of the wave length. However, the utility of spectroscopy extends beyond mere structural characterization, since it also serves as a technique to gauge molecular properties, whose nature generally depends on the wave length of light. [START_REF] Norman | Principles and Practices of Molecular Properties: Theory, Modeling and Simulations[END_REF]Chapter 8] For example, in the lower energy range of the spectrum, i.e. microwave and IR, the radiation induces molecular rotations and vibrations, whereas the UV-Vis regime is associated with electronic excitations. [START_REF] William | Molecular quantum mechanics[END_REF]Chapters 10 and 11] Further decreasing the wave length gives rise to X-rays, whose application in molecular spectroscopy has been pioneered by de Broglie [START_REF] De | Sur une nouveau procédé permettant d'obtenir la photographie des spectres de raies des rayons Röntgen[END_REF], Moseley [START_REF] Moseley | XCIII. The high-frequency spectra of the elements[END_REF], Siegbahn [START_REF] Siegbahn | Röntgenspektroskopische Präzisionsmessungen[END_REF][START_REF] Siegbahn | LVIII. Precision-measurements in the X-ray spectra[END_REF] and Stenström. [START_REF] Stenström | Experimentelle Untersuchungen der Röntgenspektra[END_REF] X-rays have sufficient energy to reveal the shell structure of the inner electrons, which are buried deep inside the core and otherwise unaffected by UV-Vis radiation. Furthermore, above a certain threshold, X-rays may even ionize the system, which manifests itself as sudden vertical jumps in the absorption spectrum, also referred to as the edges. The edges form a particularly useful tool in structural analysis, as they are centered at the ionization energies of core electrons, thus being element-specific and well-separated.

However, the more interesting structural features are not easily deduced from the edges alone, but rather from its neighbouring regions. [START_REF] Bianconi | Surface X-ray absorption spectroscopy: Surface EXAFS and surface XANES[END_REF][START_REF] Stumm Von Bordwehr | A History of X-ray absorption fine structure[END_REF] In the region beyond the edge, subtle oscillations can be observed, caused by interferences of photoelectrons that are back-scattered from their local surroundings. The technique to measure these oscillations, also referred to state and local coordination environment of metal complexes. [START_REF] Westre | A Multiplet Analysis of Fe K-Edge 1s → 3d Pre-Edge Features of Iron Complexes[END_REF] Therefore, the advances in experimental techniques should go hand-in-hand with the development of theory and implementation thereof.

To simulate X-ray spectroscopy, it seems only natural to apply the well-established theory developed for the UV-Vis regime. However, a large discrepancy with experiment is observed if we blindly transfer this theory to the X-ray domain. To correct this discrepancy, it is thus necessary to include several effects beyond what is typically required for UV-Vis spectroscopy. First of all, since relativistic effects are generated in the core region, they need consideration in the simulation of core excitations already for quite light elements. [START_REF] Pyykko | Relativistic effects in structural chemistry[END_REF][START_REF] Fransson | Carbon X-ray absorption spectra of fluoroethenes and acetone: A study at the coupled cluster, density functional, and static-exchange levels of theory[END_REF] Secondly, after exciting the core electron, the system is left with a core-hole that reduces the screening of the nuclear charge. As a result, the effective nuclear charge increases, which amounts to an additional stabilization of the excited state, referred to as core-hole relaxation. [START_REF] Siegbahn | Electron spectroscopy for atoms, molecules, and condensed matter[END_REF][START_REF] Drescher | Time-resolved atomic inner-shell spectroscopy[END_REF] Third, in the UV-Vis regime, the wavelength is considerably larger than the extent of the molecule, suggesting that the fields appear to be uniform from the perspective of the molecule. Due to this difference in scale, the light-matter interaction can effectively be described by the first terms of a multipole expansion, yielding the electric-dipole approximation for linearly polarized light and the rotational strength for circular dichroism, the latter of which depends on both the electric-and magnetic-dipole moment. Considering that in both cases only dipolar contributions are needed, it seems natural to use dipole approximation as an umbrella term to designate both of these schemes. In the X-ray regime, however, the dipole approximation come into question. In this thesis, it will be assessed whether the dipole approximation holds in the simulation of X-ray absorption spectroscopy.

In general, there are two possible approaches to include non-dipolar effects: either the semi-classical light-matter interaction is treated exactly, or the multipole expansion is truncated beyond zeroth order. However, truncated expansions inherently introduce gauge-origin dependency in the calculation, a problem which Bernadotte et al. [START_REF] Bernadotte | Origin-independent calculation of quadrupole intensities in X-ray spectroscopy[END_REF] solved by expanding the absorption cross-section rather than the light-matter interaction itself. Furthermore, schemes of this type may also suffer from negative absorption cross-sections [START_REF] Lestrange | The consequences of improperly describing oscillator strengths beyond the electric dipole approximation[END_REF] and appearant divergences, which are, in principle, both alleviated if enough terms are included in the multipole expansion, although the latter problem may require an unreasonably large amount. [START_REF] Holmgaard List | Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory[END_REF] There are several other examples of implementations, besides the ones from the abovementioned references, that are capable to compute absorption intensities using either a truncated interaction, [START_REF] Kragh Sørensen | Applications to metal K pre-edges of transition metal dimers illustrate the approximate origin independence for the intensities in the length representation[END_REF][START_REF] Kragh Sørensen | Gauge origin independence in finite basis sets and perturbation theory[END_REF] or the full interaction. [START_REF] Holmgaard List | Beyond the electric-dipole approximation: A formulation and implementation of molecular response theory for the description of absorption of electromagnetic field radiation[END_REF][START_REF] Holmgaard List | Rotationally averaged linear absorption spectra beyond the electric-dipole approximation[END_REF][START_REF] Kragh Sørensen | Implementation of a semiclassical light-matter interaction using the Gauss-Hermite quadrature: A simple alternative to the multipole expansion[END_REF] Some of these schemes have even been extended to model more complicated phenomena such as X-ray magnetic circular dichroism, [START_REF] Nicolás | Going beyond the electric-dipole approximation in the calculation of absorption and (magnetic) circular dichroism spectra including scalar relativistic and spin-orbit coupling effects[END_REF] X-ray scattering [START_REF] Mickaël G Delcey | Exact semi-classical light-matter interaction operator applied to two-photon processes with strong relativistic effects[END_REF] or transient absorption. [START_REF] Einar Aurbakken | Transient spectroscopy from time-dependent electronic-structure theory without multipole expansions[END_REF] However, all of these implementations have a limited spectral range of applicability, since they are derived from a non-relativistic framework, having at most perturbative relativistic corrections. In the hard X-ray regime, where non-dipolar effects are most relevant, the light typically excites core orbitals of heavy elements, which demand a fully relativistic treatment. To answer the main question in a way that also holds for the hard X-ray regime, it is thus mandatory to work within such a framework.

In this thesis, I will present an implementation to simulate light-matter interactions using both the full-and truncated interaction in a four-component relativistic framework. This implementation is a part of Dirac, [START_REF] Saue | The DIRAC code for relativistic molecular calculations[END_REF] a quantum chemical code specialized in relativistic molecular calculations either at the two-or four-component level. However, it should be noted that even this level of theory has a limited range of validity. At extreme energy scales, such as the ones encountered in highly charged cations, the effects of quantum electrodynamics may play a role in X-ray absorption spectra, either affecting the core electron through vacuum polarization/self-energy effects, or pair-creation stemming from high-energy photons. In addition, I have not included the effects of the core hole, the importance of which has been alluded to in previous paragraph. However, the treatment of these effects is beyond the scope of this thesis and will not be further pursued. This thesis will be divided into four chapters. The first three discuss the required theory, whereas the fourth chapter discusses the Dirac implementation itself together with some applications. In particular, each chapter contains the following subjects:

• In Chapter 1, I will discuss the conventional methodology to calculate UV-Vis absorption spectra.

• Chapter 2 will be devoted to the basic principles of relativistic quantum chemistry.

• Subsequently, in Chapter 3, it will be discussed how light-matter interactions can be treated beyond the dipole approximation.

• Using the results from previous three chapters, I will discuss the implementation in Dirac and apply it to various examples in Chapter 4.

• The last section is devoted to the conclusion and outlook.

The various quantities appearing throughout this work are expressed in SI units and unless stated otherwise, the Einstein summation convention will be applied.

Chapter 1

Conventional Methods to Simulate UV-Vis Absorption Spectroscopy

Introduction

In this chapter, I will illustrate conventional quantum chemical methods to calculate UV-Vis absorption spectra. The ideas and derivations expressed in this chapter generally follow the book Principles and Practices of Molecular Properties [START_REF] Norman | Principles and Practices of Molecular Properties: Theory, Modeling and Simulations[END_REF], which should be consulted for further details. To classify the type of approximations typically made in the simulation of UV-Vis spectroscopy, I will divide the calculation of absorption intensities into three main components: the ground state calculation, excited-state calculation and the treatment of the light-matter interaction.

To simplify the ground state calculation, approximations can be applied to either the electronic-structure method or the Hamiltonian. For example, the valence orbitals typically involved in UV-Vis spectroscopy are the least susceptible to relativistic effects, suggesting that a non-relativistic Hamiltonian, such as the Schrödinger-or Pauli Hamiltonian, is usually sufficient. After having decided which Hamiltonians to use, an electronic structure method has to be chosen to approximate the complexity of the many-body wave function. There is a plethora of electronic structure methods to choose from, each of which having a distinct cost-to-accuracy ratio. For example, the ground state can most reliably be obtained using wave function methods such as coupled cluster (CC) or configuration interaction (CI), albeit at the expense of computational costs. The computational costs are less for the Hartree-Fock (HF), although its accuracy is rather low as well. An improved cost to accuracy ratio can be obtained by employing density-functional theory (DFT), which roughly comes at the cost of a HF calculation and can even approach the accuracy of wave function methods in some cases. In this thesis, HF and DFT will be the methods of choice to describe the ground state. Hereafter, I will use the term self-consistent-field methods to encompass both approaches.

Using the ground-state as a starting point, there is a variety of methods available to treat excited states, as will become more clear in this chapter. For exact states, and CI wave functions, this procedure is rather straightforward, because excited states can be obtained directly from diagonalization of the Hamiltonian. Following this route at the SCF level is more cumbersome, because HF and DFT are essentially ground state theories. However, explicit calculation of excited states can be avoided by applying linear response theory. In this chapter, it will be shown how the poles and residues of the linear response functions can be related to excitation energies and transition moments, which in turn, can be used to obtain absorption intensities. As a main drawback, this method does not take core-hole effects into consideration, which should not matter too much in the UV-Vis regime.

What thus remains is to find a suitable description of the light-matter interaction. In this work, a semi-classical description will be pursued, although there are examples in literature in which the quantum mechanical nature of light is taken into account. [START_REF] Flick | Atoms and molecules in cavities, from weak to strong coupling in quantum-electrodynamics (QED) chemistry[END_REF][START_REF] Rosario R Riso | Molecular orbital theory in cavity QED environments[END_REF] A particular feature of the UV-Vis regime is the large size of the wavelength compared to the molecular system, which allows for the application of the dipole approximation.

To better understand the origins of this approximation, let us consider the following example. Suppose we want to model the absorption of linearly polarized red light (λ = 700 nm) by a benzene molecule. The electromagnetic fields describing this type of light are shown below

E (r, t) = E ω sin [k • r -ωt + δ] ; B (r, t) = E ω ω (k × ) sin [k • r -ωt + δ] , (1.1)
where appears the wave vector k with length

k = ω c = 2π λ , (1.2)
the polarization vector and the phase δ. [START_REF] David | Introduction to electrodynamics[END_REF]Chapter 9][41, Chapter 7] Light can be polarized in other forms than the one shown above, but that discussion will be postponed to Chapter 3.

For dilute solutions, the absorption of light is given by the Lambert-Beer law

I I 0 = e -N σ(ω) . (1.3)
In this expression, I and I 0 are the out-and ingoing intensities, whereas the exponent gives the effective number of absorbing molecules, expressed in terms of the number density of absorbing molecules, N , the length traversed by the light, , and the absorption cross-section, σ(ω). The latter can be regarded as a surface spanned by all molecules that have successfully absorbed the incoming light, hence forming a measure of the absorption intensity. The quantities expressed in the Lambert-Beer law are macroscopic in nature and are in principle, only available from experiment, whereas quantum chemical simulations typically provide microscopic quantities. In the context of light absorption, the two scales can be connected by the following relation

Iσ(ω f i ) = ω f i w i→f , (1.4)
where w i→f is the transition rate and ω f i the energy difference between the initial-and final state. Both sides of this equation give a measure of the energy that is absorbed by the system. The transition rate can be calculated using quantum mechanical methods, thus avoiding the need to directly calculate the absorption cross-section. Before we proceed to calculate this quantity, let us first take a step back and examine the system under consideration (Figure 1.1).

Figure 1.1: schematic representation of a benzene molecule interacting with linearly polarized red light (λ = 700 nm) consisting of an electric (red) and magnetic (blue) component. To make the magnetic field amplitude visible in this picture, it is not depicted at its true relative scale with respect to the electric field. The benzene molecule is only visible after zooming into a small region of space and in this region, the molecule effectively experiences a uniform field, which is dominated by its electric component. The mathematical expressions describing the electromagnetic fields are given at the bottom of the box. The propagation-and polarization direction are described by the unit vectors e k and , respectively, with the latter being defined by the relation e k = c ω k.

What stands out is the vast difference in scale between the wavelength and the benzene molecule, the latter of which can only be perceived in a zoomed-in frame. In this frame, most of the curvature of the plane wave is lost, implying that the molecule effectively experiences a uniform field, whose magnetic component can safely be neglected due to its prefactor of 1/c. Formulated more mathematically, the light-matter interaction is expanded in orders of the wave vector and truncated at first order, justified by the limit kr 1. At this order, our interaction is given by the electric-dipole operator, corresponding to a uniform electric field. These considerations form the basis of the electric-dipole approximation, which is widely used in UV-Vis spectroscopy. To describe ECD, however, the electric-dipole approximation is not sufficient, since this interaction also depends on the magnetic-dipole operator, appearing an order higher in our expansion.

Let us now direct our attention to the Hamiltonian that describes this system. Without the loss of generality, our Hamiltonian can be separated into a static part (given by a nonrelativistic Hamiltonian) and a time-dependent interaction operator

Ĥ = Ĥ0 + V (t); V (t) = ∞ -∞ dte -iωt V (ω), (1.5)
the latter of which is conveniently expressed as a Fourier transform. Furthermore, hermiticity of the interaction operator implies the following relation in the frequency domain

V † (t) = V (t) → V † (ω) = V (-ω). (1.6)
I will postpone the explicit construction of this operator to Chapter 3, as it suffices for now to assume that this operator is periodic. In turn, this implies that the Fourier transform of the interaction operator is discretized into frequency components that are integer multiples of a fundamental frequency

V (t) = V (t + T ) → ω = nω T ; ω T = 2π T , n ∈ N. (1.7)
Therefore, the interaction operator can be expressed as

V (t) = N y=-N
λ ωy α Vα (ω y )e -iωyt+ t , (

where the indices y and α represent the Fourier-and Cartesian components. For the Cartesian components, the Einstein summation convention is applied, whereas the sum over frequency components is written explicitly. As can be concluded from the hermiticity of the overall operator, the summation over the frequency components contains positive-and negative indices, corresponding to equal but opposite pairs of frequencies ω 0 , (ω 1 , -ω 1 ), (ω 2 , -ω 2 ), (ω 3 , -ω 3 ), • • • .

(1.9)

Here, the zeroth frequency is zero (ω 0 = 0), thus allowing for the possibility of static contributions to our perturbation. Furthermore, in this expression, is an infinitesimal number ensuring that the perturbation is switched on slowly, i.e. an adiabatic switch. Within the electric-dipole approximation, the frequency components are given by the electric-dipole moment operator Vα (ω y ) = Q [START_REF] Holmgaard List | Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory[END_REF] α = -er α ;

λ ωy α = 1 2 E ω α , (1.10)
where -e is the electron charge. In our particular case of monochromatic light, only one pair of frequencies contributes, allowing us to express our interaction operator in a more simple form

V (t) = E ω α Q[1]
α cos(ωt)e t .

(1.11)

However, in the following, the representation from Eqn. (E.8) is preferred to preserve generality of the formalism.

1.2 Exact-State Formalism

Absorption Intensities

Using our Hamiltonian as a starting point, the time evolution of the wave function can be found by solving the time-dependent Schrödinger equation i ∂ ∂t |ψ(t) = Ĥ|ψ(t) .

(1.12)

Approximative schemes are needed to find a solution to this equation, because in general it is not analytically solvable. To keep our analysis simple, it will be assumed for now that the exact eigenfunctions of the time-independent Hamiltonian are known

Ĥ0 |n = E n |n . (1.13)
This set of functions is complete and can thus form a basis for the time-dependent wave function

|ψ(t) = n c n (t)e -iEnt |n , (1.14)
where the time-dependent phase factors of each eigenfunction is included in the expansion. Equations of motion can be obtained for these coefficients by inserting this ansatz into the time-dependent Schrödinger equation V mn (t) = m| V (t)|n .

(1.17)

However, solving this equation still forms a major obstacle. The problem can be simplified by expanding the coefficients, and thus the wave function, in orders of λ = {λ ωy α } c n (t) = c (0) n + c (1) n (t) + c (2) n (t) + • • • ; |ψ(t) = |ψ (0) + |ψ (1) + |ψ (2) + • • • . (1.18) This expansion is the essence of perturbation theory and it is valid provided that λ 1. At t = -∞, before the perturbation has been switched on, the wave function is in the ground state, which translates to the following boundary condition c (0) n = δ n0 . To first order in the perturbation, the equation of motion reads ∂c

m (t) ∂t = n 1 i e iωmnt V mn (t)c (0) n (t) (1.19)
from which the first-order coefficients can be isolated upon integration c (1)

m (t) = 1 i t -∞
dt e iωmnt V mn (t )c (0) n = - In the integration step, the adiabatic switch shows its true virtue, because the lower bound of this integral would be ill-defined without its presence. Furthermore, due to its infinitesimal nature, it can be neglected in the final expression. To arbitrary order, the coefficients can be obtained by applying this procedure recursively

c (N ) m (t) = 1 i t -∞ dt e iωmnt V mn (t )c (N -1) n (t). (1.21)
However, for the purposes of this thesis, the first-order equations are sufficient, as the light intensity of conventional spectrometers is typically too small to evoke high-order effects. We thus arrive at the following expression for the first-order perturbed wave function |ψ (1) = n N y=-N -λ ωy α n| Vα (ω y )|0 ω n0ω y e -iωyt e -i E 0 t |n , (1.22) in which the adiabatic switch is neglected. Turning back our attention to Eqn. (E.4), the absorption rate from the ground state to a final state, |f , can be found using the first-order coefficients

w 0→f = d|c (1)
f (t)| 2 dt = d dt - N y=-N λ ωy α f | Vα (ω y )|0 ω y -ω f 0 + i e (-iωy+ )t 2 (1.23)
If we assume resonance conditions, i.e. ω y = ω f 0 , and let the adiabatic switch go to zero, then the y th term in the summation diverges. Evidently, an infinite transition rate does not make physical sense. However, for now, let us tolerate the presence of this divergence and use it to our advantage. The fact that one term in this summation is infinite, implies that all other terms can be neglected. From further manipulations of the above expression, we arrive at the following result

w 0→f = 2π 2 | f |λ ω f 0 α Vα (ω f 0 )|0 | 2 f (ω; ω f 0 , )e 2 t ; f (ω; ω f 0 , ) = 1 π (ω f 0 -ω) 2 + 2 . (1.24)
As the adiabatic switch approaches zero, the frequency-dependent factor reduces to a Diracdelta function

lim →0 f (ω; ω f 0 , ) = δ(ω f 0 -ω), (1.25)
which confirms the presence of a divergence, as expected from previous analysis. Strictly speaking, the delta function is a distribution rather than an actual function, implying that it can only be properly defined inside of an integral: f (a) = dxδ(xa)f (x). In section 1.2.2, it will be illustrated how divergences of this type can be useful in the calculation of excitation energies and transition moments. However, before we arrive there, I will first demonstrate how to avoid these divergences altogether.

The origin of this problem can be traced back to the lack of essential physics in our model. The perturbing operator in the Hamiltonian only allows for stimulated emission, whereas in more realistic systems, emission can also occur spontaneously under the influence of environmental factors such as thermal collisions or vacuum fluctuations. [START_REF] Weisskopf | Probleme der neueren Quantentheorie des Elektrons[END_REF] These effects can be taken into account phenomenologically by the following change of variables ω n0 → ω n0 + iγ n0 in the denominator

f (t) = - N y=-N λ ωy α f | Vα (ω y )|0 ω f 0 -ω y + iγ f 0 e (-iωy+iω f 0 )t e t . (1.26)
Here, γ n0 represents a damping factor that introduces exponential decay. The damping factor pushes the poles into the complex plane, thus making the coefficient well-defined on the real line. Using these adjusted coefficients, the transition rate can be expressed as

lim →0 w 0→f = 2π 2 | f |λ ω f 0 α Vα (ω f 0 )|0 | 2 f (ω; ω f 0 , γ f 0 ); f (ω; ω f 0 , γ f 0 ) = 1 π γ f 0 (ω f 0 -ω) 2 + γ 2 f 0 .
(1.27) Due to the finite value of the damping factor, f (ω, ω f 0 , γ f 0 ) becomes a Lorentzian linewidth function. By comparing with Eqn. (E.4), we find the following expression for the absorption cross-section

σ(ω) = ω I |V ω f 0 f 0 | 2 f (ω; ω f 0 , γ f 0 ) (1.28)
The above result is referred to as Fermi's golden rule and it will be the central equation in this thesis to evaluate absorption intensities. [START_REF] Adrien | The quantum theory of the emission and absorption of radiation[END_REF][START_REF] Fermi | Nuclear physics: a course given by Enrico Fermi at the University of Chicago[END_REF] In practice, the auxiliary quantity ω I |V ω f 0 | 2 is calculated first, giving rise to discrete peaks, which are then broadened according to the Lorentzian distribution. Alternatively, absorption intensities can be calculated using response functions, which will be discussed in the next section.

Response Functions

Having established the time-dependence of the first-order perturbed wave function, we can proceed to express the time evolution of expectation values. Let us in this example study the time evolution of the αth component of the electric-dipole moment of our system Q [1] α (t) = ψ(t)| Q [START_REF] Holmgaard List | Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory[END_REF] α |ψ(t) .

(1. [START_REF] Kragh Sørensen | Applications to metal K pre-edges of transition metal dimers illustrate the approximate origin independence for the intensities in the length representation[END_REF] This expression will be simplified by developing it in orders of the perturbation Q [1] α (t) = ψ (0) | Q [START_REF] Holmgaard List | Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory[END_REF] α |ψ (0) + ψ (1) | Q [START_REF] Holmgaard List | Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory[END_REF] α |ψ (0) + ψ (0) | Q [START_REF] Holmgaard List | Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory[END_REF] α |ψ (1) + • • • .

(1.30)

To describe linear absorption processes, it suffices to only keep the first few terms in this expansion

Q [1] α (t) = Q [1] 0;α + N y=-N λ ωy β Q[1] α ; Vβ (ω y ) e -iωyt , (1.31)
where appears the permanent electric-dipole moment

Q [1] 0;α = ψ (0) | Q[1] α |ψ (0) (1.32)
and the linear response function

Q[1] α ; Vβ (ω) = - 1 n 0| Vβ (ω)|n n| Q[1] α |0 ω n0 + ω + 0| Q[1] α |n n| Vβ (ω)|0 ω n0 -ω . (1.33)
In this expression, I have dropped the frequency index y, since the response function only depends on one frequency component of the interaction operator. For a given perturbation frequency, the linear response function tells us how the electric-dipole moment changes under the influence of the external perturbation. More generally, the response function Â; B gauges the response of the expectation value of  under the influence of B as a perturbation, or the other way around due to their symmetric nature. Furthermore, by only considering the zeroth frequency mode of a perturbation, i.e. ω 0 = 0, response functions can also be used to describe static perturbations. In the following I will denote static response functions as Â; B 0 . Similar to the findings of previous section, response function diverges under resonance conditions, i.e. ω n0 = ±ω y . However, this artifact should not be seen as a nuisance, but rather as a useful tool that provides us with excitation energies and transition moments. [START_REF] Oddershede | Polarization propagator methods in atomic and molecular calculations[END_REF][46, Section 6.4] From response functions, we can assert whether a certain frequency is a transition frequency, because a divergence appears in this case. Therefore, the poles serve as a diagnostic that can be used to find transition frequencies. Once this pole has been identified, transition moments can be extracted from the residues of the response functions.

lim ω→ω n0

(ω n0ω) Q [START_REF] Holmgaard List | Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory[END_REF] α ; Vβ (ω) = -1 0| Q [START_REF] Holmgaard List | Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory[END_REF] α |n n| Vβ (ω)|0 .

(1.34)

These transition moments can then be inserted into Fermi's golden rule to obtain the absorption cross-section. Although these relations seem obvious in an exact-state-formalism, they prove to be essential to calculate excitation energies and transition moments in approximate theories.

Alternatively, the divergences can be eliminated by introducing an imaginary damping factor Q [START_REF] Holmgaard List | Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory[END_REF] α ; Vβ (ω) = -

1 n 0| Vβ (ω)|n n| Q[1] α |0 ω n0 + ω + iγ n0 + 0| Q[1] α |n n| Vβ (ω)|0 ω n0 -ω -iγ n0 (1.35)
that push the poles into the imaginary plane. This response function is thus valid over the entire spectral range. [START_REF] Norman | Nearresonant absorption in the time-dependent self-consistent field and multiconfigurational selfconsistent field approximations[END_REF][START_REF] Orr | Perturbation theory of the non-linear optical polarization of an isolated system[END_REF] Due to their general structure, damped response functions can provide information on a plethora of molecular properties such as the refractive index, [START_REF] Norman | Nearresonant absorption in the time-dependent self-consistent field and multiconfigurational selfconsistent field approximations[END_REF] electronic circular dichroism [START_REF] Jiemchooroj | Electronic circular dichroism spectra from the complex polarization propagator[END_REF][START_REF] Jiemchooroj | Near-edge x-ray absorption and natural circular dichroism spectra of L-alanine: A theoretical study based on the complex polarization propagator approach[END_REF] and absorption spectra. [START_REF] Ekström | X-ray absorption spectra from the resonant-convergent first-order polarization propagator approach[END_REF][START_REF] Ekström | Polarization propagator for x-ray spectra[END_REF] An absorption spectrum can be obtained from this response function by tracing its imaginary part over the spectral range of interest. To facilitate this calculation, it is often assumed that there is a universal damping factor for all terms in the response function

Im[ Q[1] α ; V ω β ] = γ n 0| Q[1] α |n n| Vβ (ω)|0 (ω n0 -ω) 2 + γ 2 - 0| Vβ (ω)|n n| Q[1] α |0 (ω n0 + ω) 2 + γ 2 (1.36)
In this work, we will pursue the former method that uses the poles and residues of the response functions to calculate absorption intensities, which will be generalized to SCF methods in the next section.

SCF Theory

Energy Optimization

In the previous section, it was assumed that the exact eigenstates of the time-independent Hamiltonian are known. However, in practice, this is seldom the case due to the sheer complexity of the many-body wave function. To perform calculations, it is thus necessary to resort to approximative schemes. I recall that this work is based on SCF methods, i.e. HF and DFT, both of which essentially involve the optimization of a single Slater determinant

1 √ N ! φ 1 (r 1 ) φ 2 (r 1 ) • • • φ N (r 1 ) φ 1 (r 2 ) φ 2 (r 2 ) • • • φ N (r 2 ) . . . . . . . . . . . . φ 1 (r N ) φ 2 (r N ) • • • φ N (r N ) , (1.37)
where {φ i } are one-electron functions, r i the coordinates of electron i and N the number of electrons in the system. At first glance, HF and DFT seem to resemble quite a lot, although the physical origins of the two theories differ significantly.

The foundations of DFT are given by the two Hohenberg-Kohn theorems. [START_REF] Hohenberg | Inhomogeneous Electron Gas[END_REF] The theorem states that all properties of the wave function can be obtained from the electron density alone. In particular, the energy can be expressed as a functional of the density

E[ρ] = T [ρ] + V en [ρ] + V ee [ρ], (1.38)
where T [ρ], V en [ρ] and V ee [ρ] are the kinetic energy, electron-nucleus attraction and the electron-electron repulsion functionals, respectively. The second theorem states that this functional obeys a variational principle

E 0 ≤ E[ρ], (1.39)
where ρ is some trial density. If the density functional is known, it thus becomes possible to obtain the ground state density using optimization schemes. However, the first theorem is an existence proof and thus neither prescribes how the exact functional should be constructed, nor how approximate functionals can be systematically improved. Notably, the kinetic energy functional is difficult to construct. Kohn and Sham made a huge leap forward with their follow-up paper, in which they represented the problem as a fictitious non-interacting system. [START_REF] Kohn | Self-Consistent Equations Including Exchange and Correlation Effects[END_REF] This fictitious system partially eliminates the complications associated with the kinetic energy functional, since the kinetic energy of this system can be obtained exactly. To compensate for this drastic approximation, the system is subjected to an effective one-electron potential ensuring that the non-interacting one-electron density is equivalent to the density of the exact manybody wave function. More specifically, this potential corrects for exchange and correlation effects and the approximative nature of the kinetic energy. The one-particle functions that constitute the determinant can be obtained by solving

T + v H [ρ] + v xc [ρ] φ i = φ i , (1.40)
where appears the kinetic energy operator, T , the exchange-correlation potential, v xc [ρ], and the Hartree potential, v H [ρ] = d 3 r ρ(r )/|r -r|. Using this fictitious system, the exact energy can be obtained from the functional

E[ρ] = T s [ρ] + V en [ρ] + J[ρ] + E xc [ρ], (1.41)
where T s [ρ] is the kinetic energy of the fictitious system, J[ρ] the energy associated with the Hartree potential and E xc [ρ] the exchange-correlation energy. To this day, the main challenge associated with DFT is to find approximative exchange-correlation potentials, since there is no systematic method to improve this potential. HF theory, however, is much more simple in nature. In this theory, the wave function is approximated by a single determinant, while the Hamiltonian is exact. Given that the exact many-body wave function is an infinite expansion of such determinants, the HF determinant is in some cases the dominant contribution to the wave function. Therefore, the HF determinant can be understood as an effective one-determinantal wave function. It should be stressed, however, that the Kohn-Sham determinant does not share this interpretation, because in this case the density is physically meaningful and not the determinant on its own. Regardless of the applied method, it is convenient to represent SCF methods using the second-quantization formalism. [START_REF] Helgaker | Second quantization[END_REF] In this formalism, the occupation numbers of each orbital are contained in a vector. Electrons can be added or removed from orbitals by the action of creation and annihilation operators

a p |k = a p |i 1 , i 2 , • • • i p , • • • , i N = 0 if i p = 0 Γ k p |i 1 , i 2 , • • • 0, • • • , i N if i p = 1 (1.42) a † p |k = a † p |i 1 , i 2 , • • • i p , • • • , i N = Γ k p |i 1 , i 2 , • • • 1, • • • , i N if i p = 0 0 if i p = 1 ,
where {i p } are occupation numbers of orbitals and Γ k p is a phase factor depending both on the index p and the overall occupation, represented by the the label k. It can be shown that this phase factor is required to correctly reproduce the Slater-Condon rules. In the Slater determinant representation, the fermionic nature of electrons is taken into account by the anti-symmetry of determinants upon exchange of rows. The second-quantization formalism, however, naturally incorporates this feature into its algebraic properties. The algebra of these operators can be summarized as

{a p , a q } = 0; {a † p , a q } = δ pq {a † p , a † q } = 0, (1.43)
from which we can derive the relation

|1 p , 1 q = a † p a † q |vac = -a † q a † p |vac = -|1 q , 1 p . (1.44)
In the following, it will be assumed that the indices (pqrs) represent general orbitals, (ijkl) occupied orbitals and (abcd) virtual orbitals. In this convention, the second-quantized electronic Hamiltonian assumes the form

Ĥ0 = h pq a † p a q + 1 2 (pq|rs)a † p a † r a s a q , (1.45)
where appears the one-electron integrals

h pq = d 3 rφ † p (r) T + V en φ q (r); h * pq = h qp (1.46)
and the two-electron integrals

(pq|rs) = d 3 r 1 d 3 r 2 φ † p (r 1 )φ q (r 1 )φ † r (r 2 )φ s (r 2 ) r 12
.

(1.47)

For some generic wave function, ψ, the energy can be obtained from the expectation value of the Hamiltonian

E = ψ| Ĥ0 |ψ = h pq D pq + 1 2 (pq|rs)d pq,rs , (1.48)
where in the second equality, I have introduced the one-body

D pq = ψ|a † p a q |ψ (1.49)
and two-body density matrix

d pq,rs = ψ|a † p a † r a s a q |ψ . (1.50)
In the special case of Hartree-Fock theory, which is based on a single determinant state, the two-body density matrix can be expressed in terms of the one-body density matrix, yielding the following expression for the Hartree-Fock energy

E HF = h pq D pq + 1 2 (pq|rs) -(ps|rq) D pq D rs . (1.51)
A similar expression can be found for DFT by relating the one-body density matrix to its corresponding density function [START_REF] Saue | Four-component relativistic Kohn-Sham theory[END_REF]

ρ(r) = D pq φ † p (r)φ q (r) (1.52)
and inserting it in Eqn. (1.41)

E DF T = h pq D pq + 1 2 (pq|rs) -γ(ps|rq) D pq D rs + E xc [ρ]. (1.53)
To allow for the possibility of hybrid functionals, the factor γ is included which mixes in Hartree-Fock exchange. [START_REF] Becke | A new mixing of Hartree-Fock and local density-functional theories[END_REF] Both SCF theories are variational, so the orbitals can be obtained by minimizing the energy. For this procedure, it is convenient to express the determinant in a set of parameters. In this work, an exponential parametrization will be chosen to describe the SCF determinant

| 0 = exp[-κ]|0 ; κ = κ pq â † p âq . (1.54)
Here, |0 is a reference state which, in current context, is the wave function from previous SCF iteration. As a starting guess, orbitals are typically given as linear combinations of atomic orbitals, which may be obtained from equivalent basis set calculations, [START_REF] Van Lenthe | Starting SCF calculations by superposition of atomic densities[END_REF] the extended Huckel approach [START_REF] Norman | Phosphorescence parameters for platinum (II) organometallic chromophores: A study at the non-collinear four-component Kohn-Sham level of theory[END_REF] or numerical calculations. [START_REF] Lehtola | Efficient implementation of the superposition of atomic potentials initial guess for electronic structure calculations in Gaussian basis sets[END_REF] The action of the exponential operator is to rotate between orbitals. Unitarity is ensured in this operator by imposing anti-hermiticity in the parameter matrix: κ * pq = -κ qp . Using the properties of this operator, it can be shown that orbitals transform according to

ãp = U pq â † q ; U pq = exp[-κ] pq . (1.55)
Written in this form, the wave function parameters can assume any value without breaking orthonormality. Alternatively, a constrained optimization can be applied using the method of Lagrange multipliers, [61][62, p. 407] but this will not be further pursued in this work.

Using the exponential parametrization, the one-body density matrix assumes the form

Dpq = 0|a † p a q | 0 = 0|exp[κ]a † p a q exp[-κ]|0 . (1.56)
The energy can thus be expressed as a function of the parameters by inserting this definition into the energies provided by Eqs. (1.51) and (1.53). At the minimum of the energy, the partial derivatives of the energy with respect to the parameters vanish, hence forming a condition of convergence. By extension, this condition is determined by the first partial derivatives of the one-body density matrix. However, before proceeding to further analyse this condition, it should be stressed that for a certain subset of parameters, the density matrix does not change, hence fulfilling the variational condition trivially. To understand this better, let us apply a Baker-Campbell-Hausdorff expansion to the one-body density matrix

0|exp[κ]a † p a q exp[-κ]|0 = 0|a † p a q + [κ, a † p a q ] + 1 2 κ, [κ, a † p a q ] + • • • |0 . (1.57)
When evaluated at the reference point, only the linear commutator terms can contribute to the first derivative with respect to the parameters. For the occupied-occupied and virtualvirtual block, these derivatives thus assume the form

∂ Dpq ∂κ ij κ=0 = 0| a † i a j , a † p a q |0 = 0 (1.58) ∂ Dpq ∂κ ab κ=0 = 0| a † a a b , a † p a q |0 = 0.
The non-redundant parameters are given by the occupied-virtual and virtual-occupied blocks

∂ Dpq ∂κ ai κ=0 = 0|[a † a a i , a † p a q ]|0 = 0|a † p a q |Φ a i = δ pi δ qa (1.59) ∂ Dpq ∂κ * ai κ=0 = 0|[a † i a a , a † p a q ]|0 = Φ a i |a † p a q |0 = δ qi δ pa .
The variational condition can thus be expressed as

∂E ∂κ ai κ=0 = ∂E 0 ∂κ * ai κ=0 = 0; ∀κ ai , κ * ai , (1.60)
In case of DFT, the energy derivatives also depend on derivatives of the exchange-correlation energy. These derivatives can most conveniently be expressed in terms of the exchangecorrelation density,

∂e xc (ρ, ζ) ∂κ ai κ=0 = ∂e xc ∂ρ ∂ρ ∂κ ai + ∂e xc ∂ζ ∂ζ ∂κ ai ; E xc [ρ] = d 3 re xc (ρ, ζ). (1.61)
As a general feature of GGA functionals, the exchange-correlation density is also a function of ζ = ∇ρ•∇ρ. [START_REF] Saue | Four-component relativistic Kohn-Sham theory[END_REF] For DFT, it can be shown that the variational condition can be expressed in terms of the Kohn-Sham Fock matrix

∂E DF T ∂κ ai κ=0 = -F KS ai = 0 (1.62) ∂E DF T ∂κ * ai κ=0 = -F KS ia = 0 F KS pq = h pq + (pq|ii) -γ(pi|iq) + v xc;pq , (1.63)
whereas in HF theory, we obtain the closely related Fock matrix

∂E HF ∂κ ai κ=0 = -F ai = 0 (1.64) ∂E HF ∂κ * ai κ=0
= -F ia = 0

F pq = h pq + (pq|ii) -(pi|iq). (1.65)
Therefore, the optimal wave function can be obtained by finding a set of parameters that put the secondary-inactive/inactive-secondary block of the (Kohn-Sham) Fock matrix to zero. Alternatively, the orbitals can be found by solving the pseudo-eigenvalue problem of the (Kohn-Sham) Fock operator

F (KS) φ i = i φ i , (1.66)
which amounts to a diagonalization of the Fock matrix. By extension, the variational condition is met in a diagonal basis, although this procedure is somewhat excessive, as it also sets the off-diagonal of the inactive-inactive and secondary-secondary blocks to zero, corresponding to redundant parameters. Nevertheless, the pseudo-eigenvalue problem is particularly useful in atomic physics, where spherical symmetry renders the problem one-dimensional, thus allowing the problem to be solved on a numerical grid. [START_REF] Froese-Fischer | Computational atomic structure: an MCHF approach[END_REF]Chapter 3] Using the grid-based methods, it is possible to nearly approach the complete basis set limit and it is thus very suitable for benchmark calculations. However, like any method, it does have significant drawbacks. For instance, the finite-difference methods is less applicable for molecules, generally requiring three dimensional grids. Furthermore, using a finite-difference method it is only possible to compute occupied orbitals, which complicates its extension to correlated-or excited-state methods. Therefore, a basis set expansion is the preferred method for the purposes of this thesis.

However, before proceeding it should be noted that Eqs. (1.62) and (1.64) do not guarantee a global minimum, because they can also mark a local minimum, a saddle point, or even a local maximum. By using second-order optimization schemes, the possibility of maxima and saddle points can be ruled out. However, even second-order schemes are not capable of finding a global minimum with absolute certainty.

Besides its utility in energy optimization, the variational condition can also be used to determine static molecular properties. To arrive at this point, the Helmann-Feynman theorem needs to be invoked, [START_REF] Hellmann | Einführung in die Quantenchemie[END_REF][START_REF] Feynman | Forces in molecules[END_REF] which is an exact-state theorem that can be reproduced for variational, approximate theories, such as SCF theory

dE(λ, κ, κ * ) dλ 0 α = ∂E ∂λ 0 α + ai ∂E ∂κ ai ∂κ ai ∂λ 0 α + ai ∂E ∂κ * ai ∂κ * ai ∂λ 0 α = ∂E ∂λ 0 α = Φ 0 | V 0 α |Φ 0 , (1.67)
where V 0 α is a static perturbation with λ 0 α being its associated coupling strength. In the above equation, it is assumed that the variational condition holds at every field strength and the orbitals do not explicitly depend on the perturbation, the latter of which does not hold for geometric distortion or magnetic properties with the use of London atomic orbitals. [START_REF] London | Théorie quantique des courants interatomiques dans les combinaisons aromatiques[END_REF] Further differentiation of this condition yields the static response function

d 2 E dλ 0 α dλ 0 β λ=0 = V 0 α ; V 0 β 0 . (1.68)
However, in current context, we are interested in light-matter interactions, thus implying that the static response functions have limited interest. In the following, I will generalize above findings to the time-dependent domain.

Quasi-Energy

We start of with our SCF wave function and subject it to the influences of a time-dependent interaction operator. However, by introducing time-dependence into our Hamiltonian, we bump into the immediate problem that the energy is not a conserved quantity. Without the energy and a variational condition, it is not straightforward how to apply Eqn. (1.68) and find time-dependent response functions. Notwithstanding, in the presence of a periodic perturbation, the perturbed state does share some similarities with stationary states. If the system is exposed sufficiently long to a time-periodic perturbation, the influences of the adiabatic switch can be neglected and after this point, the wave function will continue oscillating periodically. This steady-state behaviour can be inferred from Eqn. (1.22) and it holds more generally by virtue of Floquet's theorem. [START_REF] Shirley | Solution of the schrödinger equation with a hamiltonian periodic in time[END_REF][START_REF] Young | Quasi-periodic states of an oscillatory hamiltonian[END_REF][START_REF] Sambe | Steady States and Quasienergies of a Quantum-Mechanical System in an Oscillating Field[END_REF] The existence of such steady-states suggest that there might be other conserved quantities apart from the energy.

To further investigate this idea, I will draw inspiration from the static problem, hence assuming that only the zeroth frequency component in Eqn. (E.8) contributes. However, the Chapter 1. Conventional Methods to Simulate UV-Vis Absorption Spectroscopy perturbation is strictly speaking quasi-static, as it is turned on using an adiabatic switch. Using the adiabatic theorem, it can thus be shown that in the limit t → ∞, the wave function becomes

|ψ = n c n (t)e -iEnt/ |n → |ψ = e -iE t/ n c n (t)|n , (1.69)
where E is the exact energy of the full Hamiltonian, including the perturbation. It can be shown that the coefficients, c n , are related to c n by

c n (t) = c n (t)e -i E -En t/ . (1.70)
The perturbative expansion of these coefficients is complicated because the exact perturbed energy appearing in the exponent needs to be expanded as well. Furthermore, the additional terms arising from this exponent diverge. Therefore, it seems to be more practical to isolate the exponential phase from our ansatz

|ψ = e -iE t/ | ψ (1.71)
and apply a perturbative series based on the phase-isolated wave function. However, if one insists, expression (1.70) can be used, because the diverging terms do not contribute to expectation values, as pointed out by Langhoff et al. [START_REF] Langhoff | Aspects of Time-Dependent Perturbation Theory[END_REF] Therefore, the divergences introduced by the exponential phase are secular in nature, meaning that they can be removed with the proper mathematical manipulations.

Besides its capability to isolate secular divergences, the phase-isolated ansatz is also particularly useful when defining time-dependent molecular properties. The time-dependent generalization of the phase-isolated ansatz can be expressed as

|ψ = e iφ(t)/ | ψ , (1.72)
where the phase, φ(t), is now a general time-dependent function. This decomposition is made unique by imposing that φ(t) is real and that the projection of phase-isolated wave function onto the unperturbed wave function is zero. The following equation can be obtained by inserting our ansatz in the time-dependent Schrödinger equation

Ĥ -i ∂ ∂t | ψ = Q(t)| ψ ; Q(t) = ∂φ(t) ∂t , (1.73)
where the second equation defines the quasi-energy. However, this equation is seldom used, as it is much more convenient to obtain the quasi-energy from projection with the phaseisolated wave function

Q(t) = ψ| Ĥ -i ∂ ∂t | ψ . (1.74)
The quasi-energy deserves its name due to its connection to the energy in the static limit

Q(t) → E.
Due to this connection, the quasi-energy seems like a suitable candidate for the generalization of Eqn. (1.68) to the time-dependent domain. However, considering that Eqn. (1.67) is derived from the variational properties of the energy, it should be assessed whether the quasi-energy is variational as well before making this connection. I will thus apply the following variations to the wave function

| ψ → | ψ + |δ ψ , (1.75)
which should preserve the norm of the wave function

( δ ψ|+ ψ|)(| ψ + |δ ψ ) = ψ| ψ + δ ψ| ψ + ψ|δ ψ . (1.76)
Therefore, the allowed variations assume the form

|δ ψ = | ψ⊥ + i | ψ ; ∈ R, (1.77)
where | ψ⊥ and | ψ are parallel and perpendicular to the phase isolated wave function.

Applying these variations to the quasi-energy yields the result

δQ(t) = -i ∂ ∂t ψ|δ ψ , (1.78)
suggesting that the quasi-energy is not variational. Furthermore, differentiation with respect to the field strength gives the time-dependent generalization of the Helmann-Feynman theorem

dQ dλ ωy α = ψ| V ωy α | ψ -i ∂ ∂t ψ| d ψ dλ ωy α
.

(1.79)

However, we would like to connect quasi-energy derivatives with expectation values, and by extension to time-dependent response functions. Fortunately, the time-periodicity of the system can be used to our advantage. Due to this property, the time derivative term is periodic, suggesting it vanishes upon time-averaging over a full period

1 T T /2 -T /2 dt ḟ (t) = 0; f (t + T ) = f (t).
(1.80)

Using these relation we can retrieve a variational condition for the time-averaged quasi-energy

δQ T = 0 (1.81)
and a more useful expression for the time-dependent Helmann-Feynman theorem [START_REF] Christiansen | Response Functions from Fourier Component Variational Perturbation Theory Applied to a Time-Averaged Quasienergy[END_REF] dQ T dλ

ωy α = 1 T t+T /2 t-T /2 dt ψ| V ωy α | ψ -i 1 T t+T /2 t-T /2 dt ∂ ∂t ψ|δ ψ = Φ 0 | V ωy α |Φ 0 δ ωy , (1.82)
where δ ωy is a generalized Kronecker delta that equals 1 if ω y = 0 and 0 otherwise. In analogy with the static case, the time-dependent Helmann-Feynman theorem can be exploited to compute response functions. For these purposes, I will further differentiate Eqn. (1.82)

d 2 Q T dλ ωy α dλ ωz β λ=0 = 1 T t+T /2 t-T /2 dt d ψ dλ ωz β | Vα (ω y )e -iωyt | ψ λ=0 + 1 T t+T /2 t-T /2 dt ψ| Vα (ω y )e -iωyt | d ψ dλ ωz β λ=0
, (1.83) which indeed gives the time-dependent response functions upon insertion of the first-order perturbed wave function.

d 2 Q T dλ ωy α dλ ωz β λ=0 = Vα (ω y ); Vβ (ω z ) δ ωy+ωz (1.84)
Response functions can thus be computed by evaluating field strength derivatives of the quasi-energy.

Response Theory

To evaluate the quasi-energy derivatives, it is necessary to expand SCF theory to the timedependent domain. For HF theory, this is a straightforward exercise, because the HF determinant directly represents the wave function. In exponentially parametrized form, the time-dependent generalization of the HF wave function can be expressed as

| ψ(t) = exp[-κ(t)]|Φ 0 ; κ(t) = κ ai (t)a † a a i -κ * ai (t)a † i a a , (1.85)
where the generator is given by non-redundant rotations. Contrary to Eqn. (1.54), the reference state in this ansatz is the optimized HF ground state, while the parameters are time-dependent to accommodate the influences of the perturbing operator. The action of the generator is to (de-)excite orbitals, thus including contributions from excited determinants. Using this ansatz, the quasi-energy can be expressed as

Q T = Q 0:HF + N y=-N λ ωy α Q α (ω y ), (1.86)
in terms of the two quantities

Q 0;HF = {E HF } T -ψ|i ∂ ∂t | ψ T ; Q α (ω y ) = ψ| Vα (ω y )e -iωyt | ψ T , (1.87)
where the subscript T denotes time-averaging. Due to previously established similarities between HF and DFT, it is tempting to assume that a time-dependent generalization of the latter (TDDFT) is based on a similar ansatz. However, demonstrating the validity of this assumption turns out to be a non-trivial task.

After all, the Hohenberg-Kohn theorems makes use of a variational principle of the energy, thus only applying to the ground state. A major breakthrough was provided in a seminal article by Runge and Gross, where they generalized the Hohenberg-Kohn theorems to the time-dependent domain by formulating the problem as an action integral. [START_REF] Runge | Density-functional theory for time-dependent systems[END_REF] However, this formulation can be rather cumbersome, as the initial state needs to be known to establish a one-to-one correspondence between the density and potentials. Here, I will rather follow a procedure based on the variational properties of the time-averaged quasi-energy, which avoids problems associated with boundary conditions due to the assumed periodicity of the system. [START_REF] Deb | Schrödinger fluid dynamics of many-electron systems in a time-dependent density-functional framework[END_REF][START_REF] Aiga | Frequency-dependent polarizabilities, hyperpolarizabilities, and excitation energies from time-dependent density-functional theory based on the quasienergy derivative method[END_REF] Using the variational properties of the time-averaged quasi-energy, it can be demonstrated that it can be expressed as a functional of the density, in analogy with the Hohenberg-Kohn theorems. Furthermore, this density can be expressed using fictitious system of non-interacting particles, thus yielding the following functional

Q T [ρ] = Q 0;DF T [ρ] + N y=-N λ ωy α d 3 r Vα (ω y )ρ(r, κ) T (1.88) Q 0;DF T [ρ] = {T s [ρ] + V en [ρ] + J[ρ] + Q xc [ρ]} T -ψ|i ∂ ∂t | ψ T . (1.89)
It seems tempting to identify the first four terms in Q 0:DF T with the time-averaged DFT energy (Eqn. (1.41)). However, the observant reader should have noticed that the exchange correlation functional is different. Indeed, in the time-dependent case, this functional does not only correct for the lack of exchange, correlation and the approximate nature of the kinetic energy, but also for the approximate nature of the time-derivative term. [START_REF] Salek | Linear response at the 4-component relativistic density-functional level: application to the frequency-dependent dipole polarizability of Hg, AuH and PtH2[END_REF] In general, the time-dependent exchange-correlation functional is difficult to obtain. For that reason, the adiabatic approximation is often applied, where the general exchange correlation potential is approximated by the time-independent one, [START_REF] Gross | Local density-functional theory of frequency-dependent linear response[END_REF] i.e.

Q xc [ρ] → {E xc [ρ]} T .
Within this approximation, the quasi-energy thus becomes

Q T [ρ] = {E DF T [ρ]} T -ψ|i ∂ ∂t | ψ T + N y=-N λ ωy α d 3 r Vα (ω y )ρ(r, κ) T . (1.90)
Having derived expressions for the quasi-energy at the SCF level, the most difficult theoretical step has been taken, with the remainder of the derivation leaning on the general properties of the quasi-energy, thus being mostly equivalent for both HF and DFT. In the following, I will closer inspect the dependency of the quasi-energy on the rotation parameters to derive working equations for its field strength derivatives. Due to time-periodicity, the parameters in our ansatz can be expanded in Fourier components that are integer multiples of a fundamental frequency [START_REF] Shirley | Solution of the schrödinger equation with a hamiltonian periodic in time[END_REF][START_REF] Young | Quasi-periodic states of an oscillatory hamiltonian[END_REF][START_REF] Sambe | Steady States and Quasienergies of a Quantum-Mechanical System in an Oscillating Field[END_REF]

κ ai (t) = N k=-N κ (1) ai (ω k )e -iω k t + N k,l=-N κ (2) ai (ω k , ω l )e -i(ω k +ω l )t + • • • (1.91) κ * ai (t) = N k=-N κ (1) * ai (-ω k )e -iω k t + N k,l=-N κ (2) * ai (-ω k , -ω l )e -i(ω k +ω l )t + • • • .
Here, the superscripts on the parameters denote the order of the response. To distinguish the frequency components of the response with the ones of the interaction, I have labelled the former using the indices k, l and the latter using y, z. Within this parametrization, the time averaged quasi-energy becomes a function of the field strengths and the Fourier amplitudes

Q T (λ ω k α , κ (1) n (ω k ), κ (1) * n (ω k ), κ (2) n (ω k , ω l ), κ (2) * n (-ω k , -ω l ), • • • ), (1.92)
where the index pair ai is now represented as a super index n. In linear response, only the first-order response parameters are considered, which can be conveniently represented in vector form

K(ω k ) = κ (1) (ω k ) κ (1) * (-ω k ) ; K m (ω k ) = κ (1) m (ω k ) if m > 0 κ (1) * m (-ω k ) if m < 0 , (1.93)
where the indices of this vector are positive for the upper half and negative for the lower half. The quasi-energy derivatives can now be found using the chain rule

d 2 Q T dλ ωy α dλ ωz β λ=0 = ∂ 2 Q T ∂λ ωy α ∂λ ωz β λ=0
(1.94)

+ N k=-N ∂ 2 Q T ∂λ ωz β ∂K n (ω k ) ∂K n (ω k ) ∂λ ωy α λ=0 + N k=-N ∂Q T ∂K n (ω k ) ∂ 2 K n (ω k ) ∂λ ωy α ∂λ ωz β λ=0 + N k,l=-N ∂K n (ω k ) ∂λ ωz β ∂ 2 Q T ∂K n (ω k )∂K m (ω l ) ∂K m (ω l ) ∂λ ωy α + ∂ 2 Q T ∂λ ωy α ∂K n (ω k ) λ=0 .
Several simplifications can be applied to above results. Firstly, the perturbation only appears linear in the Hamiltonian, implying that the first term in Eqn. (1.94) vanishes. Secondly, as demonstrated in previous section, the time averaged quasi-energy is variational for all field strengths

∂Q T ∂K n (ω k ) λ = 0; ∀λ, (1.95)
rendering the third term zero as well. Further simplifications can be made by differentiation of the variational condition with respect to the field strength

∂ 2 Q T ∂K n (ω k )∂λ ωy α λ=0 + N l=-N ∂ 2 Q T ∂K n (ω k )∂K m (ω l ) ∂K m (ω l ) ∂λ ωy α λ=0 = 0. (1.96)
This expression necessarily equals zero because the variational condition holds for all field strengths. Upon inspection of Eqn. (1.94), the differentiated variational condition can be recognized in the last two terms. Therefore, Eqn. (1.94) simplifies to

d 2 Q T dλ ωy α dλ ωz β λ=0 = N k=-N ∂ 2 Q T ∂λ ωz β ∂K n (ω k ) ∂K n (ω k ) ∂λ ωy α λ=0
.

(1.97)

To evaluate the double derivatives of the quasi-energy, and hence response functions, it is required to evaluate the property gradient

E V ωy β δ ω k +ωy =   ∂ 2 Q T ∂λ ωy β ∂κ (1) n (ω k ) λ=0 ∂ 2 Q T ∂λ ωy β ∂κ (1) * n (-ω k ) λ=0   (1.98)
and the solution vectors

X V ωy α (ω k ) = Z Y * =   ∂κ (1)
n (ω k ) ∂λ ωy α λ=0 ∂κ (1) * n (-ω k ) ∂λ ωy α λ=0   .
(1.99)

Note that both of these quantities are stored in vector format for convenience. Using this notation, the response functions can be expressed in the elegant form below

Â; B = E † A X B . (1.100)
The solution vectors can be obtained by inverting the differentiated variational condition. However, solving this equation requires the property gradient and the double derivative of the quasi-energy with respect to the first-order Fourier components. Therefore, I will first proceed to calculate these quantities. To evaluate the property gradient, I will exploit the fact that its evaluated at zero field strength, implying that only the linear terms with respect to the field strength and first-order response parameters are non-vanishing. To gather contributions to the quasi-energy in a transparent manner, it proves to be useful to apply a Baker-Campbell-Hausdorff expansion to the perturbed density matrix and the time-derivative term

Φ 0 |exp[κ]a † p a q exp[-κ]|Φ 0 = Φ 0 |a † p a q |Φ 0 + Φ 0 |[κ, a † p a q ]|Φ 0 + 1 2 Φ 0 | κ, [κ, a † p a q ] |Φ 0 + • • • (1.101) Φ 0 |exp[κ]i ∂ t exp[-κ]|Φ 0 = Φ 0 |i ∂ t |Φ 0 + Φ 0 |[κ, i ∂ t ]|Φ 0 + 1 2 Φ 0 | κ, [κ, i ∂ t ] |Φ 0 + • • • .
Note that unlike the expansion from Eqn. (1.57), these expressions are defined with respect to the SCF ground state and the response parameters. The property gradient can be evaluated by first inserting these terms into the quasi-energy and then gathering the terms of the right order. Therefore, the gradient reads

∂ 2 Q T ∂K m (ω k )∂λ ωy α λ=0 = -Φ 0 | V ωy α , a † a a i |Φ 0 δ ω k +ωy m > 0 Φ 0 | V ωy α , a † i a a |Φ 0 δ ω k +ωy m < 0 , (1.102)
which has inherited the overall vector structure from Eqn. (1.93). In vector notation, it can be rewritten as

E V ωy α = g V ωy α g * V ωy α ; g V ωy α ;ai = -Φ 0 | V ωy α |Φ a i = -d 3 rφ * i (r) V ωy α φ a (r). (1.103)
We now proceed to calculate the double derivative of the quasi-energy with respect to the response parameters. This quantity is ideally represented as a matrix having the following block-structure

Q [2] T = ∂ 2 Q T ∂κ * n (-ω k )∂κm(ω l ) ∂ 2 Q T ∂κ * n (-ω k )∂κ * m (-ω l ) ∂ 2 Q T ∂κn(ω k )∂κm(ω l ) ∂ 2 Q T ∂κn(ω k )∂κ * m (-ω l )
.

(1.104)

To evaluate these derivatives, we can follow a similar approach as the one for the property gradient, although the situation is somewhat more complicated for DFT, due to the additional burden to evaluate the double derivatives of the exchange-correlation density. The evaluation of these terms boil down to calculating the exchange-correlation kernel (see ref. [START_REF] Salek | Linear response at the 4-component relativistic density-functional level: application to the frequency-dependent dipole polarizability of Hg, AuH and PtH2[END_REF] for more detail)

W xc;pq,rs = d 3 r ∂ 2 e xc ∂ρ 2 λ=0 Ω pq Ω rs + 2 ∂ 2 e xc ∂ρ∂ζ λ=0 (∇ρ 0 • ∇Ω pq )Ω rs + Ω pq (∇ρ 0 • ∇Ω rs ) (1.105) + 4 ∂ 2 e xc ∂ζ 2 λ=0 (∇ρ 0 • ∇Ω pq )(∇ρ 0 • ∇Ω rs ) + 2 ∂e xc ∂ζ λ=0 (∇Ω pq • ∇Ω rs ) Ω pq = φ † p (r)φ q (r)
. Therefore, we end up with the following general structure

Q [2] T = (E [2] 0 -ω l S [2] )δ ω k +ω l , (1.106)
where appears the generalized metric

S [2] = Σ ∆ -∆ * -Σ * Σ ai,bj = 0| -a † i a a , a † b a j |0 = δ ab δ ij ∆ ai,bj = 0| a i a a , a † j a b |0 = 0 (1.107)
and the electronic Hessian, E [START_REF] Martin Van Horn | Probing chirality across the electromagnetic spectrum with the full semi-classical light-matter interaction[END_REF] 0 . Depending on the level of theory, the Hessian assumes different forms. In Hartree-Fock theory, [START_REF] Dunning | Nonempirical Calculations on Excited States: The Ethylene Molecule[END_REF] the Hessian reads

E [2] 0 = A B B * A * A ai,bj = 0| -a † i a a , [a † b a j , Ĥ0 ] |0 = δ ij F ab -δ ab F ji + (ai|jb) -(ab|ji) B ai,bj = 0| a † i a a , [a † j a b , Ĥ0 ] |0 = (ai|bj) -(aj|bi) (1.108) whereas in DFT it assumes the form[78] E [2] 0 = A B B * A * A ai,bj = δ ij F KS ab -δ ab F KS ji + (ai|jb) -γ(ab|ji) + W xc;ai,jb B ai,bj = (ai|bj) -γ(aj|bi) + W xc;ai,bj .
(1.109) Using these quantities, Eqn. (1.96) can be written in its conventional form

E [2] 0 -ωS [2] X V ω α (ω) = -E V ω α , (1.110)
which is also referred to as the response equation. Note that in this equation, the action of the Kronecker symbol ensures that all different frequency components stemming from various sources are the same. For ease of notation, this index has thus been dropped. As eluded to before, the solution vectors can be obtained by inverting the matrix expression

X V ω α (ω) = -E [2] 0 -ωS [2] -1 E V ω α . (1.111)
In practice, however, such a matrix inversion is rarely performed explicitly due to the sheer size of the electronic Hessian. Instead, the Hessian is projected in a basis of trial vectors and solved iteratively, which is explained more thoroughly in Appendix A.1.

With the explicit form of the solution vectors, the response functions can be expressed as

Â; B = -E † A (E [2] 0 -ωS [2] ) -1 E B . (1.112)
However, I recall that we started this venture in response theory to find transition moments and excitation energies at the SCF level. In the exact-state case, it is a straightforward exercise to identify the poles and hence find these quantities. From expression (1.112), however, it is less clear where the poles are located. From elementary linear algebra, it is known that the inverse of any matrix is inversely proportional to its determinant. It thus follows that the resolvent matrix in Eqn. (1.112)

induces divergences if det E [2]
0 -ωS [2] = 0.

(1.113)

This condition is nothing more than the characteristic equation of the generalized eigenvalue problem E [START_REF] Martin Van Horn | Probing chirality across the electromagnetic spectrum with the full semi-classical light-matter interaction[END_REF] 0 X n = ω n0 S [2] X n .

(1.114)

In above expression ω n0 should be understood as the specific value of ω that induces a divergence, i.e. the transition frequency. From the structure of the electronic Hessian and the generalized metric, it can be shown that the eigenvectors come in pairs having equal but opposite frequencies

ω + = + |ω n | , X n,+ = Z n Y * n ∪ ω -= -|ω n | , X n,-= Y n Z * n . (1.115)
Expressing the response function in the set of these eigenvectors thus allows us the identify the poles. To further pursue this idea, I define a unitary matrix

X = X 1 , X 2 , • • • , X N ; X † X = I (1.116)
which has the eigenvectors as columns. [START_REF] Bast | Relativistic adiabatic timedependent density functional theory using hybrid functionals and noncollinear spin magnetization[END_REF] The response functions can be transformed by inserting the identity matrix

-E † A XX † (E [2] 0 -ωS [2] ) -1 XX † E B = -E † A X(XE [2] 0 X † -ωXS [2] X † ) -1 X † E B (1.117)
Brought in diagonal form, the resolvent can be expressed as

(XE [2] 0 X † -ωXS [2] X † ) -1 =         1 ω 1,0 -ω 0 0 • • • 0 0 1 ω 2,0 -ω 0 • • • 0 0 0 . . . . . . 0 . . . . . . . . . 1 ω N -1,0 + ω . . . 0 0 0 • • • 1 ω N,0 + ω        
, (1.118) which further establishes the analogy between the resolvent and the frequency denominators in the exact response functions (Eqn. (1.33)). Written in this diagonal representation, the approximate response functions become .119) In this form, the residues can be readily evaluated

Â; B = - n (E † A X n )(X † n E B ) ω n0 -ω + (E † A X n )(X † n E B ) ω n0 + ω . ( 1
lim ω B →ω n0 (ω B -ω n ) Â; B = - 1 (E † A X n )(X † n E B ), (1.120)
implying that the transition moments can be written as

f | Â|i SCF = E † A X n = -Z ai Φ a i | Â|Φ 0 -Y * ai Φ 0 | Â|Φ a i . (1.121)
Using the well-defined hermiticity of the interaction operator, the transition moments can be further simplified by utilizing the paired structure of the solution vectors

X h = 1 2 (X m,+ + X m,-) = 1 2 Z + Y (Z + Y ) * ; X a = 1 2 (X m,+ -X m,-) = 1 2 Z -Y -(Z -Y ) * ,
(1.122) which makes the transition moment either purely real or imaginary

X † h E [1] A = Re (Z + Y ) † g A ; X † a E [1] A = iIm (Z -Y ) † g A . (1.123)
To calculate absorption intensities, the eigenvalue problem from Eqn. (1.114) is first solved, yielding the transition frequencies and the solution vectors. The transition moments, which may be frequency-dependent, are then constructed from relation (1.121). Absorption intensities are then calculated by inserting these transition moments into Fermi's golden rule. By comparison with exact-state theory, it is tempting to associate the solution vectors with excited states

Xn |Φ 0 = Y ai a † i a a + Z * ai a † a a i |Φ 0 . (1.124)
However, this comparison does not hold, because |X n does not obey the so-called killercondition. [START_REF] Dunning | Nonempirical Calculations on Excited States: The Ethylene Molecule[END_REF][START_REF] Rowe | Equations-of-motion method and the extended shell model[END_REF] This condition applies to excited states

|n = Ôn |0 , (1.125)
generated from the ground state by the operator Ô. According to the killer-condition, the Hermitian conjugate of Ô should annihilate the ground-state

Ô † n |0 = 0. (1.126)
However, when applying X † n to the ground state, this condition is violated

X † n |Φ 0 = Y ai |Φ a i . (1.127)
Therefore, the approximate methods describes in this section can only provide transition moments and excitation energies.

Using the techniques developed in this chapter, we are now in a position to calculate UV-Vis absorption spectra at the SCF level of theory. In principle, these techniques can also be used to calculate X-ray absorption intensities, although the results differ severely from experiment. These discrepancies can be explained by the limitations of the models that we have been using throughout this chapter. The next two chapters will describe the theory required to adapt our model for the simulation of X-ray absorption spectroscopy.

Chapter 2

Relativistic Quantum Chemistry

Introduction

In the formation of bonds, the core electrons typically do not need special considerations, since they are not easily polarized and thus largely retain their atomic character. Notwithstanding, the core electrons take center stage in X-ray spectroscopy, which operates at high enough energies to probe these electrons. The simulation of X-ray spectroscopy is met with difficulties, because the conventional theory from UV-Vis spectroscopy (see Chapter 1) cannot accurately reproduce X-ray experiments. At least some of these shortcomings are rooted in the special characteristics of core electrons. Due to their close proximity to the nucleus, core electrons experience a much stronger attraction than valence electrons, thus significantly increasing their velocities (see Fig. 2.1). An analogous effect occurs in the solar system, where the planet Mercury orbits the sun at much higher velocities than the more distant planets. In general, the velocity of the electrons increases with higher atom number. Already for elements of moderate size, the core electrons move so fast that classical quantum mechanics fails to properly account for their motion. However, for elements in the lower half of the periodic table, the influences of relativistic effects extend beyond the core, since the orbitals of all other electrons need to remain orthogonal to the core orbitals, hence effectively contracting the valence shell. To even obtain a qualitatively correct wave function of molecules that contain such elements, relativistic effects must be included. [START_REF] Pyykko | Relativistic effects in structural chemistry[END_REF] Together with quantum mechanics, special relativity has revolutionized the paradigm of modern physics. The latter theory requires a drastic re-evaluation of the concepts of space and time, although it only gains relevance at extremely high energy scales. However, this theory is classical in the sense that it concerns the deterministic motion of point particles, whereas quantum mechanics is fundamentally probabilistic in nature. To include the effects of relativity in our calculations, it has to be extended to the quantum mechanical realm. Fortunately, this work has already been done for us by Paul Dirac, who formulated a relativistic extension of the Schrödinger equation. [START_REF] Adrien | A theory of electrons and protons[END_REF] The merits of the Dirac equation are its more accurate energy values, its natural description of spin and spin-orbit coupling and its prediction of anti-matter. As alluded to before, the Dirac equation is particularly useful when describing properties that depend on the core electrons, such as NMR-, [START_REF] Pyykkö | On the relativistic theory of NMR chemical shifts[END_REF][START_REF] Proton | carbon, and cadmium nmr measurements and relativistic calculation of the cadmiumcarbon coin tensor in dimethyl cadmium[END_REF][START_REF] Jokisaari | The r α -structure and anisotropy of the Hg-C indirect coupling constant of methylmercury nitrate dissolved in nematic and lyotropic liquid crystals[END_REF] Mössbauer- [START_REF] Knecht | Mössbauer spectroscopy for heavy elements: a relativistic benchmark study of mercury[END_REF][START_REF] Donovan Hedegård | Theoretical 57 Fe Mössbauer spectroscopy: isomer shifts of [Fe]-hydrogenase intermediates[END_REF][START_REF] Zelovich | Relativistic coupled cluster calculation of Mössbauer isomer shifts of iodine compounds[END_REF] or X-ray spectroscopy. [START_REF] Siegbahn | Electron spectroscopy for atoms, molecules, and condensed matter[END_REF] In this work, a relativistic approach will be pursued to properly describe the core electrons. Therefore, this chapter will be devoted to the methods of relativistic quantum chemistry.

Special Relativity

The Postulates of Special Relativity

As a point of departure, I will first discuss the theory of special relativity, thus illustrating the basic principles to understand the more advanced topics that are introduced later in this chapter. The foundations of this theory are given by the two postulates of special relativity, the first of which, originally formulated by Galileo, states that the laws of physics are the same regardless of the inertial frame from which it is described. Here, an inertial frame is any frame that does not undergo accelerated motion. Therefore, the axis of two inertial frames may be oriented differently and move at a constant relative speed. Suppose that we have two of such frames S and S that are described by parallel axes, where the former is stationary and the latter is moving with a speed of v in the direction of the positive part of the x-axis. In the framework of classical mechanics, the coordinates of both systems are related by the transformation rules

t = t (2.1) x = x -vt y = y z = z,
where it has further been assumed that both frames coincide at t = 0. In transformations of this type, referred to as Galilean transformations, time enters as a parameter that is universally agreed on. It can be shown straightforwardly that Newton's second law is indeed invariant with respect to transformations of this type

F = m d 2 r dt 2 = m d 2 dt 2 (r -vt) = m d 2 r
∇ • E = 1 0 ρ (2.2) ∇ × E = - ∂B ∂t ∇ • B = 0 ∇ × B = µ 0 J + µ 0 0 ∂E ∂t ,
where ρ and J are the charge-and current densities, 0 the electric constant and µ 0 the magnetic constant. To demonstrate the behaviour under Galilean transformations, the partial derivatives appearing in these equations should be expressed in the coordinates of S using the chain rule. Furthermore, it should be noted that in a moving frame, we may have additional currents that are not present in the stationary frame. For example, a charge distribution, ρ 0 (r) at rest in S, gains velocity in the moving frame S , thus creating a current according to j (r , t) = -vρ 0 (r, t).

(2.3) At this point, invariance of the Maxwell equations can be demonstrated if and only if a substitution can be made

E → E ; B → B (2.4) ρ → ρ ; j → j
that reduce the equations to their untransformed form. However, for Galilean transformations, such a substitution does not exist, implying that either something is wrong with the Maxwell equations or the Galilean transformations. Here, I will assume the latter to be the case, so in the following, I will lay out a transformation rule that respects the structure of the Maxwell equations.

However, using aforementioned procedure to find a correct transformation is a rather lengthy one and does not reveal much of the physics. To obtain a more intuitive picture of the correct transformation, I will simplify our problem by considering a specific instance of the Maxwell equations: the homogeneous case

∇ • E = 0 (2.5) ∇ × E = - ∂B ∂t ∇ • B = 0 ∇ × B = µ 0 0 ∂E ∂t ,
which is valid in the absence of charge-and current densities. Although these equations are less complicated than the inhomogeneous counterparts, it is desirable to further simplify them by decoupling the electric and magnetic field. This can be achieved by first applying the cross-product to the second-and fourth equation

∇ × (∇ × E) = - ∂∇ × B ∂t (2.6) ∇ × (∇ × B) = µ 0 0 ∂∇ × E ∂t , followed by insertion of the identity ∇ × (∇ × F ) = ∇(∇ • F ) -∇ 2 F , yielding the result ∇ 2 E = -µ 0 0 ∂ 2 E ∂t 2 (2.7) ∇ 2 B = -µ 0 0 ∂ 2 B ∂t 2 .
These equations can be identified as three-dimensional wave equations with a propagation speed of c = 1 √ µ 0 0 . Possible solutions to these equations are given by the linearly polarized plane waves from previous chapter (Eqn. (E.1)). The first postulate thus creates a paradox: the speed of light is expressed in fundamental constants, so it should be the same regardless of the inertial frame, whereas from the Galilean transformations one would expect the speed to be c + v if the emitter of light is moving. If we insist that the Galilean transformations are not entirely correct, we need to introduce the second postulate, stating that the speed of light is the same regardless of the inertial frame. [START_REF] Einstein | Zur elektrodynamik bewegter körper[END_REF]

Lorentz Transformations

Based on the two relativistic postulates, I will derive new transformation laws that respect the structure of the Maxwell equations. For simplicity, I will derive these transformation rules for the previously introduced frames S and S . The universality of the speed of light can only be explained if the notion of time and space is different depending on the inertial frame. Therefore, to avoid any prior assumptions, I will ascribe the space-time coordinates {x, y, z, t} to S and {x , y , z , t } to S .

Suppose that we have a trolley carrying a point-like light source that is following the origin of S . At t = t = 0, the light source will start emitting spherical waves which, according to the second postulate, travel at the speed c regardless of the inertial frame. Accordingly, to each spatial interval traversed by the light, |r 2r 1 | in S and |r 2r 1 | in S , we can associate a time interval, t 2t 1 and t 2t 1 , required for the light to propagate said regions in space. Therefore, the following relation should hold in both frames

|r 2 -r 1 | |t 2 -t 1 | = |r 2 -r 1 | |t 2 -t 1 | = c. (2.8)
From simple manipulations, we arrive at [START_REF] Einstein | Zur elektrodynamik bewegter körper[END_REF][START_REF] Saue | Principles and applications of relativistic molecular calculations[END_REF]

(x 2 -x 1 ) 2 +(y 2 -y 1 ) 2 +(z 2 -z 1 ) 2 -c 2 (t 2 -t 1 ) 2 = (x 2 -x 1 ) 2 +(y 2 -y 1 ) 2 +(z 2 -z 1 ) 2 -c 2 (t 2 -t 1 ) 2 = 0, (2.9
) which can be further simplified by introducing the compactified notation

∆x µ ∆x µ = ∆x µ ∆x µ = 0.
(2.10)

In this expression, the position four-vector is defined as

x µ =          ct µ = 0 x µ = 1 y µ = 2 z µ = 3 ; x µ =          -ct µ = 0 x µ = 1 y µ = 2 z µ = 3 , (2.11)
where the zeroth component represents time and the remaining three space. Hereafter, I will assume that Greek indices represent the index of a four vector, whereas Latin indices represent its spatial component. Furthermore, according to this convention, raising and lowering the index switches the sign of the zeroth component. This operation can be represented tensorially by the Minkowski metric [START_REF] David | Introduction to electrodynamics[END_REF] x µ = η µν x ν ; η = diag(-1, 1, 1, 1).

(2.12)

The space-time interval in expression (2.10) is particularly useful to find the class of transformations that preserve the speed of light. Given a transformation between the two frames

x µ = Λ µ ν x ν , (2.13)
the speed of light is preserved if the following relation holds

Λ ρ µ η µν Λ σ ν = η ρσ , (2.14)
because in this case, we have

∆x µ ∆x µ = ∆x ν η νµ ∆x µ = ∆x ρ Λ ρ ν η νµ Λ σ µ ∆x σ = ∆x ρ η ρσ ∆x σ = ∆x σ ∆x σ . (2.15)
Transformations of this type form a group and are referred to as Lorentz transformations. The action of these transformations mixes the space-time coordinates of the two frames. Lorentz transformations that only mix spatial components (∃i, j s.t. Λ j i = 0) can be thought of as ordinary rotations, whereas translations, or boosts (∃µ s.t. Λ µ 0 = 0 and Λ 0 µ = 0), are associated with the admixture of spatial and temporal coordinates and vice versa. In line with the group properties of Lorentz transformations, two consecutive Lorentz transformation can generally be decomposed into a rotation and a boost. The admixture of space with time resulting from boosts does not seem all that unfamiliar, since this also occurs in the Galilean transformations (Eqn. (2.1)) to describe relative motion between frames. However, the admixture of time with space seems extremely counter-intuitive and it highlights an essential feature of special relativity: time is not absolute.

In our specific example we have Λ j i = 0, thus implying that our Lorentz transformation is a boost, described by the following non-trivial relations [90, chapter 2]

ct = Λ 0 1 x + Λ 0 0 ct ; ct = -Λ 1 0 x + Λ 0 0 ct (2.16) x = Λ 1 1 x + Λ 1 0 ct ; x = Λ 1 1 x -Λ 0 1 ct.
Using the second postulate of relativity, the coefficients of this linear transformation can be obtained. In both frames, the spherical wave travels the following distance along the x/x -axis

x = ct; x = ct .
(2.17)

Substituting these relations into the transformation rule gives

t t = Λ 1 1 + Λ 1 0 ; t t = Λ 1 1 -Λ 0 1 , (2.18)
which can be combined to yield the following equation

(Λ 1 1 ) 2 -(Λ 1 0 ) 2 = 1. (2.19)
To solve for these elements, we need an additional relation, which can be found by evaluating the coordinates of S at the origin

0 = Λ 1 1 x -Λ 0 1 ct → x t = v = cΛ 0 1 Λ 1 1 . (2.20)
Therefore, the elements read

Λ 1 1 = 1 1 -v 2 c 2 = γ (2.21) Λ 0 1 = v c 1 -v 2 c 2 = βγ.
From similar manipulations, it can be shown that Λ 0 0 = γ, which is hereafter referred to as the Lorentz factor. From these coefficients we arrive at the final form of the Lorentz transformations [START_REF] Einstein | Zur elektrodynamik bewegter körper[END_REF]

t = γ(t + v c 2 x ) (2.22) x = γ x + vt ) y = y z = z .
In this example, we chose to consider a transformation to a frame that is moving along the x-axis, although these relations can be readily generalized to arbitrary directions. As alluded to before, the admixture of space and time provided by Lorentz transformations drastically changes the properties of space-time. For instance, in a relativistic framework, the concept of simultaneity needs re-evaluation. To illustrate this, I have created a schematic representation of our Lorentz transformation in the form of a space-time diagram (Fig. 2.2). In such a diagram, time is represented by a coordinate axis that may mix with the other axes upon transformation, contrary to classical mechanics where time is a universal parameter. An important feature of Fig. 2.2 is that the ct -and x -axis are tilted towards the speed of light. In the unprimed frame, all simultaneous events occur in the xy-plane perpendicular to the ct-axis, whereas all simultaneous events in the primed frame lie in the x y-plane. Therefore, two events that are simultaneous in frame S, are not necessarily simultaneous in frame S . Furthermore, when observed from different intertial frames, length intervals are not equivalent. For example, some interval along the x-axis in frame S

= x 2 -x 1 , (2.23)
is not the same when evaluated in the moving frame S

= x 2 -x 1 = γ x 2 -vt) -γ x 1 -vt) = γ(x 2 -x 1 ), (2.24)
which is referred to as length contraction. [START_REF] Antoon | De relatieve beweging van de aarde en den aether[END_REF][START_REF] Franscis | The Ether and the Earth's Atmosphere[END_REF] An analogous phenomenon occurs for time intervals

∆t = 1 γ ∆t , (2.25)
referred to as time-dilation. [START_REF] Larmor | A dynamical theory of the electric and luminiferous medium[END_REF][START_REF] Cohn | Zur Elektrodynamik bewegter Systeme I[END_REF] Furthermore, to every moving object, we can associate the following quantity

dτ = 1 c η µν dx µ dx ν = dt 2 - dx 2 c 2 - dy 2 c 2 - dz 2 c 2 (2.26) = 1 - 1 c 2 dx 2 dt 2 - dy 2 dt 2 - dz 2 dt 2 dt = 1 γ dt.
In the rest frame of this object, it evaluates to

dτ = dt , (2.27)
thus describing the passage of time as perceived from this frame. Accordingly, this quantity is called the proper time. [START_REF] Minkowski | Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern[END_REF] The major advantage of the proper time is its invariance under Lorentz transformation, thus being independent of the frame from which it is observed.

Potentials and Fields: a Relativistic Formulation

I recall that our initial goal was to find a transformation that leaves the Maxwell equations invariant. Before proceeding, I will evaluate whether our transformation obeys this constraint, thereby also deriving transformation rules for the electromagnetic fields in the process. Following a similar procedure as before, invariance can be demonstrated by first transforming the partial derivatives using the chain rule and making the following substitution for the electromagnetic fields[96]

E x = E x B x = B x (2.28) E y = γ(E y -vB z ) B y = γ(B y + v c 2 E z ) E z = γ(E z + vB y ) B z = γ(B z - v c 2 E y ),
where the (un)primed quantities are evaluated in the (un)primed coordinates. To avoid clutter of notation, I left out explicit coordinate dependence, although this feature should not be taken lightly, as it has profound consequences on the nature of the electromagnetic interactions. However, I will postpone the discussion on the coordinates until we have found proper transformation rules for the charge-and current densities as well. Since these distributions are extended objects, they occupy a certain interval in threedimensional space, thus implying that their shape will be deformed according to Eqn. (2.24). The Lorentz contraction, however, will only manifest itself in one direction, thus implying the following relation between a moving charge, ρ (r , t ), and its value at rest, ρ 0 (r, t) [START_REF] David | Introduction to electrodynamics[END_REF]Section 12.3.4] ρ (r , t ) = γρ 0 (r, t).

(2.29)

Inserting this deformed charge density into Eqn. (2.3), allows us to find an expression for the transformed current density

j (r , t ) = γvρ 0 (r, t) = uρ 0 (r, t); u = dr dτ , (2.30)
where in the second equality, I have expressed the velocity in terms of the proper time.

Comparing these two equations, it follows that we can gather the charge-and current density in a four vector

ρ µ = ρ 0 u µ cρ µ = 0 j i µ = 1, 2, 3 , (2.31)
which, in turn, is defined in terms of the four-velocity

u µ = dx µ dτ = γc µ = 0 u i = γv i µ = 1, 2, 3
.

(2.32) However, in above example, I have assumed that the charge is at rest in S, whereas more general relations can be inferred from the transformation properties of four vectors

ρ = γ(ρ + v c 2 j x )
(2.33)

j x = γ j x + vρ) j y = j y j z = j z .
By making the substitutions in Eqn. (2.28) and (2.33), we can thus preserve the structure of the Maxwell equations. Furthermore, from the four-current, it can be clearly demonstrated that the continuity equation is Lorentz invariant

∂ µ j µ = 0; ∂ µ = (- 1 c ∂ t , ∇). (2.34)
What remains to discuss are the coordinate substitutions in the transformed fields and densities. For example, if we take the second line in Eqn. (2.28) and explicitly write the coordinate dependence

E y (r , t ) = γ(E y (r, t) -vB z (r, t)), (2.35)
the right-hand-side can be written in terms of the primed coordinates by inserting the Lorentz transformation

E y (r , t ) = γ E y γ(x +vt ), y , z , γ(t + v c 2 x ) -vB z γ(x +vt ), y , z , γ(t + v c 2 x ) . (2.36)
The consequences of this can be better understood by the following example. Suppose that we are observing a charge distribution from a certain point. Any change in the charge distribution can be observed from the emitted electromagnetic fields. However, due to the finite speed of light, it will take some time before this field will reach the observation point, suggesting that the observed fields in fact display a delayed image of the charge distribution, also referred to as retardation. Because Lorentz transformations inherently take the finiteness of the speed of light into account, the effects of retardation are included by making this coordinate substitution. Another interesting feature of these transformation rules is that apparently the electric-and magnetic field mix. Similar to the fact that two observers do not always agree on the notion of space and time, neither will they agree on the definition of the electric and magnetic fields.

Although the transformation rules in Eqn. (2.28) are completely valid, they are rather unwieldy. More elegant relations can be derived by considering the potential formulation of electrodynamics. The central idea to this formulation is the Helmholtz decomposition, which states that any vector field can be decomposed into a curl free (irrotational) and divergence free (solenoidal) component, provided that the curl and divergence of this field approach zero faster than 1 r 2 as r → ∞. [START_REF] Helmholtz | Über Integrale der hydrodynamischen Gleichungen[END_REF][START_REF] Gabriel | On the dynamical theory of diffraction[END_REF][40, Appendix B] For a vector field, F , to which this condition applies, the Helmholtz decomposition assumes the form

F = -∇U + ∇ × W , (2.37)
where the scalar function and vector function, U and W , are given by

U (r) = 1 4π d 3 r D(r ) |r -r | ; W = 1 4π d 3 r C(r ) |r -r | (2.38)
with the integrands defined as

D(r) = ∇ • F ; C(r) = ∇ × F . (2.39)
Because the curl of every gradient and the divergence of every curl vanishes, it is straightforward to demonstrate that the first-and second terms in the decomposition are indeed irrotational and solenoidal. According to the third Maxwell equation (Eqn. (2.2)), the magnetic field is always divergence free, suggesting that this field is purely solenoidal.[40, Chapter 10][41, Chapter 6] Its Helmholtz decomposition thus reads

B = ∇ × A, (2.40)
where A is referred to as the vector potential. The electric field, on the other hand, is only divergence-free in the absence of charge density. In addition, the second Maxwell equation implies that the electric field is generally not curl-free. However, by substituting Eqn. (2.40) in the second Maxwell equation, it is possible to obtain an auxiliary vector field that is irrotational

∇ × E + ∂A ∂t = 0, (2.41)
suggesting that it can be expressed as the gradient of a scalar potential

-∇φ = E + ∂A ∂t → E = -∇φ - ∂A ∂t . ( 2
∇ 2 φ + ∂ ∂t (∇ • A) = - 1 0 ρ (2.43) ∇ 2 A -µ 0 0 ∂ 2 A ∂t 2 -∇ ∇ • A + µ 0 0 ∂φ ∂t = -µ 0 j.
Therefore, the potential formulation comes with the benefit of reducing the amount of required vector components (from six to four), albeit at the cost of more complicated differential equations.

The additional complexity of these equations can be somewhat reduced by exploiting the redundancies of the potential formulation. It can be shown that this formulation is not unique: we can add the gradient of any real function to the vector potential without changing the magnetic field

∇ × (A + ∇χ) = ∇ × A, (2.44)
because the curl of a gradient always vanishes. The magnetic field is thus invariant under transformations of the form A → A + ∇χ. However, this does not hold for the electric field

E = -∇φ - ∂A ∂t - ∂∇χ ∂t . (2.45)
By extending our definition of our transformation

A → A + ∇χ (2.46) φ → φ - ∂χ ∂t ,
it can be shown that the electric field is invariant

E = -∇φ + ∇ ∂χ ∂t - ∂∇χ ∂t - ∂A ∂t = -∇φ - ∂A ∂t . (2.47)
Transformations of this type are called gauge transformations and their form should be kept in mind, because they are applied extensively throughout this thesis. [START_REF] Jackson | Historical roots of gauge invariance[END_REF] To fix the additional degrees of freedom provided by gauge transformations, a gauge condition ought to be imposed. In most quantum chemical applications the condition ∇ • A = 0 is imposed, thus implying that it does not have a longitudinal component. From this condition, we obtain the following equations

∇ 2 φ = - 1 0 ρ (2.48) ∇ 2 A -µ 0 0 ∂ 2 A ∂t 2 -µ 0 0 ∇ ∂φ ∂t = -µ 0 j,
where the solutions of the first equation gives the well-known Coulomb potential. Accordingly, this gauge condition is called Coulomb gauge. However, imposing Coulomb gauge does not fix all additional degrees of freedom, as this gauge condition indicates a family of transformations, having the following constraint on the gauge function ∇ 2 χ = 0.

At first sight, it seems that the instantaneous nature of the scalar potential contradicts the second postulate of relativity. Furthermore, the gauge condition manifestly breaks Lorentz invariance, as it does not treat space and time on equal footing. However, from careful analysis, it follows that all effects related to retardation are captured by the vector potential, thus respecting relativistic laws. [START_REF] Jackson | From Lorenz to Coulomb and other explicit gauge transformations[END_REF] In a relativistic context, the analysis is simplified if the gauge condition itself is also Lorentz invariant. Therefore, a more suitable condition is given by the Lorentz gauge

∇ • A + 1 c 2 ∂φ ∂t = 0, (2.49)
providing us with more symmetric equations for the potentials

∇ 2 φ - 1 c 2 ∂ 2 φ ∂t 2 = - 1 0 ρ (2.50) ∇ 2 A -µ 0 0 ∂ 2 A ∂t 2 = -µ 0 j.
Note that this expression depends on the d'Alembertian

= ∂ µ ∂ µ = ∇ 2 - 1 c 2 ∂ 2 ∂t 2 , (2.51)
which is a Lorentz invariant quantity by construction. Using the definition of the current four vector, these two equations can be combined in one

A µ = -µ 0 J µ . (2.52)
From the structure of this equation, it can be argued that the four-potential is a valid fourvector, since the d'Alembertian is Lorentz invariant, whereas the right-hand-side is given by the four-current. The right-and left-hand-side can only be made consistent if the fourpotential A µ = (φ/c, A), is indeed a four-vector. Finding transformation rules for the fields is thus much more efficient by first transforming the potentials and then compute the fields from these potentials. For example, let us consider a point charge at rest in S. Being at rest, the vector potential is zero, while the scalar potential is given by the Coulomb potential. In the moving frame, S , the charge is in motion, thus implying the following potentials φ (r , t ) = γφ(r, t)

(2.53)

A (r , t ) = γvφ(r, t)
Therefore, the field is purely electric in the rest frame, whereas it gains a magnetic component in the moving frame. This effect can be identified with magnetic induction, whose discovery long predates the development of relativistic theory. In Section B.1 of the appendix, it will be shown that this effect is fundamental to spin-orbit coupling.

Relativistic Kinematics

Having thus established a coordinate transformation that preserves the Maxwell equations, one may wonder why the invariance of the Maxwell equations should be favoured over the invariance of Newton's second law as prescribed by the Galilean transformations. After all, Galilean transformations intuitively make sense from everyday experiences. Every train enthusiast can testify that from the outside of a train, a walking passenger indeed seems to move at the sum of his/her own speed and the speed of the train. These seemingly different transformations can be connected by recognizing that the velocity of everyday objects does not even reach a fraction of the speed of light, thus hinting at the fact that Galilean transformations are a limiting case of Lorentz transformations. At these velocity scales, it is safe to assume that the following limit applies c → ∞, hereafter referred to as the non-relativistic limit. Within this limit, it can be shown that γ → 1 and β → 0, which gives rise to the Galilean transformation upon substitution in Eqn. (2.15), implying that these transformations effectively hold true at low speeds. Provided that the speeds appearing in our system are small enough compared to the speed of light, it is justified to utilize the familiar concepts of classical kinematics. At elevated speeds, however, we are forced to generalize these concepts to a relativistic framework.

Using the previously introduced definition of the four-velocity (Eqn. (2.32)), it is possible to generalize the well-known concept of momentum to the relativistic domain

p µ = m 0 u µ = γm 0 c = mc µ = 0 γm 0 v i = mv i µ = 1, 2, 3 . (2.54)
Here, I have defined the rest mass, m 0 , and the instantaneous mass m = γm 0 . Using these relations, it can be argued that massive objects become even heavier when put in motion.

From the structure of the Lorentz factor, γ, it thus follows that at the light-speed, massive objects gain an infinite mass, thus prohibiting anything except light or any other massless object to travel at this speed. Although the spatial components of the four-momentum can be recognized straightforwardly as the relativistic generalizations of classical momentum, the zeroth component has a less clear meaning. To reveal the physical interpretation of this component, I will exploit that the square of the four momentum

p µ p µ = m 2 (v 2 -c 2 ) = -m 2 0 c 2 (2.55) is conserved in time d(p µ p µ ) dτ = 2p µ dp µ dτ = 0. (2.56)
In this expression we can further identify the four-force

F µ = dp µ dτ = dmc dτ µ = 0 dp i dτ µ = 1, 2, 3. (2.57)
By expanding the dot product from Eqn. (2.56) into its individual components, we arrive at the following expression

2mu • F -2mc dmc dτ = 0. (2.58)
Further eliminating the common factors yields the result

dmc 2 dτ = F • v = F • dr dτ = dE dτ , (2.59)
where in the last line, I have used the relation between work and energy. Therefore, we can make the identification

E = mc 2 → p 0 = mc = E c , (2.60)
which implies that the zeroth component is energy and even more surprisingly that energy is equivalent to mass. [START_REF] Einstein | Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?[END_REF] This relation is particularly useful to describe nuclear decay, where the total mass of the decay products is slightly smaller than the mass of the original nucleus, thus implying that the mass difference has been converted into energy. Even though this difference is often tiny, the factor of c 2 implies that possibly a vast amount of energy can be released in such processes.

From the square of the four-momentum, we can derive an alternative expression for the relativistic energy that separates the rest energy from the kinetic energy

E = ± c 2 p 2 + m 2 0 c 4 . (2.61)
A cumbersome feature of this equation is that it allows for negative energies. However, in the current context, the negative energies do not form a problem. After all, the positive and negative energies are separated by a barrier of 2mc 2 and for classical point-particles, energies can only vary continuously. Therefore, the negative energy solutions can be discarded in a classical context. The non-relativistic kinetic energy can be obtained by subtracting the energy at rest and expanding the (positive) square root

E -m 0 c 2 = m 0 c 2 p 2 m 2 0 c 2 + 1 -m 0 c 2 = p 2 2m 0 - p 4 8m 3 0 c 2 + • • • , (2.62)
where the zeroth order term can be recognized as the non-relativistic kinetic energy.

In its current form, however, this expression has limited use, as it does not include any influences of external electromagnetic fields. By minimizing an interaction Lagrangian, as proposed by Schwarzschild in 1903 [START_REF] Schwarzschild | Zur Elektrodynamik: I. Zwei Formen des Princips der kleinsten Action in der Elektrontheorie[END_REF], it follows that the influences of these fields on a charged particle can be introduced by substituting the four-momentum with the mechanical four-momentum

p µ → π µ = p µ -qA µ , (2.63)
which is referred to as minimal substitution. [START_REF] Gell-Mann | The interpretation of the new particles as displaced charge multiplets[END_REF] Therefore, the relativistic energy of a point particle in an electromagnetic field can be expressed as

E = ± m 2 0 c 4 + c 2 π 2 + qφ; π = p -qA. (2.64)
Based on this classical energy, I will proceed to derive a relativistic wave equation.

Relativistic Wave Equation

The Dirac Equation and Its Predecessors

The classical description of a point particle is deterministic in the sense that everything there is to know can be calculated with certainty from the position and momentum of the particle. In a quantum mechanical framework, however, observable properties are associated with Hermitian operators that give the probabilities of measurement outcomes when acted on the wave function. Assuming the position representation, the following substitution has to be made to promote classical observables to Hermitian operators

E → i ∂ t ; p → i ∇.
(2.65)

However, applying this procedure to the relativistic energy of a point particle (Eqn. (2.61)) is cumbersome, because defining a square-root operator requires special consideration. A straightforward solution is to expand the square root (Eqn. (2.62)), truncate it at finite order and making the aforementioned operator substitutions. [START_REF] Kenneth | Introduction to relativistic quantum chemistry[END_REF] To first order, this gives the Schrödinger Hamiltonian

Ĥs = i ∂ ∂t = p2 2m e = (σ • p) 2 2m e , (2.66)
where in the last equality I have applied the Dirac identity,

(σ•A)(σ•B) = A•B+iσ•(A×B).
The validity of the last equality suggests that in the free-particle case, spin is a hidden degree of freedom, which is revealed by the influence of an external magnetic field. To introduce these fields into our Hamiltonian, the momentum ought to be substituted with the mechanical momentum (Eqn. (2.63)), which should then be promoted to an operator. It should be noted that minimal substitution provides a relativistic coupling between particle and fields, whereas we apply this to a non-relativistic Hamiltonian. However, strictly speaking all magnetic interactions vanish in the non-relativistic limit, as can be inferred from the Maxwell equations, leaving us with electrostatic interactions. [ Here, the field-matter interaction is mediated through five terms. The first two terms are responsible for the coupling of the electronic motion with the external fields, which can be combined into a single term ( e me A • p) if Coulomb gauge is imposed. The third term corresponds to diamagnetic interactions, whereas the fourth term is responsible for the Zeeman splitting of spin-levels as induced by the external magnetic field. The last term gives an electric interaction between the scalar potential and the electron charge, which is the Coulombic attraction if our external fields stem from the nuclei. It is important to note that spin is introduced ad hoc, because it was an arbitrary decision to use (σ • π) in our Hamiltonian instead of π.

In its current form, our Hamiltonian does not include relativistic corrections, which can be taken into consideration by including higher-order terms from Eqn. (2.62) in our Hamiltonian. The next order to include depends on a fourth power of the momentum, which corrects for the increase of electron mass for higher velocities. However, unlike the kinetic energy, which is positive-definite T = ψ| 2 2me p2 |ψ = 2 2me pψ|pψ > 0, the fourth-power term is negativedefinite for similar reasons. Any variational procedure that is based on this Hamiltonian is doomed to fail, because the fourth-power term will draw the energy towards negative infinity. Nevertheless, it is possible to include the effects of this operator as a perturbation of the Pauli Hamiltonian. To describe strong relativistic effects, this approach does not seem to be ideal.

Instead of quantizing the square-root expansion, it might be a better idea to apply this procedure to the square of the energy,

+ m e c 2 ψ KG (r, t) = 0, (2.69)
leading to the Klein-Gordon equation. [START_REF] Gordon | Der comptoneffekt nach der schrödingerschen theorie[END_REF][START_REF] Kragh | Equation with the many fathers. The Klein-Gordon equation in[END_REF] Being constructed from the Lorentz invariant d'Alembertian operator, this equation seems like a perfectly valid relativistic wave equation, although it is met with two problematic features: it allows for negative energies and negative probability densities. The problem of negative energy solutions was to be expected, since it already manifest itself in the classical case (Eqn. (2.61)). In a quantum mechanical framework, however, the presence of such solutions cannot be ignored anymore, because discontinuous jumps in energy are allowed. Upon decaying to lower and lower energy states, an infinite amount of energy is released. States of negative energy exhibit the additional paradoxical feature of requiring energy absorption to decelerate their motion, which has never been observed experimentally. [START_REF] Adrien | A theory of electrons and protons[END_REF] Furthermore, the existence of negative densities completely breaks down the probabilistic interpretation of the wave function.

Paul Dirac was especially frustrated with the second problem and thus set out to find an alternative to the Klein-Gordon equation. In his work, he proposed to factorize the Klein-Gordon equation

p0 + α 1 p1 + α 2 p2 + α 3 p3 + m e cβ -p0 + α 1 p1 + α 2 p2 + α 3 p3 + m e cβ ψ(r, t) = 0, (2.70)
in terms of the quantities {α i } and β. [START_REF] Adrien | The quantum theory of the electron[END_REF] By expanding the two factors

-p2 0 + α 1 α 1 p2 1 + α 2 α 2 p2 2 + α 3 α 3 p3 p3 + m 2 e c 2 ββ (2.71) +(α 1 α 2 + α 2 α 1 )p 2 p1 + (α 1 α 3 + α 3 α 1 )p 3 p1 +(α 2 α 3 + α 3 α 2 )p 3 p2 + m e c(α 1 β + βα 1 )p 1 +m e c(α 2 β + βα 2 )p 2 + m e c(α 3 β + βα 3 )p 3 ψ(r, t) = 0,
it can be shown that these quantities need to obey the following algebra in order for the above expression to reduce to the Klein-Gordon equation

{α i , α j } = 2δ ij (2.72) {α i , β} = 0.
Therefore, {α i } and β are non-commutative, implying that they can be represented by matrices. One possible set of matrices is given by

α i = 0 2 σ i σ i 0 2 ; β = I 2 0 2 0 2 -I 2 , (2.73)
also referred to as the Dirac matrices. We arrive at the following expression for the timedependent Dirac equation

cα • p + m e c 2 β ψ(r, t) = ĥD ψ(r, t) = i ∂ψ(r, t) ∂t . (2.74)
The relativistic character of the Dirac equation can be further highlighted by multiplying from the left by

β i γ µ ∂ µ + m e c ψ(r, t) = 0; γ µ = (β, -βα), (2.75)
where appears the gamma matrices, obeying the algebra

{γ µ , γ ν } = -2η µν I 4 . (2.76)
This notation suggests that the gamma matrices form a four-vector

i γ µ ∂ µ + m e c ψ(r, t) = γ ρ Λ ρ ν η νµ Λ σ µ pσ + m e c ψ(r, t) = 0, (2.77)
although this can only hold true if their algebra is conserved upon Lorentz transformations.

Using the algebra of the gamma matrices, this can be confirmed [START_REF] Adrien | The quantum theory of the electron[END_REF]

{γ µ , γ ν } = Λ µ ρ Λ ν σ {γ ρ , γ σ } = -2Λ µ ρ η ρσ Λ ν σ I 4 = -2η ρσ I 4 . (2.78)
Although the aforementioned covariant form of the Dirac equation is useful to demonstrate its transformation properties, it is typically not used in electronic structure theory, where the point of depart is given by the time-independent Dirac equation. This equation can be derived by expressing the time-dependent wave function as ψ(r, t) = T (t)ψ(r) and inserting it into Eqn. (2.74)

ĥD ψ(r) ψ(r) = i T (t) ∂T (t) ∂t . (2.79)
However, it should be noted that this separation does not hold for all inertial frames. Within the special frame where our separation is valid, above equation holds for all values of r and t, suggesting that it can only be true if both sides equal a constant that can be identified with the energy. Therefore, the time-independent Dirac equation reads

cα • p + m e c 2 β ψ(r) = Eψ(r). (2.80)
By applying the minimal substitution procedure and aligning the energy scale with the nonrelativistic energy (such as in Eqn. (2.64)), the latter achieved by the substitution β → β = β -I 4 , we arrive at the following result

cα • p + m e c 2 β -eφ + ecα • A ψ(r) = Eψ(r). (2.81)
Note that all light-matter interactions are mediated by two linear terms, unlike the interaction terms in the Pauli Hamiltonian in Eqn. (2.68) which may also include the spin-Zeeman term and the quadratic diamagnetic term.

Physical Properties of the Dirac Equation

In our derivation, we have introduced the Dirac matrices from the necessity to have consistent equations. The physical content of these matrices can be revealed in the Heisenberg picture, where states are time-independent, whereas observables evolve according to Ω(t) = e i Ĥt/ Ω(0)e -i Ĥt/ . Using the Heisenberg equation of motion

dr dt = -i[r, ĥD ] = cα, (2.82)
it can thus be concluded that the Dirac matrices can be identified as velocity operators. However, the algebraic properties of the Dirac matrices create a paradox, from which we have c 2 |α| 2 = 3c 2 I 4 , thus contradicting the second postulate of relativity. Fortunately, the same algebraic properties imply that the three components cannot be known simultaneously, similar to angular momenta. If one of these three components is completely known, for example cα x , its eigenvalues ±c imply that the velocity is still much higher than expected from a massive particle like the electron. It can be demonstrated that the Dirac electron undergoes highly oscillatory motion (referred to as zitterbewegung in ref. [START_REF] Schrödinger | Über die kräftefreie bewegung in der relativistischen quantenmechanik[END_REF]) superimposed on its average motion. It is the latter type of motion that is more in line with the expected behaviour of massive particles (see ref.

[109, Section 7.3] for a more detailed derivation).

From the structure of the Dirac matrices, it can be further concluded that the Dirac wave function is a four-component function

ψ(r) =     ψ 1 (r) ψ 2 (r) ψ 3 (r) ψ 4 (r)     . (2.83)
Based on the results from previous sections, it almost comes as a reflex to conclude that everything with four components is a four vector. For the Dirac wave function this is definitely not the case. Two of the four degrees of freedom are related to spin symmetry, since the Dirac wave function can be expressed in terms of two Pauli spinors[90, Section 4.5]

ψ(r) = ψ L (r) ψ S (r) , (2.84)
where ψ L and ψ S are conventionally referred to as the large-and small component. In Section B.1 of the appendix, the validity of this identification is further demonstrated. It thus remains to explain the origin of the remaining two degrees of freedom, or put in other words, why we need two Pauli-spinors in our wave function. A straightforward mathematicsoriented answer to this question is that the matrices should be at least four-dimensional to obey the algebraic rules from Eqn. (2.72). However, a more satisfying answer can be found from the operation of charge conjugation. The action of this operation allows us to relate the solutions of the minimally coupled time-dependent Dirac equation, to the same equation but with opposite external charges [

cα • p + m e c 2 β + eφ -ecα • A Ĉψ(r, t) = i ∂ t Ĉψ(r, t).
(2.85)

In Section B.2 of the appendix it is demonstrated that the explicit action of charge conjugation is given by

Ĉψ(r, t) =     ψ Sβ * (r, t) -ψ Sα * (r, t) -ψ Lβ * (r, t) ψ Lα * (r, t)     , (2.86)
which suggests that this operator consists of a matrix operator that permutes the components, and the complex conjugation operator

Ĉ = K0 Ûc . (2.87)
Since charge conjugation swaps the role of the large-and small component, it follows that these two components are needed to accommodate for this symmetry. Furthermore, in the time-independent case, charge conjugation can be used to relate the solutions of positiveand negative energies

cα • p + m e c 2 β + eφ -ecα • A Ĉψ(r) = -E Ĉψ(r), (2.88)
where the negative energy sign appears from complex conjugation of the time-dependent wave function K0 (e -iEt/ ψ(r)) = e iEt/ K0 (ψ(r)). Charge conjugation hence reveals the intricate relation between positrons and negative-energy electronic solutions.

However, before jumping to conclusions, I recall that in a quantum mechanical framework negative energy solutions are a problematic feature. The relativistic hydrogen atom, for instance, is predicted to decay within nanoseconds if we allow our system to decay to negative energy solutions. To solve this problem, Dirac proposed that all negative energy solutions are occupied, which prohibits decay by virtue of the exclusion principle. [START_REF] Adrien | A theory of electrons and protons[END_REF] In its time, this proposition was highly controversial, since it implies that all space is permeated by an infinitely dense collection of negative energy particles, more commonly referred to as the Dirac sea. However, Dirac argued that the negative energy sea will not have a significant effect on its surroundings, provided that the density of these states is homogeneous enough. For a completely filled negative energy continuum, this assumption seems to be reasonable. However, the Dirac sea should induce measurable effects in regions where homogeneity is broken. When a negative-energy electron is excited from the Dirac sea, such that it now occupies a positive energy state, a hole is left in the Dirac sea. This hole has the same physical characteristics as the electron, except that its charge is opposite. Therefore, the Dirac equation predicts the existence of anti-matter, which was experimentally confirmed by Anderson through the discovery of the positron. [START_REF] Anderson | The Positive Electron[END_REF] In the presence of charge, inhomogeneities are induced in the Dirac sea, also referred to as vacuum polarization, the effects of which are only measurable at extremely high charges. [START_REF] Salman | Quantum electrodynamic corrections in quantum chemistry[END_REF] A more rigorous description of these effects can be obtained from the theory of quantum electrodynamics. The negative energy solutions are not present anymore in this theory, but are rather replaced by positive energy positronic solutions, thus solving the problems associated with the Dirac sea. However, quantum electrodynamics is far beyond the scope of this thesis and I will rather apply Dirac's approach to negative energy solutions.

The Radial Dirac Equation

Having established the main features of the Dirac equation, I now proceed to study this equation under the influence of a central potential (V (r)), a class of systems that encompasses atoms, thus being essential for quantum chemistry. Spherical symmetry provides one-dimensional radial equations that in some cases are even solvable analytically. This section is mainly inspired from the seventh chapter in Introduction to Relativistic Quantum Chemistry. [START_REF] Kenneth | Introduction to relativistic quantum chemistry[END_REF] In the non-relativistic case, the Hamiltonian commutes with the angular momentum operator, [ Ĥs , ˆ ], thus implying that in this case the wave function is an angular momentum eigenstate

ψ m (r) = 1 r R(r)Y m (θ, φ), (2.89)
where and m are the azimuthal and magnetic quantum number. Because spin is not explicitly present in the non-relativistic Schrödinger Hamiltonian, it follows that Ĥs , ŝ = 0. However, in a relativistic formulation, these two relations do not hold anymore

ĥD , ˆ = i cα × p ; ĥD , 2 Σ = -i cα × p , (2.90)
where appears the four component spin operator Σ = σ 0 0 σ . Interestingly, these two equations exactly cancel, suggesting that the total angular momentum

ĵ = ˆ + 2 Σ (2.91)
commutes with the Hamiltonian ĥD , ĵ = 0.

(2.92)

Therefore, for central potentials, the corresponding wave functions are rather eigenfunctions of the total angular momentum. This function is given by a four component spinor, which in turn can be expressed in terms of two Pauli spinors. To learn more about the structure of the four component eigenfunctions, it turns out to be useful to start from the two component case.

The two component functions are also total angular momentum eigenfunctions, although the associated total angular momentum operator is defined in terms of the more conventional spin operator in terms of the Pauli matrices. The structure of these functions can be found by applying a Clebsch-Gordan series

| s; jm j = m =- 1 2 ms=-1 2 | m sm s m sm s | s; jm j , (2.93)
in terms of the uncoupled basis, | m sm s = | m ⊗|sm s . This basis implies that the resulting function is also an eigenfunction of ŝ2 and ˆ 2 , which generally holds for two-component states, but not for four component spinors. In the current context, it is preferred to express these states in the position representation

ξ jm j (θ, φ) = m =- 1 2 ms=-1 2 r| m sm s m sm s | s; jm j = m =- 1 2 ms=-1 2 m sm s | s; jm j Y m |sm s , (2.94
) which defines the spherical spinors. From the rules of the Clebsch-Gordon decompositions, it follows that the total angular momentum assumes the values ± 1 2 . However, from this result, it can be concluded that j is not a unique label, because at least two combinations of and s can give the same value of j. For example, j = 3 2 , can be obtained from parallel coupling of = 1; s = 1 2 or anti-parallel coupling of = 2; s = 1 2 . A more useful quantum number can be found from the action of the operator

κ = -σ • ˆ -= 1 ˆ 2 + ŝ2 -ĵ2 - (2.95)
onto the spherical spinors

κξ j,m j (θ, φ) = 1 ˆ 2 + ŝ2 -ĵ2 -ξ j,m j (θ, φ) (2.96)
= ( + 1) -j(j + 1) - 1 4 ξ j,m j (θ, φ) = κξ j,m j (θ, φ).
(2.97)

The κ quantum number can assume positive and negative integer values

κ = for j = -1 2 --1 for j = + 1 2 . (2.98)
To each of the two combinations of and s that yields the same value of j, we can assign an equal but opposite value of κ, thus resolving the ambiguity associated with j. Therefore, we arrive at a uniquely labelled expression for the spherical spinors

ξ κ,m j (θ, φ) =   sgn(-κ) κ+ 1 2 -m j 2κ+1 Y ,m j -1 2 κ+ 1 2 +m j 2κ+1 Y ,m j + 1 2   . (2.99)
In analogy with the non-relativistic hydrogen atom, it can be argued from symmetry considerations that the four-component hydrogenic wave function assumes the form

ψ L (r) ψ S (r) = 1 r P (r)ξ κ L m j (θ, φ) iQ(r)ξ κ S m j (θ, φ) (2.100)
where the values of κ are intentionally labelled differently for the large-and small component.

Because the wave function is an eigenfunction of total angular momentum, they are either equal or opposite: κ L = ±κ S . To find the correct relation between these two quantum numbers, I will insert this ansatz into the Dirac equation,

1 r V cσ • p cσ • p V -2mc 2 P ξ κ L m j iQξ κ S m j = E r P ξ κ L m j iQξ κ S m j . (2.101)
and apply the separation of variables procedure. In this equation and the ones that follow, the radial-and angular dependence have been suppressed.

To facilitate this procedure, I will split the kinetic energy term (σ • p) into a radial and angular part. From the relation (σ • e r )(σ • e r ) = I 2 , we obtain the following result

σ • p = 1 r σ r (σ • r)(σ • p), (2.102)
where appears the radial Pauli matrix

σ r = e r • σ = cos θ e -iφ sin θ e iφ sin θ -cos θ . (2.103)
Subsequently, we can obtain the desired separation by inserting the Dirac identity

σ • p = 1 r σ r (r • p) + 1 r σ r iσ • (r × p) = σ r (p r + 1 r iσ • ˆ ) (2.104) = -iσ r ( ∂ ∂r - 1 r σ • ˆ ) = -iσ r ∂ ∂r - 1 r (κ + ) ,
where appears the radial momentum pr = e r • p and the κ operator. Inserting this result into the Dirac equation yields the following system of equations

-c 1 r (κ S + ) - ∂ ∂r 1 r Qσ r ξ κ S m j = E -V 1 r P ξ κ L m j (2.105) c 1 r (κ L + ) - ∂ ∂r 1 r P σ r ξ κ L m j = E -V + 2mc 2 1 r Qξ κ S m j .
In these equations all angular dependence is contained in the spherical spinors and radial Pauli matrix, which implies the relations

σ r ξ κ S m j = -ξ κ L m j (2.106) σ r ξ κ L m j = -ξ κ S m j .
From the identity σ r ξ κm j = -ξ -κm j , [START_REF] Szmytkowski | Recurrence and Differential Relations for Spherical Spinors[END_REF] it can thus be concluded that κ S = -κ L , implying that the small component is opposite in parity compared to the large component. By projecting out the spherical spinors, we can isolate differential equations for the radial wave function

-c 1 r (κ S + ) - ∂ ∂r 1 r Q = E -V 1 r P (2.107) c 1 r (κ L + ) - ∂ ∂r 1 r P = E -V + 2mc 2 1 r Q.
In case of hydrogenic systems, where the potential is given by V (r) = -Ze 2 4π 0 r , this equation is analytically solvable. The solution of these equations is beyond the scope of this thesis and the reader is referred to refs. [START_REF] Grant | The Dirac Equation[END_REF]Section 3.3] and [START_REF] Reiher | The Dirac Hydrogen Atom[END_REF]Chapter 6] for further detail. A problematic feature of these solutions is their singular behaviour at the origin for |κ|= 1. However, this singularity can be classified as a weak singularity, because the wave function can still be normalized. In the calculation of many-electron system, this singularity is avoided by applying other nuclear distributions than the conventional point-charge distribution, for example a Gaussian distribution [START_REF] Visscher | Dirac-Fock atomic electronic structure calculations using different nuclear charge distributions[END_REF] or the Fermi nuclear model. [START_REF] Parpia | Relativistic basis-set calculations for atoms with Fermi nuclei[END_REF] 2.4 Four-Component Relativistic SCF Methods

Similarities and Differences with Non-Relativistic Theory

Here, the findings from previous sections will be extended to many-electron systems. In a four-component relativistic framework, the Hamiltonian can still be written as the general form in Eqn. (1.45), although the integrals are expressed in terms of four-component orbitals. [START_REF] Saue | Relativistic Hamiltonians for Chemistry: A Primer[END_REF] The one-electron integrals follow straightforwardly from Eqn. (2.81)

h pq = d 3 rψ † p cα • p + mc 2 β + V ψ q . (2.108)
To derive a two-electron interaction operator, one possible route is to establish the relativistic interaction between two moving classical point charges and quantize this expression.

However, already at the classical level, formulating this expression is complicated. I recall that from the finiteness of the speed of light, it follows that the electromagnetic interaction is retarded. The presence of retardation implies that the full history of the point particles needs to be known to find the interaction between them. For this reason, it is not possible express this interaction in closed form, thus requiring us to expand it in orders of the 1/c. At lowest order, this expansion yields the familiar Coulomb repulsion between electrons (see.

Eqn. (1.47)), whereas the next order corresponds to the Breit operator

ĝB (r ij ) = - e 2 8π 0 c 2 r ij cα i • cα j + (cα i • r ij )(cα j • r ij ) r 2 ij (2.109)
which captures effects related to magnetic interactions and retardation. [START_REF] Breit | The Effect of Retardation on the Interaction of Two Electrons[END_REF] The first-and second term in this operator are also referred to as the Gaunt and gauge term, the latter of which depends on the choice of gauge. [START_REF] Gaunt | IV. The triplets of helium[END_REF][START_REF] Huang | Orbit-orbit interaction[END_REF] Here, Coulomb gauge is imposed, whereas the Lorentz gauge, for instance, removes the gauge term. Throughout this thesis, I will neglect such interactions and only consider the Coulomb repulsion between the electrons. Due to the structural similarities between the relativistic and non-relativistic manyelectron Hamiltonian, most electronic structure theories can be carried over the the relativistic domain, albeit with the proper assumptions. Indeed, the Hartree-Fock determinant can be readily generalized to the relativistic domain by defining a Slater determinant consisting of four-component orbitals. A relativistic extension to DFT, however, requires more elaborate theory. To generalize the fundamental theorems of DFT, it is required to invoke the theory of quantum electrodynamics, where the negative energy electrons are converted into positive energy positrons. However, even within this framework, it is difficult to formulate the Hohenberg-Kohn theorems, since a variational procedure of QED is not as well-established as in the non-relativistic regime. However, the variational properties of QED can be postulated by the observation that everyday matter (consisting of electrons, protons, atoms etc.) is stable. [START_REF] Engel | Relativistic density functional theory[END_REF] Using these results, it can be shown that relativistic DFT is very similar to its non-relativistic counterpart. In principle, relativistic DFT requires functionals that are optimized for relativistic calculations, although in practice, functionals from non-relativistic theory are used, which is justified from a pragmatic point of view.

With the similarities between relativistic and non-relativistic DFT further established, I can now proceed to define the generator of orbital rotations. In the relativistic case, rotations involving negative-energy orbitals require consideration as well. The rotations between negative-energy orbitals do not change the energy, as this quantity is defined in terms of occupied positive-energy solutions. The negative-energy orbitals can thus be thought of as an additional type of virtual orbitals. Therefore, in line with Section 1.3, the relativistic extension of the generator of orbital rotations can be defined as

κ = ai κ a + i + a † a + a i + -κ * a + i + a † i + a a + + κ i + a -a † i + a a --κ * i + a -a † a -a i + . (2.110)
Likewise, when extending this machinery to response theory, it follows that the positivenegative/negative-positive rotations need to be included to fully relax the perturbed state. Moreover, in the context of magnetic interactions, it can even be argued that these rotations correspond to diamagnetic interactions. [START_REF] Aucar | On the origin and contribution of the diamagnetic term in four-component relativistic calculations of magnetic properties[END_REF] However, these additional rotations should be treated with care, to avoid variational collapse towards the negative-energy continuum. In fact, it has been shown that the energy should be minimized with respect to the positivepositive rotations and maximized with respect to the positive-negative rotations, thus corresponding to a minmax principle. [START_REF] Talman | Minimax Principle for the Dirac Equation[END_REF] Alternatively, the relativistic SCF procedure can be understood as an excited state search, which aims to find the lowest positive energy solution on top of the negative-energy solutions. Formally, this method can be solved by projecting out the lower states in each iteration, creating a reduced matrix which is subsequently diagonalized. Although the reduced problem is bounded from below, this bound can fall below the exact eigenvalue, because in the full problem the eigenvalue increases due to the interlace theorem. This feature of the Dirac equation is conventionally referred to as prolapse. [START_REF] Faegri | Relativistic Gaussian basis sets for the elements K-Uuo[END_REF] In practice, however, explicit projectors are seldom used, because the same effect can be achieved by discarding the negative-energy solutions and only use positive-energy solutions to construct the Fock matrix in accordance with the aufbau principle. This procedure is facilitated by the large energy gap of 2mc 2 between the positive-and negative-energy solutions, which makes it easy to distinguish either type of solutions at every step of the SCF cycle.

In the special case of atomic symmetry, the relativistic analogue of Eqn. (1.66) can be solved using grid-based methods. Here, the orbitals assume the general structure of Eqn. (2.100), where the angular part is fully known from symmetry, whereas the radial functions are to be determined on the grid. Furthermore, the pseudo-eigenvalue equation assumes the general form of Eqn. (2.107), with the averaged two-electron interaction from HF theory, or the exchange-correlation potential from DFT included in the potential. [START_REF] Grant | The Dirac Equation[END_REF] This procedure will be used in Section 4.3 to generate reference data effectively corresponding to the complete basis set limit.

Kinetic Balance

We proceed to expand the large-and small component in a set of functions {χ i (r)}

ψ X (r) = i c X i χ i (r); X = L, S, (2.111)
and find the coefficients of by following the minmax procedure. However, the first basis set calculations based on the Dirac equation were met with problems associated to the lower bounds of the energy. [START_REF] Kim | Relativistic Self-Consistent-Field Theory for Closed-Shell Atoms[END_REF] For instance, basis set calculations of the relativistic hydrogen atom fell much below its corresponding value from a non-relativistic framework. [START_REF] Asaad | Relativistic K Electron Wave Functions by the Variational Principle[END_REF] These problems are rooted in the basis set choice, which was essentially the same for the largeand small component functions. To better understand the shortcomings of our basis set expansion, I will write the expansion in a more general form

ψ(r) = χ L 0 0 χ S c L c S ; χ L i = χ Lα i χ Lβ i ; χ S i = χ Sα i χ Sβ i , (2.112)
where the basis functions of the large-and small component are allowed to be expanded in terms of different functions. Expressed in this basis, the Dirac Hamiltonian assumes the form

H = V LL cπ LS cπ SL V SS -2mc 2 S SS , (2.113)
which is constructed from the matrices

V XX ij = d 3 rχ X † i V χ X j ; X = L, S (2.114)
π LS ij = d 3 rχ L † i σ • pχ S j S XX ij = d 3 rχ X † i χ X j .
In matrix form, the eigenvalue problem thus reads

H c L c S = E S LL 0 0 S SS c L c S , (2.115)
which can be decoupled to yield a similar relation as Eqn. (B.3)

T LL = 1 2m e π LS S SS + ES SS -V SS 2mc 2 -1 π SL ≈ 1 2m e π LS π SL ; c 1. (2.116)
Note that in the non-relativistic limit, above expression should reduce to the non-relativistic kinetic energy. Let us closer inspect the expression on the right-hand-side of (2.116). In the complete basis set limit, this term can indeed be related to the non-relativistic kinetic energy

1 2m e π LS π SL ik = 1 2m e j d 3 r d 3 r χ L † i (r)σ • pχ S j (r)χ S † j (r )σ • p χ L k (r ) (2.117) = 1 2m e d 3 r d 3 r χ L † i (r)σ • pδ(r -r )σ • p χ L k (r ) = 1 2m e d 3 rχ L † i (r)(σ • p)(σ • p)χ L k (r) = 1 2m e d 3 rχ L † i (r)p 2 χ L k (r),
implying that the non-relativistic limit can only yield the correct result if a complete basis set is used. In practice, however, a complete basis set amounts to an infinite set of functions, which is not feasible in any computational scheme. Let us now suppose that our basis set is not complete anymore, yielding the expression below

1 2m e π LS π SL ik = 1 2m e d 3 rχ L † i (r)p 2 χ L k (r) (2.118) - 1 2m e j d 3 r d 3 r χ L † i (r)σ • pχ C j (r)χ C † j (r )σ • p χ L k (r ).
Here the functions {χ C i } form the orthogonal complement of {χ S i } whose union {χ C i }∪{χ S i } is complete. If we express the small component functions in the basis [START_REF] Schwarz | Basis set expansions of relativistic molecular wave equations[END_REF][START_REF] Grant | Conditions for convergence of variational solutions of Dirac's equation in a finite basis[END_REF][START_REF] Wallmeier | Basis-set expansion of the Dirac equation without variational collapse: Numerical test of the forth-back free-particle Foldy-Wouthuysen transformation[END_REF][START_REF] K G Dyall | Matrix representation of operator products[END_REF]

χ S i = σ • pχ L i , (2.119)
the second term in Eqn. (2.118) reduces to zero by orthogonality

1 2m e j d 3 r d 3 r χ L † i (r)σ • pχ C j (r)χ C † j (r )σ • p χ L k (r ) (2.120) = 1 2m e j d 3 r d 3 r χ S † i (r)χ C j (r)χ C † j (r )χ S k (r ) = 0.
The choice of small component space in Eqn. (2.119), referred to as restricted kinetic balance, thus restores the deficit in the kinetic energy. [START_REF] Stanton | Kinetic balance: A partial solution to the problem of variational safety in Dirac calculations[END_REF] As implied by Eqn. (2.119), the condition of restricted kinetic balance requires a two-component basis. Expressed in a scalar basis, Eqn. (2.119) translates to the weaker condition of unrestricted kinetic balance φ Sx i = ∂ j φ Lx i ; x = α, β, which has the undesirable feature of introducing spurious solutions. It is possible, however, to combine restricted kinetic balance with a scalar basis, but this requires an alternative scheme based on the so-called modified Dirac equation, which is the preferred method in this thesis. However, I will not delve into further details, because the aim of this section is to outline the main idea of kinetic balance. The interested reader is referred to refs. 37, 132. Regardless of the specifics, the kinetic balance procedure allows basis set calculations to be performed at the four-component relativistic level of theory, especially providing a drastic improvement for heavy-element compounds.

Symmetry Reductions from Time-Reversal Symmetry

However, these improvements in accuracy are met with elevated computational costs, since the wave function consists of four components, contrary to the one or two components appearing in non-relativistic calculations. Several approximations to the Dirac Hamiltonian are known that aim to reduce the amount of components, such as the exact two-component relativistic Hamiltonian (X2C), [START_REF] Jørgen | BSS / DKH infinite order the easy way ! (Talk from REHE[END_REF][START_REF] Iliaš | Theoretical study of PbO and the PbO anion[END_REF][START_REF] Kutzelnigg | Quasirelativistic theory equivalent to fully relativistic theory[END_REF][START_REF] Iliaš | An infinite-order two-component relativistic Hamiltonian by a simple one-step transformation[END_REF] the zeroth-order regular approximation (ZORA) [START_REF] Chang | Regular Two-Component Pauli-Like Effective Hamiltonians in Dirac Theory[END_REF][START_REF] Van Lenthe | Relativistic regular two-component Hamiltonians[END_REF][START_REF] Van Lenthe | The zero-order regular approximation for relativistic effects: The effect of spin-orbit coupling in closed shell molecules[END_REF] and the Douglas-Kroll-Hess (DKH) [START_REF] Douglas | Quantum electrodynamical corrections to the fine structure of helium[END_REF][START_REF] Hess | Applicability of the no-pair equation with free-particle projection operators to atomic and molecular structure calculations[END_REF][START_REF] Hess | Relativistic electronic-structure calculations employing a two-component nopair formalism with external-field projection operators[END_REF] Hamiltonians. However, the discussion of these Hamiltonians is beyond the scope of this thesis and the reader is referred to ref. [START_REF] Saue | Relativistic Hamiltonians for Chemistry: A Primer[END_REF] for more details. Instead of approximating the Dirac equation, I will discuss how the computational costs can be reduced using the time-reversal symmetry of the Dirac equation.

Let us first consider some general features of this symmetry. I will define K as the time-reversal operator, which is anti-linear and anti-unitary

K(aψ 1 + bψ 2 ) = a * Kψ 1 + b * Kψ 2 (2.121) Kψ 1 | Kψ 2 = ψ 1 |ψ 2 * = K ψ 1 |ψ 2 .
In general, any anti-unitary operator can be expressed as

K = ÛT K0 , (2.122)
where K0 is the complex conjugation operator introduced in Eqn. (2.87), and ÛT a unitary operator, whose representation depends on the form of the wave function, i.e. two-or four-component (see Section B.3 for an explicit representation in the four component case). Systems are time-reversal symmetric, if the Hamiltonian commutes with the time-reversal operator K, Ĥ = 0, which is generally the case for systems with an even amount of electrons in the absence of an external magnetic field. For such systems, Kramers theorem states that the action of the time-reversal operator defines pairs of functions (Kramers partners) [START_REF] Antoon | Théorie générale de la rotation paramagnétique dans les cristaux[END_REF]

Kψ = ψ; K ψ = -ψ, (2.123)
that are degenerate

E ψ = ψ| Ĥ|ψ = ψ| K-1 K Ĥ K-1 K|ψ = ψ| Ĥ| ψ = E ψ.
(2.124)

Kramers partners are not exclusive to relativistic formulations, since the two possible spin functions are related by time-reversal

K|α = |β ; K|α = -|β , (2.125)
thus forming a pair of Kramers partners. However, it should be kept in mind that spin symmetry is much more restrictive, because for any spin-free operator, Ô, the matrix elements between functions of opposite spin are necessarily zero

φ α p | Ô|φ β q = 0; ∀p, q (2.126) |φ α p = χ p (r)|α ; |φ β q = χ q (r)|β
implying a block-diagonal structure of the matrix representation

Ω = Ω αα 0 0 Ω ββ Ω αα pq = φ α p | Ω|φ α q Ω ββ pq = φ β p | Ω|φ β q . . (2.127)
Furthermore, from the structure of the spin-orbitals, it can be shown that these two blocks are identical

Ω αα pq = d 3 rχ * p Ωχ q = Ω ββ pq . (2.128)
Therefore, only one sub-matrix is needed in this simple example, thus reducing the computational cost roughly by a factor of four.

For general Kramers partners, however, these reductions are not as impressive. Let us construct the matrix representation of a general time-reversal symmetric operator, i.e. Ω = K-1 Ω K, in a basis of Kramers parters

Ω = ψ p | Ω|ψ q ψ p | Ω| ψq ψp | Ω|ψ q ψp | Ω| ψq (2.129)
Here, the off-diagonal blocks only vanish if the two functions in the matrix element are Kramers partners

ψp | Ω|ψ p = ψ p | Ω| ψp = 0. (2.130)
However, some symmetry gains can be achieved by relating the off-diagonal and on-diagonal blocks through time-reversal

K ψ p | Ω|ψ q = Kψ p | K Ω K-1 | Kψ q = ψp | Ω| ψq = ψ p | Ω|ψ q * (2.131) K ψp | Ω|ψ q = K ψp | K Ω K-1 | Kψ q = -ψ p | Ω| ψq = ψp | Ω|ψ q * ,
yielding the expression

Ω = A B -B * A * A pq = ψ p | Ω|ψ q B pq = ψ p | Ω| ψq . (2.132)
In this expression, the blocks A and B are unique, suggesting that only these two blocks need to be stored for calculations. Throughout each SCF cycle, it is thus desirable to preserve the structure in Eqn. (2.132). Maintaining this structure can either be achieved by expressing the generator of orbital rotations in terms of Kramers replacement operators, [START_REF] Ga Aucar | Operator representations in kramers bases[END_REF] or by employing the quaternion scheme. [START_REF] Saue | Quaternion symmetry in relativistic molecular calculations: The Dirac-Hartree-Fock method[END_REF] In this work, the quaternion scheme is preferred, because it simultaneously allows for the efficient treatment of point-group symmetry. In Appendix B.3 the quaternion scheme is discussed more thoroughly. Moreover, the treatment of transition moments benefits heavily from the structure provided by time-reversal symmetry. If our interaction operator is time-reversal symmetric, it follows that the property gradient assumes the same form as Eqn. (2.132). Operators that are anti-symmetric with respect to time-reversal, TA , can be made time-reversal symmetric by extracting an imaginary phase, i.e. TA = -i(i TA ) = -i TS , although the hermiticity of the operator is now inverted. By exploiting both the structure provided by time-reversal and hermiticity, it can be shown that the property gradient assumes the structure [START_REF] Saue | Linear response at the 4-component relativistic level: Application to the frequency-dependent dipole polarizabilities of the coinage metal dimers[END_REF]

E Ω =             C D -D * C * hC * hD * -hD hC             C ia = ψ i | Ω|ψ a D ia = ψ i | Ω| ψa . . (2.133)
where h is (-1), 1 for (anti-)Hermitian operators. From the structure of the response equations, which generally preserve time-reversal symmetry, it follows that the solution vectors can be expressed in a similar form [START_REF] Saue | Linear response at the 4-component relativistic level: Application to the frequency-dependent dipole polarizabilities of the coinage metal dimers[END_REF] X

n =             V n W n -W * n V * n hV * n hW * n -hW n hV n            
(2.134)

V n;ai = Z n;ai + hY n;ai ; W n;ai = Z n;a ī + hY n;a ī (2.135)
Transition moments calculated from such vectors thus read

X † n E Ω = = 4Re[C † V n ] + 4Re[D † W n ] if h E h X = 1 = 0 if h E h X = -1 , (2.136)
which vanishes for opposite hermiticity of both vectors and gives rise to real transition moments for the same hermiticity.

Using the tools described in this chapter, we are in a position to generalize all findings from Chapter 1 to the relativistic domain, thus enabling us to properly describe relativistic effects in our simulations of X-ray spectroscopy. However, there is still much room left for improvements in our model. In particular, based on the large discrepancy between the wave length and the extent of the system, I have applied the dipole approximation. However, in the X-ray domain, this approximation might not hold anymore. Therefore, in the following chapter, these assumptions will be challenged.

Chapter 3

Light-Matter Interactions Beyond the Dipole Approximation

Introduction

We have come a long way since we started our discussion about the basic principles of SCF linearresponse theory, followed by relativistic quantum chemistry. With the theoretical tools of the preceding chapters in place, we are almost ready to discuss some practical examples. However, before proceeding I need to address one key assumption made in Chapter 1. I recall that in the UV-Vis regime, the spatial extent of the target molecule is typically much smaller than the wave length of light. For that reason, it is justified to apply the dipole approximation, according to which the electromagnetic fields can be approximated by a uniform field. However, in the X-ray regime, this approximation comes into question because the wave length is of comparable size relative to the molecule. Figure 3.1 depicts a similar situation as in Figure 1.1, although in this case, the benzene molecule is interacting with X-rays, having a much shorter wavelength (λ = 2.6 nm).

Considering that the dipole approximation essentially boils down to the first order multipole expansion of the electromagnetic fields, there are generally two methods to correct for the shortcomings of this approximation. The first and most straightforward method avoids a multipole expansion altogether and bases its interaction on the exact electromagnetic fields, thus retaining their sinusoidal character in the interaction operator. Using the full interaction operator, all nondipolar effects are exactly included, although the formalism is somewhat more complicated due to the frequency dependence of this operator.

Alternatively, the multipole expansion can be truncated beyond first order, thus taking higherorder multipole moments into account. Multipole moments can be understood as idealizations of charge-and current distributions. For example, the electric-dipole moment is a vectorial quantity that gives the overall direction in which the charge distribution is polarized. The electric-quadrupole moment, on the other hand, describes the overall broadness of a charge distribution, akin to the variance of statistical distributions. Further inhomogeneities of the distributions are described by higher-order multipole moments, which, in Cartesian form, are defined as By including the contributions of all multipole moments, the charge-or current distributions can be recovered exactly. However, in practice, this is often not necessary, because typically the most dominant contributions stem from the lowest-order multipole moments. Complicated charge-or current distributions can thus be simplified by only considering these dominant contributions. This approach turns out to be particularly useful when describing intermolecular interactions. [START_REF] Stone | The Theory of Intermolecular Forces[END_REF] In the current context, however, I will use this machinery to approximate light-matter interactions.

Q [n] j 1 •••jn = d 3 rr j 1 r j 2 • • • r jn ρ(r, t); m [n] j 1 •••j n-1 ;i = n n + 1 d 3 rr j 1 r j 2 • • • r j n-1 r × j(r, t) i . (3.1)
Using either the full-or truncated light-matter interaction, I will describe two types of spectroscopy: near-edge X-ray absorption fine structure (NEXAFS) and X-ray natural circular dichroism (XNCD). NEXAFS can straightforwardly be calculated from the absorption cross-section derived from linearly polarized light. The calculation of XNCD, however, is slightly more complicated, as it involves the difference between the absorption cross-section of left-and right-handed circularly polarized light

∆σ(ω) = σ L (ω) -σ R (ω). (3.2)
From considerations of parity inversion, it can be argued that the circular dichroism of isotropic samples, e.g. gases and solutions, is non-zero only for chiral molecules (see Section C.1). [START_REF] Barron | A historical review of optical activity phenomena[END_REF] In fact, for enantiomers, the circular dichroism is equal but opposite, thus making circular dichroism one of the few spectroscopic techniques to distinguish enantiomers. Similar effects can be achieved for non-chiral systems by either imposing anisotropic conditions, [START_REF] Isborn | The optical rotatory power of water[END_REF] such as in crystals, or by applying an additional external magnetic field. However, the latter suggestion creates a different type of interaction, also referred to as X-ray magnetic circular dichroism. [START_REF] Nicolás | Going beyond the electric-dipole approximation in the calculation of absorption and (magnetic) circular dichroism spectra including scalar relativistic and spin-orbit coupling effects[END_REF] Leaning on the results from the previous chapter, all derivations will be performed in a fourcomponent relativistic framework. These derivations are mainly based on articles with my coauthorship (see. refs. 1, 2, 3 for more detail). Alternatively, it is also possible to derive a similar formalism in a non-relativistic framework, although it is more complicated, due to the many terms required to describe the light-matter interaction. Therefore, the non-relativistic formalism will not be further pursued here. For more details see refs. 31, 32, 150. Furthermore, unless stated otherwise, I will use notation from exact-state theory throughout my derivations, since the generalization to SCF theory follows straightforwardly from the procedure outlined in Section 1.3.3.

General Expressions for Monochromatic Light

To derive interaction operators that correct for the shortcomings of the dipole approximation, we have to start from the solutions of the homogeneous Maxwell equations. I would like to emphasize the versatility of these equations. On one hand they are indispensable in the formulation of relativistic theory, whereas in current context, they serve as a starting point to derive more accurate lightmatter interaction operators. The most general expression for monochromatic light is given below

E(r, t) = E 1 1 sin(k • r -ωt + δ 1 ) + E 2 2 sin(k • r -ωt + δ 2 ); E 1 = E cos θ; E 2 = E sin θ (3.3) B(r, t) = E 1 c 2 sin(k • r -ωt + δ 1 ) - E 2 c 1 sin(k • r -ωt + δ 2 ); E = E 2 1 + E 2 2 ,
where the fields now have two sinusoidal components, each of which having a distinct phase shift, δ 1 and δ 2 , amplitude, E 1 and E 2 and polarization direction, 1 and 2 . The relative size of the amplitudes can be parametrized by the angle θ. Furthermore, the polarization vectors span a righthanded coordinate system together with the wave vector, i.e. 1 × 2 = c ω k = e k . In passing, I note that due to the linear nature of the homogeneous Maxwell equations, any linear combination of two or more solutions is itself also a solution, allowing the possibility of complicated fields. The interaction of non-standard fields with molecules has been explored in ref. [START_REF] Yang | Spectroscopy in sculpted fields[END_REF]. Here, I will restrict myself to the more simple monochromatic case.

By imposing that the phases differ by an integer multiple of π, i.e. δ 2 = δ 1 + mπ; m ∈ Z, we obtain expressions for linearly polarized light

E(r, t) = E sin(k • r -ωt + δ); B(r, t) = E ω k × sin(k • r -ωt + δ), (3.4)
in which appears the polarization vector

= cos θ 1 + (-1) m sin θ 2 . (3.5)
Therefore, the parameter θ defines the polarization direction. Figure 3.2 shows a graphical representation of linearly polarized light.

Circularly polarized light can be obtained by setting θ = 1 4 π (which implies that

|ε 1 | = |ε 2 | = 1 √ 2 ε
), while simultaneously imposing that the difference between the phases is a half-integer multiple of π, i.e.

δ 2 = δ 1 + 1 2 π + mπ; m ∈ Z
E(r, t) = 1 √ 2 E 1 sin(k • r -ωt + δ) + (-1) m 1 √ 2 E 2 cos(k • r -ωt + δ) (3.6) B(r, t) = E c √ 2 2 sin(k • r -ωt + δ) -(-1) m E c √ 2 1 cos(k • r -ωt + δ).
When observed against its propagation direction, the polarization vector traces a circle, hence its name. Because of this, circularly polarized light is generally shaped as a helix (see Figure 3.2). Therefore, this type of light is chiral, which implies that it cannot be superimposed onto its mirror image. Consequently, a distinction has to be made between right-and left-handed circularly polarized light. According to the IUPAC convention, the handedness is defined by the direction of rotation of the electric field vector at -ωt+δ = 0 against the propagation direction. [START_REF] Braslavsky | Glossary of terms used in photochemistry, 3rd edition (IU-PAC Recommendations[END_REF] Accordingly, circularly polarized light is right-handed if the vector rotates anti-clockwise and vice versa. It turns out that the integer m determines the handedness: even for left-handedness and odd for righthandedness.

Besides its polarization, another important feature of electromagnetic radiation is its intensity, which can be calculated by time-averaging the magnitude of the Poynting vector over a full oscillation

I = S T ; S = 1 µ 0 E × B. (3.7)
The Poynting vector expresses the energy transfer that is carried by the fields. By inserting the general definition of monochromatic light into this vector, we obtain

S = E 2 1 cµ 0 e k sin 2 (k • r -ωt + δ 1 ) + E 2 2 cµ 0 e k sin 2 (k • r -ωt + δ 2 ), (3.8)
which, upon time-averaging, yields the following expression for the intensity

I = 1 T T 0 dtS = 1 2 c 0 E 2 , (3.9)
thus reflecting that the intensity is independent of the polarization type.

Full Semi-Classical Light-Matter Interaction

To derive an interaction operator based on these fields, we first have to convert them to potentials and then couple these potentials to the Hamiltonian using minimal substitution. As explained in Section 2.2.3, a Helmholtz decomposition should be applied to convert the fields from Eqn. (3.3) into potentials. I recall that the Helmholtz theorem holds for fields that go to zero faster than 1 r 2 , which is clearly not the case for the sinusoidal fields in Eqn. (3.3). However, in this case, a weaker condition applies, stating that any differentiable vector field can be decomposed in terms of a solenoidal and irrotational part, although these two components are not described in terms of D(r) and C(r) (Eqn. (2.38)). [START_REF] David | Introduction to electrodynamics[END_REF]Appendix B] As outlined in Section 2.2.3, potentials are defined up to a gauge transformation, suggesting that we have to impose a gauge condition to construct our potentials. Here, I will impose Coulomb gauge, which implies that the vector potential is purely solenoidal. Furthermore, by virtue of the first and third homogeneous Maxwell equation, i.e. ∇ • E = 0 and ∇ • B = 0, the fields are purely solenoidal as well. It thus follows that they can be expressed as

E = - ∂A ∂t ; B = ∇ × A, (3.10)
where the scalar potential term, ∇φ, necessarily vanishes, because it is purely irrotational by construction. Therefore, the scalar potential can assume any constant value, thus implying that we can set it to zero. From comparison of Eqn.(3.10) and Eqn. (3.3) it follows that the potentials in the Coulomb gauge assume the form

A(r, t) = - E 1 ω 1 cos(k • r -ωt + δ 1 ) - E 2 ω 2 cos(k • r -ωt + δ 2 ); φ(r, t) = 0. (3.11)
By inserting these potentials into the Dirac Hamiltonian, the following interaction operator can be derived

V (t) = - e ω E 1 (cα • 1 ) cos(k • r -ωt + δ 1 ) - e ω E 2 (cα • 2 ) cos(k • r -ωt + δ 2 ). (3.12)
As demonstrated in Section 1.2, the frequency components of this operator

V (ω) = - eE 2ω (cα • ˜ )e ik•r ; ˜ = 1 cos θe iδ 1 + 2 sin θe iδ 2 . (3.13)
can be used to find the absorption-cross section

σ f ull (ω) = 2πω I | f | eE 2ω (cα • ˜ )e ik•r |i | 2 f (ω; ω f i , γ f i ) = πω 0 c | f | e ω (cα • ˜ )e ik•r |i | 2 f (ω; ω f i , γ f i ), (3.14
) where in the second equality the field-strength dependency was eliminated by inserting Eqn. (3.9). In practice, however, the dimensionless oscillator strength is preferred over the absorption crosssection. The former can be obtained from the latter by removing the linewidth function and applying the prefactor substitution πω 0 c → 2meω e 2 . The oscillator strength thus reads

f f ull (ω) = 2m e ω e 2 | f | e ω (cα • ˜ )e ik•r |i | 2 . (3.15)
Therefore, we need to calculate transition moments from the following effective interaction operator

T (ω) = f | T (ω)|i ; T (ω) = e ω cαe ik•r . (3.16)
However, this operator has neither a well-defined hermiticity, nor is it time-reversal symmetric (α is time-reversal anti-symmetric), thus suggesting that the simplifications described in Section 2.4.3 do not apply. Nonetheless, it is possible to decompose this operator in a Hermitian and anti-Hermitian component through the relation

T (ω) = 1 2 T + T † + 1 2 T -T † = TH + TA , (3.17)
yielding the following result

TH = e ω cα cos(k • r) (3.18) TA = e ω icα sin(k • r).
Even though TA has the desirable properties of being time-reversal symmetric and anti-Hermitian, TH is anti-symmetric upon time-reversal. The Hermitian component can be made time-reversal symmetric by extracting an imaginary phase

f | TH |i = -i f |i TH |i (3.19)
and inserting it back at the end of the calculation. Due to the generality of the polarization vector, Eqn. (3.15) applies both to the absorption of linearly polarized light

f f ull (ω) = 2m e ω e 2 | • T (ω)| 2 = 2m e ω e 2 | • iT H (ω)| 2 + 2m e ω e 2 | • T A (ω)| 2 (3.20)
and circularly polarized light

f f ull L/R (ω) = 2m e ω e 2 | L/R • T (ω)| 2 ; L/R = 1 √ 2 ( 1 ± i 2 ) = * R/L , (3.21)
where the polarization vector, , is defined in Eqn. (3.5). Using the definition of circular dichroism (Eqn. (E.13)), we arrive at the compact expression below

∆f f ull (ω) = 2m e ω e 2 ( L • T )( R • T * ) -( L • T * )( R • T ) (3.22) = 2m e ω e 2 ( L × R ) • (T × T * ) = -i 2m e ω e 2 e k • (T × T * ) = 4m e ω e 2 e k • (iT H × T A ),
which is a relativistic extension of the previously reported result by Hansen and Avery. [START_REF] Hansen | A fully retarded expression for the rotatory strength[END_REF] In the second equality, the identity

(a × b) • (c × d) = (a • c)(b • d) -(a • d)(b • c
) has been applied. I stress again that this operator contains all non-dipolar corrections, from order zero to infinity. It can be connected to the electric-dipole approximation by applying the limit kr 1

T (ω) = f | e ω cαe ik•r |i kr 1 ---→ Q [1] = f | e ω cα|i , (3.23)
where Q [START_REF] Holmgaard List | Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory[END_REF] is the electric-dipole moment operator in the velocity representation. This representation deserves its name due to the velocity interpretation of the Dirac matrices, as prescribed by the Heisenberg equation of motion (Eqn. (2.82)). In addition, this relation can be used to make a connection with the electric-dipole operator in its more conventional form introduced in Eqn. (E.10)

f | e ω cα p |i = - i f | e ω r p , ĥD |i = -i f | Q[1] p |i , (3.24)
where I have assumed exact-state conditions, i.e. Ĥ|n = E n |n . It is common convention to refer to Q [START_REF] Holmgaard List | Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory[END_REF] and Q [START_REF] Holmgaard List | Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory[END_REF] as the velocity-and length gauge. However, this nomenclature is rather misleading because it suggests the existence of some special gauge condition, whereas the Coulomb gauge is imposed for the velocity representation and multipolar gauge (see section 3.5) for the length representation. Therefore, I will insist to refer to them as representations. The velocity representation has its non-relativistic analogue, since we have

f | e ω cα q |i limc→∞ -----→ f nr | e ωm e pq |i nr , (3.25)
which can be converted to the length representation using an equivalent commutator relation

pq m e = - i r q , Ĥs . (3.26)
Here, the left-hand-side can be identified with the non-relativistic velocity operator, thus further establishing the analogies between relativistic and non-relativistic theory. As a final note, it should be stressed that in its current form, the above conversion only works under exact-state conditions. Its generalization to approximate state-theory requires special considerations, which will be discussed in section 3.6.2. The electric-dipole moment operator introduces selection rules, because transition moments are only non-vanishing if the integrand contains at least one totally symmetric component. It can thus be concluded that this operator only allows transitions that transform as one of the Cartesian coordinates Q [1] = Q [1] x e x (Γ x ) (3.27)

+ Q [1] y e y (Γ y ) + Q [1] z e z (Γ z ).

These limitations, however, do not apply to the full interactions operator. The symmetry content of this operator can be revealed by applying the plane wave expansion

e ik•r = 4π ∞ =0 m=- i j (kr)Y m (e k )Y m * (e r ). (3.28)
This expansion contains contributions from all possible symmetries of the full rotation group, since the spherical harmonics can be thought of as position space irreps (irreducible representations) of this group. However, the overall symmetry of the full interaction operator is also influenced by the Dirac matrices, which transform as Cartesian coordinates. Therefore, the symmetries generated from the plane wave expansion need to be combined with the Cartesian symmetry stemming from the Dirac matrices. From this procedure, which is discussed in more detail in Section C.2, it follows that the full interaction contains contributions from all symmetries, except the totally symmetric irrep of the full rotation group. The lack of this symmetry can be understood from the directionality imposed by the Dirac matrices. In lower point groups, which are subgroups of the full rotation group, different irreps may coalesce. In the context of molecular calculations, it proves to be particularly useful to express this operator in terms of the irreps of the D 2h group

e ±ik•r = cos(k x x) cos(k y y) cos(k z z) (Γ 0 ) (3.29) -sin(k x x) sin(k y y) cos(k z z) (Γ Rz ) -sin(k x x) cos(k y y) sin(k z z) (Γ Ry ) -cos(k x x) sin(k y y) sin(k z z) (Γ Rx ) ∓ i sin(k x x) sin(k y y) sin(k z z) (Γ xyz ) ± i cos(k x x) cos(k y y) sin(k z z) (Γ z ) ± i cos(k x x) sin(k y y) cos(k z z) (Γ y ) ± i sin(k x x) cos(k y y) cos(k z z) (Γ x ),
which again demonstrates how the full interaction operator allows for more symmetries than the electric-dipole moment operator.

Exactly which of the symmetry components contributes to the oscillator strength depends on the orientation of the molecular system with respect to the light. In these examples, however, it is assumed that the molecular orientation is fixed in space, whereas in more realistic samples, such as gases and liquids, molecules can tumble freely, hence inducing isotropic conditions. These effects will be discussed in detail in Section 3.7, but for now it suffices to assume that all components of either Eqn. (3.27) or Eqn. (3.28) contribute under isotropic conditions.

From this analysis it thus follows that within the electric-dipole approximation, isotropic oscillator strengths are non-vanishing if the transition transforms as one of the Cartesian coordinates. The full interaction operator, however, is much less restrictive, as it only forbids transitions of spherical symmetry, i.e. j = 0 → j = 0. Furthermore, in a relativistic framework, spin-selection rules do not apply anymore, implying that most of the conventional selection rules derived in the non-relativistic electric-dipole approximation are abolished. However, these selection rules are still useful to make global predictions about the relative intensities of transitions. A dipole-forbidden transition, for instance, is expected to be less intense than a dipole-allowed transition.

We thus have the machinery at our disposal to either calculate transitions within the electricdipole approximation, or include all non-dipolar effects exactly. However, we may also want to describe these interactions at an intermediate level.

Truncated Light-Matter Interactions: Generalized

Velocity Representation

Origin Independence

The general strategy to derive such a formalism is to expand the full interaction with respect to the wave-vector and truncate it at some finite order

T (ω) ≈ N n=0 T [n] (ω); T [n] (ω) = 1 n! e ω cα(ik • r) n . (3.30)
However, truncated expansions of this kind inherently introduce origin dependence in the interaction operator. By shifting the origin according to O → O + a, we have

T [n] (ω; O) → T [n] (ω; O + a) = n m=0 1 m! (ik • a) m T [n-m] (ω; O). (3.31)
This origin-dependency is problematic because spectroscopic quantities, such as the absorption cross-section, should not depend on the choice of origin, which is a mathematical artifact. It should be noted, however, that this origin is not necessarily the same as the origin of the coordinate system. For example, the authors from Refs. 154 and 155 cleverly make use of this distinction to derive origin-independent magnetic properties. To avoid confusion, I will hereafter designate the former as gauge-origin and the latter as coordinate-origin.

For the full interaction operator, this dependence on gauge-origin does not pose a problem

T (ω; O) → T (ω; O + a) = e ω cαe ik•r e ik•a , (3.32)
because the gauge-origin dependence of this operator manifests itself as a phase factor, which does not contribute to the oscillator strength. Therefore, we should proceed cautiously when deriving observable quantities from truncated expansions. Inserting the operator from Eqn. (3.30) into the oscillator strength yields the result

f f ull (ω) = 2m e ω e 2 ∞ n=0 ∞ m=0 T [n] (ω)T [m] * (ω); T [n] (ω) = f |˜ • T [n] (ω)|i . (3.33)
Having two indices, the elements in this summation can be thought of as the elements of a matrix. It turns out to be beneficial to use the fact that we can sum these terms in any order that we want. In the following, I will extensively use techniques that exploit the freedom in choice of summation index. These techniques are explained in full detail in Section C. [START_REF] Martin Van Horn | Transition moments beyond the electric-dipole approximation: Visualization and basis set requirements[END_REF].

In its current form, the elements are summed column-wise, whereas it would be completely valid to sum these terms along the anti-diagonals of this matrix. A summation of this type is particularly useful because along the anti-diagonal, all terms are of the same order in the wave vector. Eqn. (3.33) can hence be rewritten as such a sum by using n + m as an index for the anti-diagonals, as it assumes a distinct positive integer value along each anti-diagonal

f f ull (ω) = ∞ n=0 f [n] (ω); f [n] (ω) = 2m e ω e 2 n m=0 T [n-m] (ω)T [m] * (ω). (3.34)
It has been shown by Bernadotte et al. [START_REF] Bernadotte | Origin-independent calculation of quadrupole intensities in X-ray spectroscopy[END_REF] that for each order, the oscillator strength is independent on the gauge-origin. To explicitly demonstrate this, I will follow an alternative approach that is being proposed in ref. [START_REF] Holmgaard List | Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory[END_REF]. As a starting point, I will insert definition of the shifted operators (Eqn. (3.31)) into the n-th order oscillator strength

f [n] (ω; O + a) = 2m e ω e 2 n m=0 T [n-m] (ω; O + a)T [m] * (ω; O + a) (3.35) = 2m e ω e 2 n m=0 n-m p=0 m q=0 (-1) q p!q! (ik • a) p+q T [n-m-p] (ω; O)T [m-q] * (ω; O).
If we think of the first two summation indices as matrix indices, it can be argued that the corresponding summations fill the upper left diagonal of a matrix. Using similar machinery as in Eqn. (3.34), the summation can be carried out over the anti-diagonal, thus requiring us to redefine the summation according to the index y = m + p

f [n] (ω; O + a) = 2m e ω e 2 n y=0 y m=0 m q=0 (-1) q (y -m)!q! (k • a) y-m+q T [n-y] (ω; O)T [m-q] * (ω; O). (3.36)
Subsequently, I will make the substitution y → u = ny, yielding the result below

f [n] (ω; O + a) = 2m e ω e 2 n u=0 n-u m=0 m q=0 (-1) q (n -u -m)!q! (ik • a) n-u-m+q T [u] (ω; O)T [m-q] * (ω; O). (3.37)
Here, the second and third summation fill the lower triangle and the diagonal of a matrix. Along the diagonal and each sub-diagonal, the index v = mq is constant and assumes a distinct value. By using v as a summation index, we can thus write the n-th order oscillator strength as

f [n] (ω; O + a) = 2m e ω e 2 n u=0 n-u v=0 T [u] (ω; O)T [v] * (ω; O) (ik • a) n-u-v (n -(u + v))! M. (3.38)
Here, the factor

M = n-(u+v) q=0 n -(u + v) q (-1) q = (1 -1) n-(u+v) (3.39) is only non-zero if n -(u + v) = 0, yielding the end result f [n] (ω; O + a) = 2m e ω e 2 n u=0 T [u] (ω; O)T [n-u] * (ω; O), (3.40)
which is clearly equivalent to Eqn. (3.34) upon relabelling of the summation indices. Therefore, order-by-order the oscillator strength is gauge-origin independent because the origin-dependent terms cancel exactly. It thus follows that the n-th order oscillator strength is a meaningful physical quantity, unlike, for example, the individual transition moments that are used to construct this quantity.

Simplifications from Time-Reversal Symmetry

Although Eqn. (3.30) makes sense physically, there is still much room for improvement. By exploiting time-reversal symmetry of the interaction operators, it can be shown that some orders in Eqn. (3.30) vanish. For these purposes, I will re-express the transition moments

T [n] = (-i) n-1 T [n] (ω) = f | 1 n! e ω ˜ • icα(k • r) n |i , (3.41)
such that the operator inside the integral is time-reversal symmetric. From the results of Section 2.4.3, it thus follows that the corresponding transition moments are real. The simplifications that follow from this property, are generally different for absorption of linearly polarized light and circular dichroism. Therefore, I will start to analyze the former case. Expressed in these redefined transition moments, the n-th order oscillator strength reads

f [n] (ω) = i n 2m e ω e 2 n m=0 (-1) m T [n-m] (ω)T [m] * (ω). (3.42)
It should be noted that the terms in this summation come in pairs. To further pursue this idea, it proves to be useful to separate the even-from the odd-order terms

f [2n] (ω) = (-1) n 2m e ω e 2 2n m=0 (-1) m T [2n-m] (ω)T [m] * (ω) (3.43) f [2n+1] (ω) = (-1) n i 2m e ω e 2 2n+1 m=0 (-1) m T [2n+1-m] (ω)T [m] * (ω)
Let us start by closer inspecting the terms of even order, which can be expressed as

f [2n] (ω) = 2m e ω e 2 T [n] (ω)T [n] * (ω) (3.44)
+ (-1) n 2m e ω e 2 n-1 m=0 (-1) m T [2n-m] (ω)T [m] * (ω) + (-1) n 2m e ω e 2 2n m=n+1 (-1) m T [2n-m] (ω)T [m] * (ω).
The two summations have common terms, although their summation indices do not agree. Both of these two summations can be grouped together by redefining these indices. The first summation will be redefined such that the summation order is reversed, whereas the latter summation will be shifted such that the index starts at zero. These change can be achieved by redefining the first-and second summation according to m → nm and m → mn, which yields the following result

f [2n] (ω) = 2m e ω e 2 n m=0 (-1) m (2 -δ m0 )Re[T [n+m] (ω)T [n-m] * (ω)]. (3.45)
For odd orders, the oscillator strength can be expressed as

f [2n+1] (ω) = (-1) n i 2m e ω e 2 n m=0 (-1) m T [2n+1-m] (ω)T [m] * (ω) (3.46) + (-1) n i 2m e ω e 2 2n+1 m=n+1 (-1) m T [2n+1-m] (ω)T [m] * (ω).
Similar to the results for even-order terms, the above expression can be simplified by redefining the first-and second summation according to m → nm and m → mn -1,

f [2n+1] (ω) = (-1) n 2m e ω e 2 n m=0 (-1) n-m 2Im T [n+1-m] (ω)T [n-m] * (ω) = 0, (3.47)
which always vanishes, because the transition moments are real. However, these results do not generally hold for circular dichroism, which at n-th order assumes the form

∆f [n] (ω) = f [n] L (ω) -f [n] R (ω) = -i n+1 2m e ω e 2 n m=0 (-1) m e k • (T [n-m] (ω) × T [m] * (ω)) (3.48) T [n] (ω) = f | 1 n! e ω icα(k • r) n |i
It again turns out to be useful to separate even from odd contributions

∆f [2n] (ω) = (-1) n+1 i 2m e ω e 2 2n m=0 (-1) m e k • (T [2n-m] (ω) × T [m] * (ω)) (3.49) ∆f [2n+1] (ω) = (-1) n 2m e ω e 2 2n+1 m=0 (-1) m e k • (T [2n-m+1] (ω) × T [m] * (ω)).
Following a similar strategy as previous case, the even-order contributions can be expressed as

∆f [2n] (ω) = -i 2m e ω e 2 e k • (T [n] (ω) × T [n] * (ω)) (3.50) + (-1) n+1 i 2m e ω e 2 n-1 m=0 (-1) m e k • (T [2n-m] (ω) × T [m] * (ω)) + (-1) n+1 i 2m e ω e 2 2n m=n+1 (-1) m e k • (T [2n-m] (ω) × T [m] * (ω)),
which can be simplified by applying the substitutions m → nm and m → mn to the first-and second summation

∆f [2n] (ω) = -i 2m e ω e 2 e k • (T [n] (ω) × T [n] * (ω)) - 2m e ω e 2 n m=1 (-1) m e k • i(T [n+m] (ω) × T [n-m] * (ω)) (3.51) + 2m e ω e 2 n m=1 (-1) m e k • i(T [n+m] * (ω) × T [n-m] (ω)) = 0.
This expression again vanishes by virtue of the realness of the transition moments. Likewise, the odd-order contributions can be rewritten as

∆f [2n+1] (ω) = (-1) n 2m e ω e 2 n m=0 (-1) m e k • (T [2n-m+1] (ω) × T [m] * (ω)) (3.52) + (-1) n 2m e ω e 2 2n+1 m=n+1 (-1) m e k • (T [2n-m+1] (ω) × T [m] * (ω)),
which yields the end results upon the index substitutions m → nm and m → mn -1 in the first-and second summation

∆f [2n+1] (ω) = 2m e ω e 2 e k • n m=0 (-1) m 2Re T [n+m+1] (ω) × T [n-m] * (ω) . (3.53)
Therefore, for circular dichroism, only odd contributions are relevant, whereas the even terms only contribute to the absorption of linearly polarized light.

Electric and Magnetic Contributions

We have thus derived the working equations to describe a truncated light-matter interaction at arbitrary order. However, at first sight, there does not seem to be a relation with the truncated interaction operators appearing in Eqn. (3.34) and the multipole moments introduced at the beginning of this chapter. To be more precise, in its current form, the oscillator strength does not distinguish between electric and magnetic components. In the following, I will demonstrate how such a decomposition can be achieved. As a starting point, I will rewrite the n-th order interaction operator in the following manner

T [n] (ω) = 1 (n + 1)! e ω i(˜ • cα)(k • r) n + n (n + 1)! e ω i(˜ • cα)(k • r) n . (3.54)
The reason for this decomposition becomes more clear in the subsequent step, where I will insert the identity

(k × ˜ ) • (r × cα) = (cα • ˜ )(k • r) -(r • ˜ )(k • cα) (3.55)
which yields the result

T [n] (ω) = 1 (n + 1)! e ω (˜ • icα)(k • r) + n(r • ˜ )(k • icα) (k • r) n-1 (3.56) + n (n + 1)! e ω (k × ˜ ) • (r × icα)(k • r) n-1 = k j 1 k j 2 • • • k jn ˜ p X [n] j 1 •••jn;p (ω).
In the last line, I have introduced the multipole moment operator in component form

X [n] j 1 •••jn;p (ω) = 1 (n + 1)! Q[n+1] j 1 •••jn;p - i ω 1 n! m[n] j 1 •••j n-1 ;r ε rjnp (3.57)
which is expressed in terms of the two operators

Q[n+1] j 1 •••jn;p = ie ω r j 1 • • • r j n-1 (cα p r jn + ncα jn r p ) (3.58)
and

m[n] j 1 •••j n-1 ;r = n n + 1 r j 1 • • • r j n-1 (r × ĵ) r ; ĵ = -ecα. (3.59)
Because this operator yields the velocity representation electric-dipole moment (Eqn. (3.23)) at zeroth order, Eqn. (3.57) is referred to as the generalized velocity representation. In this representation, the second operator can straightforwardly be recognized as a magnetic operator, due to its similarities with the magnetic multipole moments introduced in Eqn. (E.12). It should be noted that the compact form of the current density operator is a special feature of relativistic theory, since the non-relativistic current density may also contain explicit contributions depending on the external field and the electron spin. Having established that the second operator in Eqn. (3.57) is magnetic in nature, it seems only logical to conclude that the first operator,

Q[n+1] j 1 •••jn;p
, is an electric multipole moment operator. However, before jumping to conclusions, it should be confirmed that this operator is indeed purely electric in character.

For this proof, it is more suitable to start from

V [n] (ω) rather than T [n] V [n] (ω) = - E 2 i n-1 k j 1 k j 2 • • • k jn ˜ p X [n] j 1 •••jn;p (ω) (3.60) = - E 2 i n-1 k j 1 k j 2 • • • k jn ˜ p 1 (n + 1)! Q[n+1] j 1 •••jn;p + E 2 i n-1 k j 1 k j 2 • • • k jn ˜ p i ω 1 n! m[n] j 1 •••j n-1 ;r ε rjnp .
Furthermore, it is useful to express the electromagnetic fields in terms of their frequency components

E(r, t) = E(r, ω)e -iωt + E(r, -ω)e iωt ; B(r, t) = B(r, ω)e -iωt + B(r, -ω)e iωt (3.61)
and evaluate their n-th order derivatives at r = 0

∂ n E(r , ω) ∂r j 1 • • • ∂r jn r =0 = i n E 2i ˜ k j 1 k j 2 • • • k jn ; ∂ n B(r , ω) ∂r j 1 • • • ∂r jn r =0 = i n E 2i 1 ω (k × ˜ )k j 1 k j 2 • • • k jn . (3.62)
By comparing Eqn. (3.62) with Eqn. (3.60), it follows that

V [n] (ω) = - i (n + 1)! ∂ n E p (r , ω) ∂r j 1 • • • ∂r jn r =0 Q [n+1] j 1 •••jn;p + 1 n! ∂ n-1 B r (r , ω) ∂r j 1 • • • ∂r j n-1 r =0 m [n] j 1 •••j n-1 ;r , (3.63)
in which Q

[n+1] j 1 •••jn;p couples with the n-th order derivative of the electric field, whereas m

[n] j 1 •••j n-1 ;r
couples with the (n -1)-th order derivative of the magnetic field. Therefore, the two operators in Eqn. (3.57) are indeed electric and magnetic in nature. By expressing the n-th order oscillator strength in terms of Eqn. (3.57)

f [n] (ω) = i n 2m e ω e 2 ω c n n m=0 (-1) m ˜ p ˜ q e k;j 1 e k;j 2 • • • e k;jn f | X [n-m] j 1 •••j n-m ;p (ω)|i f | X [m] j n-m+1 •••jn;q (ω)|i * (3.64) it is possible to isolate a purely electric contribution f [n] QQ (ω) = i n 2m e ω e 2 ω c n n m=0 (-1) m ˜ p ˜ q e k;j 1 e k;j 2 • • • e k;jn (3.65) × f | 1 (n -m + 1)! Q[n-m+1] j 1 •••j n-m ;p |i f | 1 (m + 1)! Q[m+1] j n-m+1 •••jn;p |i * , a mixed electric-magnetic contribution f [n] QM (ω) = i n 2m e ω e 2 ω c n n m=0 (3.66) (-1) m ˜ p ˜ q e k;j 1 e k;j 2 • • • e k;jn f | 1 (n -m + 1)! Q[n-m+1] j 1 •••j n-m ;p |i f |- i ω 1 m! m[m] j n-m+1 •••j n-1 ;r ε rjnp |i * + (-1) m ˜ p ˜ q e k;j 1 e k;j 2 • • • e k;jn f |- i ω 1 (n -m)! m[n-m] j 1 •••j n-m-1 ;r ε rj n-m p |i f | 1 (m + 1)! Q[m] j n-m+1 •••jn;p |i * ,
and a purely magnetic contribution

f [n] M M (ω) = i n 2m e ω e 2 ω c n n m=0 (-1) m ˜ p ˜ q e k;j 1 e k;j 2 • • • e k;jn (3.67) f |- i ω 1 (n -m)! m[n-m] j 1 •••j n-m-1 ;r ε rj n-m p |i f |- i ω 1 m! m[m] j n-m+1 •••j n-1 ;s ε sjnp |i *
I stress that each of these contributions is dependent on the choice of gauge-origin, unlike the accumulated value of these three components. By changing from one origin to the other, we are effectively shifting the weight of the contribution, while keeping their sum constant. However, this should not be confused with the mixing of electric and magnetic fields upon Lorentz transformation (Eqn. (2.28)). [START_REF] Khamesian | Spectroscopy of linear and circular polarized light with the exact semiclassical light-matter interaction[END_REF] Here, the electric and magnetic fields rather mix because a truncated expansion can only describe the interaction locally. It can thus be argued that this decomposition does make little sense physically, since it inherently depends on the choice of gauge-origin. [START_REF] Kragh Sørensen | Implementation of a semiclassical light-matter interaction using the Gauss-Hermite quadrature: A simple alternative to the multipole expansion[END_REF] For centrosymmetric systems, such as atoms or metal complexes, it is possible to define a unique origin at the center. For general systems, however, such a symmetry may not exist, implying that this decomposition should be used with caution or avoided altogether.

Truncated Light-Matter Interactions: Generalized Length Representation

In the previous section, I have shown how in the generalized velocity representation, multipole moments appear in a rather unusual form. A formalism can be derived that is based on multipole moments in their more conventional form by imposing multipolar gauge. To derive an appropriate gauge condition, I will follow the procedure described by Bloch, [START_REF] Bloch | Zur Wirkung äußerer elektromagnetischer Felder auf kleine Systeme[END_REF] although there is a variety of alternative methods to reach this goal. [START_REF] Barron | The multipole interaction Hamiltonian for time dependent fields[END_REF][START_REF] Brittin | Poincaré gauge in electrodynamics[END_REF][START_REF] Donald | Gauge transformations and the electric dipole approximation[END_REF] The approach that is treated here starts from the Taylor expansion of the potentials centered around the expansion point a

φ(r, t) = ∞ n=0 1 n! (δ a • ∇ ) n φ(r , t) r =a ; δ a = r -a (3.68) Ã(r, t) = ∞ n=0 1 n! (δ a • ∇ ) n Ã(r , t) r =a
, where I applied a compactified notation that avoids the excessive use of indices. Note that the relation ∇ φ = -E -∂ ∂t à can be inserted in the expansion of the scalar potential

φ(r, t) = φ(a, t) - ∞ n=1 1 n! (δ a • ∇ ) n-1 δ a • E(r , t) r =a - ∂ ∂t ∞ n=1 1 n! (δ a • ∇ ) n-1 δ a • Ã(r , t) r =a
.

(3.69) Interestingly, this expression resembles a gauge transformation

φ(r, t) = φ a (r, t) -∂ t χ a (r, t), (3.70)
with the time-derivative term corresponding to the gauge function. Isolating φ a , thus enables us to express the scalar potential completely in terms of the electric field

φ a (r, t) = φ(a, t) - ∞ n=1 1 n! (δ a • ∇ ) n-1 δ a • E(r , t) r =a . (3.71)
Accordingly, the gauge function is defined as

χ a (r, t) = ∞ n=1 1 n! (δ a • ∇ ) n-1 δ a • Ã(r , t) r =a . (3.72)
To derive an expression for the vector potential, we use the partner relation

Ã(r, t) = A a (r, t) + ∇χ a (r, t). (3.73)
thus requiring us to evaluate the gradient of the gauge function

∇χ = ∞ n=0 1 (n + 1)! (δ a •∇ ) n Ã(r , t) r =a + ∞ n=1 n (n + 1)! (δ a •∇ ) n-1 ∇ δ a • Ã(r , t) r =a . (3.74)
The transformed vector potential reads

A a (r, t) = ∞ n=0 1 n! (δ a • ∇ ) n Ã(r , t) r =a (3.75) - ∞ n=0 1 (n + 1)! (δ a • ∇ ) n Ã(r , t) r =a - ∞ n=1 n (n + 1)! (δ a • ∇ ) n-1 ∇ δ a • Ã(r , t) r =a = ∞ n=1 n (n + 1)! (δ a • ∇ ) n-1 (δ a • ∇ ) Ã(r , t) -∇ δ a • Ã(r , t) r =a
.

As a finishing touch, I will apply the identity

δ a × B(r , t) = δ a × (∇ × Ã(r , t)) = ∇ δ a • Ã(r , t) -δ a • ∇ Ã(r , t), (3.76)
allowing us to express the vector potentials in terms of the magnetic field

A a (r, t) = - ∞ n=1 n (n + 1)! (δ a • ∇ ) n-1 δ a × B(r , t) r =a . (3.77)
Therefore, multipolar gauge provides a unique separation of electric and magnetic interactions. An intriguing feature of multipolar gauge is that it appears to eliminate gauge freedom, since we can apply any gauge transformation to ( φ, Ã) without altering the structure of (φ a , A a ). However, the additional degrees of freedom associated to gauge transformations now reside inside the choice of expansion point, suggesting that potentials corresponding to different expansion points are related by gauge transformations. To further validate this idea, I will introduce the potentials (φ b , A b ) which are expressed in multipolar gauge with the expansion point centred at b. Starting from the potentials, (φ a , A a ), the potentials ( φ, Ã) can be obtained through the relations Eqs. (3.70) and (3.73). Provided that we do not truncate our expansion, we are allowed to freely choose the expansion point in Eqs. (3.68), thus suggesting that we can make the following connection

φ(r, t) = φ b (r, t) - ∂ ∂t χ b (r, t) (3.78) Ã(r, t) = A b (r, t) + ∇χ b (r, t).
By combining both gauge transformations, we can relate

(φ a , A a ) to (φ b , A b ) φ a (r, t) = φ b (r, t) -∂ t χ a→b (3.79) A a (r, t) = A b (r, t) + ∇χ a→b ,
with the gauge function defined as In Figure 3.5, the procedure outlined above is represented schematically. However, I stress that the relations from Eqs. (3.78), which form an essential step in the transformation, are only valid if we do not truncate our expansion. Upon truncation, our gauge transformation breaks down, since we have

χ a→b = χ b -χ a = ∞ n=1 1 n! (δ b • ∇ ) n-1 δ b • Ã(r , t) r =b -(δ a • ∇ ) n-1 δ a • Ã(r , t) r =a . (3.80) ( φ, Ã) (φ a , A a ) (φ b , A b ) χ b -χ a
N n=0 1 n! (δ a • ∇ ) n φ(r , t) r =a = N n=0 1 n! (δ b • ∇ ) n φ(r , t) r =b (3.81) N n=0 1 n! (δ a • ∇ ) n Ã(r , t) r =a = N n=0 1 n! (δ b • ∇ ) n Ã(r , t) r =b
, for some finite order N . Therefore, different expansion points are only connected by a gauge transformation if the expansion is not truncated. The inequivalence of two expansion points effectively implies that we have a gauge-origin dependence, similar to Eqn. (3.31). This inherent origin dependence will haunt us throughout the rest of this chapter.

Using the potentials from Eqs. (3.71) and (3.77), the following interaction operator can be derived

Va (t) = e ∞ n=0 1 (n + 1)! (δ a •∇ ) n δ a •E(r , t) r =a -ec ∞ n=1 n (n + 1)! (δ a •∇ ) n-1 δ a ×B(r , t) •cα r =a , (3.82
) where I left out the constant term in the scalar potential, as it does not contribute to transition moments due to the orthogonality of states. However, for certain approximative schemes, such as STEX (see Section 5), this assumption might not hold. By inserting the definition of the electromagnetic fields (Eqn. (3.3)) and isolating the resulting frequency components, we obtain the following operator

Va (ω) = e ik•a e E 2i (δ a • ˜ ) ∞ n=0 1 (n + 1)! (iδ a • k) n -e ik•a ec E 2i δ a × ( 1 ω k × ˜ ) • cα ∞ n=1 n (n + 1)! (iδ a • k) n-1 .
(3.83) If we then write the dot-products in component form, we can isolate a multipole moment operator

Va (ω) = - E 2i Q[1] p ˜ p + ∞ n=1 i n ˜ p k j 1 k j 2 • • • k jn X[n] j 1 •••jn;p , (3.84)
which is expressed as

X[n] j 1 •••jn;p = 1 (n + 1)! Q[n+1] j 1 •••jn;p - i ω 1 n! m[n] j 1 •••j n-1 ;r ε rjnp , (3.85)
in terms of the electric multipole moment operator Before proceeding with the derivation of oscillator strengths, let us first appreciate the similarities between the generalized length-and velocity representation. Similar to the previous section, we arrive at some multipole moment operator that is decomposed into an electric and magnetic component. Indeed when inserting the frequency component derivatives of the fields (Eqn. (3.62)) into the interaction operator expressed in multipolar gauge (Eqn. (3.82)), the same result as in Eqn. (3.63) can be obtained. However, in multipolar gauge, these manipulations are not entirely necessary, because the separation of electric-and magnetic components could also have been inferred from the structure of the potentials (Eqs. (3.71) and (3.77)), thus reflecting how this separation is inherent to multipolar gauge.

Q[n+1] j 1 •••jn;p = -er j 1 • • • r jn r p ( 3
Due to the similarities between both representations, the n-th order oscillator strength in the generalized length representation is almost identical to Eqn. (3.64)

f [n] (ω) = i n 2m e ω e 2 ω c n n m=0 (-1) m ˜ p ˜ q e k;j 1 e k;j 2 • • • e k;jn f | X[n-m] j 1 •••j n-m ;p (ω)|i f | X[m] j 1 •••jm;q (ω)|i * ,
(3.87) with the only difference being the definition of the multipole moment operator. Furthermore, from considerations of time-reversal symmetry it follows that the absorption of linearly polarized light only contains even-order contributions, whereas circular dichroism only depends on the odd-order ones, in accordance with the results from Section 3.4.2. However, an important distinction between the two representation is that at face-value, the n-th order oscillator strength in the generalized length representation is not origin independent. The arguments proposed in Section 3.4.1, do not necessarily apply to Eqn. (3.87), due to the structure of the generalized length representation. At order n, the electric multipole moment is constructed from n + 1 position operators, whereas the magnetic multipole moment contains n position operators. This in contrast to the generalized velocity representation where both operators contain the same amount of position operators, resulting in the simple origin dependence provided by Eqn. (3.31). The more complicated origin dependence of the generalized length representation prohibits the exact cancellation between origin dependent terms, as described in Section 3.4.1. It has been shown by Lindh et al. that gauge-origin independence can be achieve by using Eqn. (3.24) (or Eqn. (3.26) in a non-relativistic framework) to convert the current density operator in the magnetic multipole moment into a position operator, which is referred to as the 'correct length gauge' by the authors. [START_REF] Kragh Sørensen | Gauge origin independence in finite basis sets and perturbation theory[END_REF][START_REF] Khamesian | Spectroscopy of linear and circular polarized light with the exact semiclassical light-matter interaction[END_REF] However, they also note that although the result is indeed origin independent, the integrals required for these calculations are rather cumbersome, which renders this approach impractical.

Connecting the Generalized Length-and Velocity Representation

At this point it seems that the generalized velocity representation is preferred over the generalized length representation, due to the gauge-origin independence of the former. However, in this section, I will show that the two representations are two sides of the same coin, although their equivalence only holds in an exact-state formalism. Indeed when applying approximate methods, such as SCF theory, both representations may give different results. Therefore, this section will be devoted to the connection between both representations. In other words, I will find the conditions that dictate when both representations produce the same oscillator strengths. In previous sections, the distinction between exact-and approximate-state theory is less relevant, as the results are equally valid in both cases. However, in the current section, this distinction becomes essential. I will thus divide this section into two parts: the first one being devoted to exact-state theory and the second to approximate-state theory.

Exact-State-Formalism: a Gauge Theoretical Approach

As a starting point, I recall that the potentials corresponding to the generalized length representation are expressed in multipolar gauge, whereas the potentials are expressed in Coulomb gauge for the generalized velocity representation. On a classical level, we know that both sets of potentials yield the same fields, since they are related by a gauge transformation. In a quantum mechanical framework, a similar effect can be achieved by the action of local unitary operators [START_REF] Jackson | Historical roots of gauge invariance[END_REF][START_REF] Fock | Über die invariante Form der Wellen-under Bewegungsgleichungen für einen geladenen Massenpunkt[END_REF][START_REF] Faddeev | Fock -Selected Works: Quantum Mechanics and Quantum Field Theory[END_REF] U (r, t) = e i eχ(r,t) ; U † (r, t)U (r, t) = Î (3.88

)
where χ is the same gauge function as the one from classical theory. This operator is local in the sense that for every point in space, a different unitary transformation is defined that is connected continuously to its local neighbourhood.

Inserting the identity operator into the time-dependent Dirac equation

ĥD -i ∂ t U † (r, t)U (r, t)ψ(r, t) = 0, (3.89)
and multiply on the left with Û (r, t) thus allows us to transform this equation according to our local unitary operator

ĥD -i ∂ t ψ(r, t) = 0 → ĥ D -i ∂ t ψ (r, t) = 0, (3.90)
where appears the transformed wave function

ψ (r, t) = U (r, t)ψ(r, t) (3.91)
and the transformed Hamiltonian

ĥ D = U ĥD U † -i U ∂ t (U † ) = U ĥD U † -e∂ t χ(r, t). (3.92)
Using the Baker-Campbell-Hausdorf expansion, introduced in Eqn. (1.57), the transformed Hamiltonian can be expressed as

ĥ D = ĥD -e∂ t χ(r, t) + e i χ(r, t), ĥD + e 2 2 2 χ(r, t), [χ(r, t), ĥD ] + • • • . (3.93)
Using the commutator identity [ p, f (r)] = -i ∇f (r), it can be shown that the first commutator term in this expansion evaluates to e i χ(r, t), ĥD = -ecα • ∇χ(r, t). (3.94) This relation also implies that any higher-order commutator vanishes, as this expression commutes with any multiplicative operator. Therefore, the transformed Hamiltonian reads

ĥ D = βm e c 2 + cα • p + e(A -∇χ(r, t)) -e(φ + ∂ t χ(r, t)), (3.95)
which is equivalent to substituting with gauge-transformed potentials. As a main advantage, this procedure allows us the track the changes of the wave function induced by the transformation of the potentials. This derivation could also have been carried out in a non-relativistic framework, which yields the same end result, although the commutators vanish at third order and higher. Because these transformations only affect the interaction operator, we can re-express Eqn. (3.93) as

V (t) = V (t) -e∂ t χ(r, t) + e i χ(r, t), ĥD , (3.96)
which can be decomposed in terms of frequency components

V (ω) = V (ω) + eiωχ(r, ω) + e i χ(r, ω), ĥD . (3.97)
What thus remains is to assess how transition moments change upon this gauge transformation. The two states involved in transition moments are stationary states of the zeroth order Hamiltonian, which does not depend on the light-matter interaction and is hence unaffected by this gauge transformation. From these considerations it can thus be concluded that gauge transformations leave transition moments unchanged, since we have

f |eiωχ(r, ω)|i + f |e i χ(r, ω), ĥD |i = f |eiωχ(r, ω)|i + e iE i f |χ(r, ω)|i (3.98) -e iE f f |χ(r, ω)|i = 0,
where I have assumed exact-state-and resonance (ω = ω f i ) conditions. When transforming the interaction operator from the Coulomb gauge to multipolar gauge, we have to use the gauge function provided by Eqn. (3.72). In this derivation, I will set a = 0, implying the following frequency component of the gauge function

χ(r, ω) = - E 2ω ∞ n=0 1 (n + 1)! (ir • k) n (r • ˜ ) = E 2eω ∞ n=0 i n (n + 1)! ˜ p k j 1 k j 2 • • • k jn Q[n+1] j 1 •••jn;p , (3.99)
where in the last equality, I have inserted the definition of the generalized length representation electric-multipole moments. Using this gauge function, Eqn. (3.97) assumes the form

Va (ω) = - E 2 ∞ n=0 i n-1 k j 1 k j 2 • • • k jn ˜ p X [n] j 1 •••jn;p (ω) + 1 (n + 1)! Q[n+1] j 1 •••jn;p + 1 ω 1 (n + 1)! Q[n+1] j 1 •••jn;p , ĥD , (3.100
) where the commutator evaluates to

1 ω ˜ p k j 1 • • • k jn Q[n+1] j 1 •••jnp , ĥD = -˜ p k j 1 • • • k jn Q[n+1] j 1 •••jnp . (3.101)
Upon insertion of this operator into the transformed operator, we obtain

Va (ω) = - E 2 ∞ n=0 i n-1 k j 1 k j 2 • • • k jn ˜ p X[n] j 1 •••jn;p (ω), (3.102)
which is equivalent to Eqn. (3.82) up to a phase factor.

It should be noted that this transformation only affects the electric multipole moment, whereas the magnetic multipole moment remains unchanged. As a shortcut to these lengthy manipulations, we can combine Eqs. (3.98) and Eqn. (3.101), which directly relates the electric-multipole moments in the two representations

f |˜ p k j 1 • • • k jn Q[n+1] j 1 •••jnp |i = 1 ω f i f | ĥD , ˜ p k j 1 • • • k jn Q[n+1] j 1 •••jnp |i = f |˜ p k j 1 • • • k jn Q[n+1] j 1 •••jnp |i . (3.
103) The gauge transformation and the commutator identity shown above can be used interchangeably to convert transition moments, since they are derived from the same principles. Even though the above results demonstrate that we can replace gauge transformations with commutator relations, such as Eqn. (3.103), the converse is not always the case. For example, the 'correct length gauge', as discussed by Lindh et al., transforms operators through commutator identities, although it is not directly related to a gauge transformation. [START_REF] Kragh Sørensen | Gauge origin independence in finite basis sets and perturbation theory[END_REF][START_REF] Khamesian | Spectroscopy of linear and circular polarized light with the exact semiclassical light-matter interaction[END_REF] Starting from the n-th order oscillator strength in the generalized length representation (Eqn. (3.87)), Eqn. (3.103) can be used to convert it into the generalized velocity representation using the above relation. From the fact that the latter is origin-independent, it thus follows that the former shares this feature. However, similar to Eqn. (3.85), these relations are only valid for exact states, whereas special considerations are needed for approximate state theories. In the following section, I will connect the generalized length-and velocity representations for SCF theory. Instead of using the gauge transformation outlined above, I will directly use Eqn. (3.103), thus significantly simplifying the derivation. For a conversion based on explicit gauge transformations, the reader should consult ref. [START_REF] Furche | On the density matrix based approach to time-dependent density functional response theory[END_REF].

Conversion at the SCF Level

In this section, I will demonstrate that the generalized length and velocity representation can be interconverted at the SCF linear-response level of theory. Besides the necessity of a complete basis set, the conversion generally depends on the level of theory, requiring special considerations for each electronic structure method. For truncated cluster expansions, for example, the two representations are not equivalent, as demonstrated by Pedersen and Koch. [START_REF] Bondo | Coupled cluster response functions revisited[END_REF] This conversion has been carried out successfully at the TD-HF, [START_REF] Linderberg | Propagators in quantum chemistry[END_REF] non-hybrid TDDFT [START_REF] Furche | On the density matrix based approach to time-dependent density functional response theory[END_REF] and MCSCF [START_REF] Olsen | Linear and nonlinear response functions for an exact state and for an MCSCF state[END_REF] level of theory. However, all these derivations assume the electric-dipole approximation, whereas here, I will extend it for electric-multipole moments to arbitrary order and for that matter, any multiplicative operator. To keep the derivation as general as possible, I will make the following substitutions:

Ω = ˜ p k j 1 • • • k jn Q[n+1] j 1 •••jnp and Γ = ˜ p k j 1 • • • k jn Q[n+1] j 1 •••jnp .
Furthermore, I will treat the TD-HF and TDDFT case separately, as it turns out that slightly different conditions apply to these two cases.

For approximate methods, the second equality in Eqn. (3.103) does not always hold. In the basis set approximation, operators are represented as matrices and commutators of such matrices are only equal to the exact commutator in the complete basis set limit (more precise conditions are given in Ref. [START_REF] Dyall | Matrix representation of operator products[END_REF]). Furthermore, approximate states are not eigenstates of the exact Hamiltonian, hence implying that the first equality in Eqn. (3.103) does not apply either. This falsely suggests that in the complete basis set limit, transition moments in the length and velocity representation should not be equal at the SCF level of theory. In practice, however, it has been observed that electric dipole transition moments in the two representations become nearly equal for large basis sets. [START_REF] Kragh Sørensen | Applications to metal K pre-edges of transition metal dimers illustrate the approximate origin independence for the intensities in the length representation[END_REF][START_REF] Kragh Sørensen | Gauge origin independence in finite basis sets and perturbation theory[END_REF][START_REF] Kragh Sørensen | Implementation of a semiclassical light-matter interaction using the Gauss-Hermite quadrature: A simple alternative to the multipole expansion[END_REF][START_REF] Holmgaard List | Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory[END_REF] Assuming that the complete basis-set limit applies, I will demonstrate under which conditions these representations are strictly equivalent at the SCF level of theory. I will start with TD-HF theory, since the discussion of this theory is less complicated than TDDFT.

Let us first multiply Eqn. (1.121) with the eigenvalue of the solution vector of the generalized eigenvalue problem (Eqn. (1.114)) and insert the square of the generalized metric [2] S [2] E Ω .

ω f i X † E Ω = ω f i X † S
(3.104)

We can now substitute the generalized eigenvalue equation in this expression

ω f i X † E Ω = Z † Y T A B B * A * g Ω -g * Ω . (3.105)
After working out the matrix multiplications and grouping together the various commutators we obtain

ω f i X † E Ω = ai Z * ai Φ a i | Ĥ, Ωov |Φ 0 + Y ai Φ 0 | Ĥ, Ωov |Φ a i , (3.106)
where appears the reduced operator

Ωov = Ω ai â † a âi + Ω * ai â † i âa (3.107)
in which occupied-occupied (oo) and virtual-virtual (vv) blocks of Ω pq are zero. In the following, it will be shown that in Eqn. (3.106) the reduced operator can be replaced with the full operator Ω. For these purposes, let us first consider the commutator of the Hamiltonian and the full operator and recognize that it assumes the structure

Ĥ, Ω = h Ω pq a † p a q + 1 4 L Ω pq,rs a † p a † r a s a q , (3.108)
where the one-index transformed integrals are defined as

h Ω pq = Ω pt h tq -h pt Ω tq L Ω
pq,rs = Ω pt L tq,rs -L pt,rs Ω tq + Ω rt L pq,ts -L pq,rt Ω ts To assess whether the full and reduced operator give the same result, I will proceed to compute the matrix elements from Eqn. (3.108) which is constructed from the full operator. These matrix elements can be conveniently expressed in terms of the Fock matrix and the anti-symmetrized two-electron integrals

Φ a i | Ĥ, Ω |Φ 0 = t Ω at F ti -F at Ω ti + Ω jt L ai,tj -L ai,jt Ω tj . (3.111)
Here one should note that in the complete basis set limit resolution of the identity, t |t t| = 1,

gives equivalence with the commutator Ĥ, Ω in first quantization. Next, due to the variational condition, i.e. F ia = 0, the terms depending on the oo and vv block of Ω vanish

Φ a i | Ĥ, Ω |Φ 0 = j Ω aj F ji - b F ab Ω bi + jb Ω jb L ai,bj -L ai,jb Ω bj . (3.112)
The remaining terms thus depend only on the occupied-virtual/virtual-occupied blocks of Ω. I have thereby demonstrated the equivalence of matrix elements Φ a i | Ĥ, Ω |Φ 0 and Φ a i | Ĥ, Ωov |Φ 0 . However, this is only valid if the summation in Eqn. (3.107) is complete over these blocks, which is not always the case when certain approximation schemes are employed. In Section C.4, I will describe the restricted excitation window approach, which is an important example that violates this condition.

In conclusion, I have made the following identification

ω f i X † E Ω = X † E [H,Ω] = X † E Γ , (3.113)
where

E [1] [H,Ω] and E [1]
Γ are the property gradients derived from the operators [ Ĥ, Ω] and Γ, respectively. Note that the second equality only holds in the complete basis set limit.

When extending this derivation to the TDDFT domain, there seems to be one major obstacle: DFT is not expressed in terms of the exact Hamiltonian (see Section 1.3). A first guess to circumvent this problem, would be to replace the Hamiltonian operator in Eqn. (3.103) with the Kohn-Sham Fock operator. In the canonical case, that is, F KS |φ p = ε p |φ p , we do find that

ϕ a | F KS , Ω |ϕ i = (ε a -ε i ) ϕ a | Ω|ϕ i . (3.114)
However, if we select -i Ω = r, the commutator F KS , Ω not only gives the velocity operator, p/m in the non-relativistic case, but for hybrid functionals also an exchange contribution, as observed already by Fock in the context of HF theory. [START_REF] Fock | Über die Anwendbarkeit des quantenmechanischen Summensatzes[END_REF] Notwithstanding, observing that the above difference of eigenvalues of the Kohn-Sham Fock operator appear in the diagonal blocks of the electronic Hessian in the canonical case together with the exchange contribution

A ai,bj = δ ij δ ab (ε a -ε i ) + (ai|jb) -γ(ab|ji) + W xc;ai,jb , (3.115)
hints at a fortuitous cancellation of terms, as realized by Starace [START_REF] Starace | Length and Velocity Formulas in Approximate Oscillator-Strength Calculations[END_REF] and Harris [START_REF] Harris | Oscillator Strengths and Rotational Strengths in Hartree-Fock Theory[END_REF] in the context of TDHF. The following section is largely based on their derivations. As a first step, I will rewrite the electronic Hessian (Eqn. (1.109)) as

A ai,bj = 0| -a † i a a , a † b a j , F KS |0 + W xc;ai,jb + (ai|jb) -γ(ab|ji) (3.116) B ai,bj = 0| a † i a a , a † j a b , F KS |0 + W xc;ai,bj + (ai|bj) -γ(aj|bi).
Following a similar procedure as in the previous paragraph, the eigenvalue equation of this Hessian (Eqn. (3.104)) can be inserted into the TDDFT transition moments, yielding the following expression

ω n0 X † E Ω = Z * ai Φ a i | F KS , Ωov |Φ 0 + Y ai Φ 0 | F KS , Ωov |Φ a i (3.117) +Z * ai W xc;ai,jb Ω bj + Z * ai (ai|jb)Ω bj -γZ * ai (ab|ji)Ω bj -Z * ai W xc;ai,bj Ω * bj -Z * ai (ai|bj)Ω * bj + γZ * ai (aj|bi)Ω * bj +Y ai W xc;jb,ia Ω bj + Y ai (jb|ia)Ω bj -γY ai (ib|ja)Ω bj -Y ai W xc;bj,ia Ω * bj -Y ai (bj|ia)Ω * bj + γY ai (ij|ba)Ω * bj .
Furthermore, it can be demonstrated straightforwardly that the reduced operator can be replaced with the full operator if the variational condition applies, i.e. F KS ai = 0

Φ a i | F KS , Ω |0 = F KS at Ω ti -Ω at F KS ti = F KS ab Ω bi -Ω aj F KS ji . (3.118)
However, it is less clear what we should do about all these additional terms involving the exchangecorrelation kernel and the two-electron integrals. As alluded to before, some of these terms may cancel with the additional contributions stemming from the commutator involving the Kohn-Sham Fock operator. To verify this, I will closer inspect this commutator in the limit of a complete basis set. By virtue of this assumption, the operators expressed in second quantization become equivalent to the corresponding first-quantized operator. Furthermore, the commutator of these two operators evaluates to a one-electron operator, reducing the many-body integral to an orbital integral according to the Slater-Condon rules

Φ a i | F KS , Ω |0 = d 3 rφ † a (r) F KS , Ω φ i (r). (3.119)
This commutator can be further simplified by recognizing that the local terms inside F KS commute with Ω, thus leaving us with the following two non-vanishing terms

d 3 rφ † a (r) F KS , Ω φ i (r) = d 3 rφ † a (r) Ts , Ω φ i (r) -γ d 3 rφ † a (r) K, Ω φ i (r), (3.120)
where appears the exchange operator, K, which is a non-local operator defined as

Kφ p (r) = j φ j (r) d 3 r φ * j (r )φ p (r ) |r -r | . (3.121)
The first of these commutators gives the same result as Eqn. (3.103), whereas the latter term contains the exchange contributions discussed earlier. Expanding the latter commutator yields the following result

-γ d 3 rφ † a (r) K, Ω φ i (r) = γ d 3 r d 3 r φ † a (r)φ j (r) Ω -Ω |r -r | φ † j (r )φ i (r ). (3.122)
Using an approach inspired by the work of Harris [START_REF] Harris | Oscillator Strengths and Rotational Strengths in Hartree-Fock Theory[END_REF], it can be shown that this expression cancels with exchange terms in Eqn. (3.117). For these purposes, consider the third violating term in this equation

(ab|ji)Ω bj = (ji|ab)Ω bj = d 3 r d 3 r d 3 r φ † j (r)φ i (r) 1 |r -r | φ † a (r )φ b (r )φ * b (r ) Ω φ j (r ). (3.
123) The resolution of the identity can be inserted in the summation over the virtual orbitals,

i φ i (r )φ † i (r ) + a φ a (r )φ † a (r ) = δ(r -r ), (3.124)
yielding the result

(ab|ji)Ω bj = d 3 r d 3 r φ † j (r )φ i (r ) Ω |r -r | φ † a (r)φ j (r) -(ji|ak)Ω kj . (3.125)
Likewise, the sixth violating term in Eqn. (3.117) can be rewritten as

(aj|bi)Ω * bj = d 3 r d 3 r φ † j (r )φ i (r ) Ω |r -r | φ † a (r)φ j (r) -(ki|aj)Ω jk . (3.126)
Subtracting these two terms and relabelling some summation indices, gives

(aj|bi)Ω * bj -(ab|ji)Ω bj = d 3 r d 3 r φ † j (r )φ i (r ) Ω - Ω |r -r | φ † a (r)φ j (r), (3.127)
which exactly cancels the additional exchange terms appearing from the commutator with the Kohn-Sham Fock operator. Therefore, all violating terms in Eqn. (3.117) containing γ cancel. The remaining terms cancel amongst each other, provided that the orbitals are real. Therefore, at the TDDFT level of theory, the generalized length-and velocity representation are equivalent if simultaneously the basis set is complete and the orbitals are real, which is not generally the case in relativistic calculations due to the presence of spin-orbit coupling. The findings of this section suggest that with a sufficiently large basis set, the generalized length representation should be origin-independent, because in that case the transition moments are equivalent to the generalized velocity representation. However, in its current formulation, origin dependence of the former has never been observed. Typically, for gauge-origins close to the coordinate origin, the two representations become equivalent with increasing basis set size, whereas equivalence is lost when the gauge-origin is displaced from the coordinate origin. At the displaced origin, the transition moments are equivalent if the following relation holds

1 ω f i f | ĥD , ˜ p k j 1 • • • k jn Q[n+1] j 1 •••jnp (O + a) |i = f |˜ p k j 1 • • • k jn Q[n+1] j 1 •••jnp (O + a)|i , (3.128)
where the length representation electric multipole moment transforms as

˜ p k j 1 • • • k jn Q[n+1] j 1 •••jnp (O) → ˜ p k j 1 • • • k jn Q[n+1] j 1 •••jnp (O+a) = n m=0 1 m! (k•a) m ˜ p k j 1 • • • k j n-m Q[n-m+1] j 1 •••j n-m p (O), (3.129
) thus implying that lower-order commutation relations should also hold to connect both representations. It is very demanding for a given basis set to obey all these commutators simultaneously, inducing small differences between the transition moments in both representations, which prohibit cancellation between the origin-dependent terms in the oscillator strength. It appears that origin independence in the length representation is only possible if alternative schemes are employed, based on either London orbitals [START_REF] Keld L Bak | Ab initio calculation of electronic circular dichroism fortrans-cyclooctene using London atomic orbitals[END_REF] or a singular value decomposition of the rotational strength tensor [START_REF] Caricato | Origin invariant optical rotation in the length dipole gauge without London atomic orbitals[END_REF][START_REF] Niemeyer | Origin invariant electronic circular dichroism in the length dipole gauge without London atomic orbitals[END_REF] (see Eqn. (3.152) for a definition of this tensor).

Rotational Averaging

Until now it has been tacitly assumed that the relative orientation between our molecule and the electromagnetic field is fixed in space. Such a model can be related experimentally to an ensemble of non-interacting molecules that are not moving and all pointing in the same direction. However, this situation is rather artificial, as molecules are allowed to rotate and tumble freely in more realistic samples such as gases and liquids (see Fig. 3.7). Therefore, due to this motion, the absorption measured from experiment is an average of all possible orientations. From a mathematical point of view, this is completely equivalent to fixing a molecule in space and averaging over every possible polarization and propagation direction of light. Within this basis, the polarization vectors can be conveniently parametrized using a third angle, χ

1 = cos χe θ + sin χe φ ; 2 = cos χe φ -sin χe θ . (3.132)
Using the definition of these unit vectors, the oscillator strength can be expressed as a function of these three angles. In general, the rotational average of any function depending on these three angles can be calculated according to the formula

f (θ, φ, χ) θ,φ,χ = 1 8π 2 π 0 dθ 2π 0 dφ 2π 0 dχ sin θf (θ, φ, χ). (3.133)
In the following I will discuss separately how these concepts apply to the full-and truncated oscillator strength. Furthermore, I will separately treat the absorption of linearly polarized light and circular dichroism.

Full Oscillator Strength

I will start off with deriving an expression for the isotropically averaged absorption of linearly polarized light using the full interaction operator. Using the parametrization from the previous section, the isotropic oscillator strength reads

f f ull iso (ω) = 2m e ω e 2 p q χ f | e ω cα p e ik(θ,φ)•r |i f | e ω cα q e ik(θ,φ)•r |i * θ,φ , (3.134)
where the averaging over χ can be factorized outside of the transition moment and hence evaluated analytically

p q χ = 1 2
(δ pqe r;p e r;q ). (3.135)

The averaging over the two other angles is more cumbersome, because the exponential form of the full interaction operator prohibits the angular dependence to be factorized outside of the transition moment. In practice, the averaging over θ and φ is carried out using Lebedev quadrature, which is a scheme to handle 2D angular integrals numerically. [START_REF] Lebedev | Values of the Nodes and Weights of Ninth to Seventeenth Order Gauss-Markov Quadrature Formulae Invariant under the Octahedron Group with Inversion[END_REF][START_REF] Lebedev | Quadratures on a Sphere[END_REF][START_REF] Lebedev | Spherical Quadrature Formulas Exact to Orders[END_REF][START_REF] Lebedev | quadrature formulas of orders 41, 47, and 53 for the sphere[END_REF][START_REF] Lebedev | a quadrature formula for the sphere of 59th algebraic order of accuracy[END_REF][START_REF] Lebedev | A Quadrature Formula for the Sphere of the 131st Algebraic Order of Accuracy[END_REF] For this type of quadrature an angular grid is defined that preserves octahedral symmetry. Accordingly, the integral is approximated by a weighted sum of the function evaluated on the grid points

f θ,φ ≈ ij w ij f (θ i , φ j ), (3.136)
where the weights are found by enforcing that this scheme exactly integrates all spherical harmonics up to a given order. These weights are then tabulated, implying that we only need to know the function at a certain amount of grid points to approximate the integral.

The rotational averaging of circular dichroism follows a very similar approach, although there is one major difference: Eqn. The independence of this expression on the angle χ reflects the helical symmetry of circularly polarized light (Fig. 3.2). As was the case for the absorption of linearly polarized light, the rotation over θ and φ is performed using Lebedev quadrature.

Truncated Oscillator Strength

The true strength of the truncated approach becomes apparent when evaluating the rotational average, which can be carried out analytically. This procedure assumes the same form for the generalized length-and velocity representation. Therefore, I will limit myself to the latter case, although it should be kept in mind that these techniques apply to both representations. For the absorption of linearly polarized light, the isotropic average assumes the form

f [2n] iso (ω) = 2m e ω c 2 ω c 2n n m=0 (-1) m (2 -δ m0 ) p q χ e k;j 1 e k;j 2 • • • e k;j 2n θ,φ (3.138)
× Re f | X [n+m] j 1 •••j n-m ;p (ω)|i f | X [n-m] j n-m+1 •••j 2n ;q (ω)|i * ,
where the angular terms completely factorize outside of the transition moment. Therefore, the rotational averaging hinges on the calculation of the tensor p q χ e k;j 1 e k;j 2

; u + v + w = 2n, (3.140)
where the indices u, v and w accumulate the powers of e k;i . In passing, it should be noted that a similar tensor is required to rotationally average the truncated circular dichroism

∆f [2n+1] = 2m e ω c 2 ω c 2n+1 n m=0 (-1) m e k;j 1 e k;j 2 • • • e k;j 2n+1 e k;i θ,φ (3.141)
× ε ipq 2Re f | X [n+m+1] j 1 •••j n+m ;p (ω)|i f | X [n-m] j n+m+1 •••j 2n+1 ;q (ω)|i * .
The integral in Eqn. (3.140) can be made more manageable by first rewriting it as

E uvw = 1 8π 2π 0 dφ cos u φ sin v φ π 0 dθ sin θ sin u+v θ cos w θ (3.142)
and then applying the relations π 0 dθ cos p θ sin q θ = 1 + (-1) p π/2 0 dθ cos p θ sin q θ (3.143) 2π 0 dφ cos p φ sin q φ = 1 + (-1) p 1 + (-1) q π/2 0 dφ cos p φ sin q φ, such that the integral only needs to be evaluated in the (+, +, +) octant of Euclidian space

E uvw = 1 8π 1+(-1) u 1+(-1) v 1+(-1) w π/2 0 dφ cos u φ sin v φ π/2 0 dθ cos u+v θ sin w θ. (3.144)
Therefore, this integral vanishes if any of the indices is odd. Further manipulations show that this integration factor can be expressed in terms of the trivariate beta function

E uvw = 1 32π 1 + (-1) u 1 + (-1) v 1 + (-1) w B( u + 1 2 , v + 1 2 , w + 1 2 ), (3.145)
which in turn can be expressed in terms of the gamma functions

B(a, b, c) = Γ(a)Γ(b)Γ(c) Γ(a + b + c) ; Γ(a) = 2 ∞ 0 dxe -x 2 x 2a-1 . (3.146)
The gamma function is readily available in math libraries, thus allowing straightforward implementation. As a final remark, it should be stressed that the method outlined above is much more practical than the traditional procedure based on linear combinations of fundamental Cartesian isotropic tensors. [START_REF] Kearsley | Linearly independent sets of isotropic Cartesian tensors of ranks up to eight[END_REF][START_REF] Andrews | Three-dimensional rotational averages in radiation-molecule interactions: an irreducible cartesian tensor formulation[END_REF][START_REF] Andrews | On three-dimensional rotational averages[END_REF][START_REF] Andrews | Eighth rank isotropic tensors and rotational averages[END_REF][START_REF] Daniel | Rotational averaging of multiphoton absorption cross sections[END_REF][START_REF] Ee | Combinatorics in tensor-integral reduction[END_REF] It is illuminating to consider the rotationally averaged oscillator strengths to lowest orders. At n = 0, the absorption of linearly polarized light reads

f [0] iso (ω) = 2m e ω c 2 p q χ θ,φ Re f | Q[0] p |i f | Q[0] q |i * , (3.147)
where the rotational factor assumes a particularly simple form

p q χ θ,φ = 1 3 δ pq , (3.148)
thus yielding the result

f [0] iso (ω) = 2m e ω 3 c 2 Q [0] • Q [0] . (3.149)
Likewise, to first order, the circular dichroism reads

∆f [1] iso = 4m e ω 3 c 2 ω c ε ipq f | X [1] i;p (ω)|i f | Q[0] q |i * = 8m e ω 3 c 2 ω c -i ω m [1] • Q [1] = R [1] iso , (3.150)
which is more commonly referred to as the rotational strength. In a non-relativistic framework, it is customary to assess the effectiveness of a certain method by applying the so-called sum rules. For example, it can be argued that the sum of the dipole oscillator strengths over all possible excitations should equal the amount of electrons in the system, which is referred to as the Thomas-Reiche-Kuhn sum-rule. [START_REF] Thomas | Über die zahl der dispersionselektronen, die einem stationären zustande zugeordnet sind. (vorläufige mitteilung)[END_REF][START_REF] Reiche | Über die Zahl der Dispersionselektronen, die einem stationären Zustand zugeordnet sind[END_REF][START_REF] Kuhn | Über die gesamtstärke der von einem zustande ausgehenden absorptionslinien[END_REF] Likewise, the sum of all rotational strengths should vanish. [START_REF] Condon | One-Electron Rotatory Power[END_REF][START_REF] Edward U Condon | Theories of optical rotatory power[END_REF] However, the derivation of these rules hinges on two key steps: the insertion of the resolution of the identity and the canonical commutation relation. The first step is problematic in a relativistic framework, as the resolution of the identity also includes negative-energy states. The second step can be made because one of the two dipole moments in Eqn. (3.147) is converted into the velocity representation, which is the momentum operator in the non-relativistic limit. However, in the relativistic case, the velocity representation is given by the Dirac matrices, which do not have the same commutation properties as the momentum operator. For these reasons, it is until this day unknown whether these sum rules even apply in a relativistic framework. Hence, they will not be further pursued in this thesis.

Angular Dependence of Truncated the Differential Oscillator Strength

In the previous section, it was shown how the anisotropic oscillator strength can be expressed as a function of the propagation-and polarization direction, which allowed us to evaluate the isotropic average. However, explicit knowledge about the angular dependence can also be used to identify directions in which the signal is unusually large, or small. Depending on the symmetry of the angular distribution, the different multipolar contribution may be isolated and identified. This information is especially relevant for ECD, where the presence of isotropic condition determine whether the signal vanishes for achiral systems, as demonstrated in Section C.1. Furthermore, due to its independence of χ, the ECD signal for all possible directions can be mapped out on the surface of a sphere. In particular, the angular dependence of the first-order truncated differential oscillator strength can be expressed in a straightforward manner ∆f [1] (θ, φ) = e k;i R

[1] ij e k;j (3.151) = R [1] iso + 1 2 (R [1] zz -R [1] iso )(3 cos 2 θ -1) + 1 2
(R [1] xz + R [1] zx ) sin 2θ cos φ + 1 2 (R [1] yz + R [1] zy ) sin 2θ sin φ + 1 2 (R [1] xx -R [1] yy ) sin 2 θ cos 2φ + 1 2 (R [1] xy + R [1] yx ) sin 2 θ sin 2φ

where I have introduced the rotational strength tensor R

[1] ij (cf. Ref. 189) R [1] ij = 4m e ω 2 ce 2 X [1] j × X [0] i ; R [1] iso = 1 3
(R [1] xx + R [1] yy + R [1] zz ).

(3.152)

From Eqn. (3.151) it can be seen that the anisotropic part of ∆f [1] (θ, φ) is a linear combination of d-orbitals weighted by the relevant elements of the rotatory strength tensor. This can be made more explicit by rewriting the first-order differential oscillator strength as ∆f [1] (θ, φ) = √ 15R

[1]

iso s + √ 3 R [1] zz -R [1] iso d z 2 + R [1] xx -R [1] yy d x 2 -y 2
+ R [1] xz + R [1] zx d xz + R [1] yz + R [1] zy d yz + R [1] xy + R [1] yx d xy ,

where I have used a common normalization for all solid harmonics. From Eqn. (3.141) it is observed that rotational strength tensors can be generalized to arbitrary odd orders. It will also be seen that the differential oscillator strength ∆f [2n+1] contains products of 2(n + 1) components of the unit wave vector and is therefore spanned by spherical harmonics of even = 0, 2, . . . , 2(n + 1). The explicit equations of these distributions in terms of these spherical harmonics is rather complicated and thus will not be shown explicitly.

Having gathered all the theoretical tools to describe X-ray spectroscopies beyond the dipole approximation, we have thus reached the end of this chapter. In the following, I will proceed to demonstrate how these techniques can be applied in practice and what are the possible pitfalls associated with them.

Chapter 4 Applications

Implementation

In contemporary quantum chemistry, the formulation of theory is only half the work, since an efficient implementation is crucial to make quantum chemical calculations feasible, which often involve many tedious mathematical operations. For that reason, this section will be devoted to the implementation of both the full-and truncated light-matter interaction operator. Both of these developments have been carried out in Dirac, which is a quantum chemical code adapted to four-component relativistic calculations. [START_REF] Dirac ; Jensen | a relativistic ab initio electronic structure program, Release DIRAC22[END_REF][START_REF] Saue | The DIRAC code for relativistic molecular calculations[END_REF] The implementation of these interactions is still a work in progress, so below follows a statement of all the available functionalities. It is possible to carry out intensity calculations using the full-and truncated interaction operators, both under isotropic and anisotropic conditions. However, in the current implementation, ECD calculations can only be carried out using the full interaction operator, whereas the truncated ECD is still under development. The truncated ECD reported in Section 4.5, was calculated using an external script with the transition moments from Dirac as input. In the following, I will give a brief overview of the implementation of the full-and truncated interaction, highlighting their unique features. In the upcoming section and the ones that follow, this part of the code will be referred to as the BED implementation, where the abbreviation stands for beyond the electric-dipole approximation. Although this abbreviation does not necessarily apply to ECD, which at first order is already beyond this approximation, it will be used regardless for brevity's sake. 4.1 depicts a flow chart of the BED implementation responsible for the calculation of the full oscillator strength. In the isotropic case, the code starts with a loop over the points on the Lebedev grid, thus sampling the various propagation directions of the incoming light. The same code applies to the anisotropic case, although in that case the loop contains only one grid point in a user-defined direction. The property integrals need to be defined on each grid point, due to the angular dependence of the full interaction operator Eqn. (3.134). These integrals are provided by the HERMIT integral package, [START_REF] Helgaker | Hermit, a molecular integral code[END_REF] which is based on a scalar Cartesian Gaussian basis

Full Interaction

G α ijk = N α ijk x i y j z k e -αr 2 ; l = i + j + k (4.1)
where N α ijk is a normalization constant. Using this basis, the integrals of the full interaction assume the form

d 3 rG α 1 i 1 j 1 k 1 e ik•r G α 2 i 2 j 2 k 2 = N α 1 i 1 j 1 k 1 N α 2 i 2 j 2 k 2 (4.2) × ∞ -∞ dxx i 1 +i 2 e -(α 1 +α 2 )x 2 e ikxx ∞ -∞ dy j 1 +j 2 e -(α 1 +α 2 )y 2 e ikyy ∞ -∞ dzz k 1 +k 2 e -(α 1 +α 2 )z 2 e ikzz ,
which can be identified with the Fourier transform of a Gaussian function. This identification has been realized independently in several works, e.g. ref. 31 and 192, the latter of which in the context of dynamic structure factors. This expression can be simplified by relabelling the exponents and discarding the normalization constants:

d 3 re ik•r x i y j z k e -ar 2 = ∞ -∞ dxe ikxx x i e -ax 2 ∞ -∞
dye ikyy y j e -ay 2 ∞ -∞ dze ikzz z k e -az 2 (4.3)

a = α 1 + α 2 ; i = i 1 + i 2 ; j = j 1 + j 2 ; k = k 1 + k 2
Because this integral is equivalent for each Cartesian component, it suffices to only treat the xcomponent. For these purposes, it proves to be useful to start with the special case i = 0, which readily evaluates to

∞ -∞ dxe -ax 2 e ikxx = e -k 2 x 4a π a . (4.4)
Starting from above equation, the general case, i.e. i = 0, can be obtained by differentiation under the integral sign

∞ -∞ dxe ikxx x n e -ax 2 = (-i) n d n dk n x [e -k 2 x 4a π a ] = (-i) n 2 √ π (4a) n+1 2 d n dQ n [e -Q 2 ] = (-i) n 2 √ π (4a) n+1 2 e -Q 2 H n (Q), (4.5
) where I have introduced the physicist's definition of the Hermite polynomials

H n (Q) = (-1) n e -Q 2 d n dQ n [e Q 2 ]. (4.6)
in terms of the dimensionless variable

Q = k x 2 √ a . (4.7)
Therefore, the full property integral evaluates to

d 3 re ik•r x i y j z k e -ar 2 = (-i) i+j+k 8π 3 2 (4a) i+j+k+3 2 e -ω 2 4c 2 a H i k x 2 √ a H j k y 2 √ a H k k z 2 √ a . (4.8)
Expressed in this form, these integrals can be readily evaluated using existing analytic schemes such as McMurchie-Davidson. [START_REF] Holmgaard List | Beyond the electric-dipole approximation: A formulation and implementation of molecular response theory for the description of absorption of electromagnetic field radiation[END_REF] Lindh et al. demonstrated an alternative scheme bases on Gauss-Hermite quadrature. [START_REF] Kragh Sørensen | Implementation of a semiclassical light-matter interaction using the Gauss-Hermite quadrature: A simple alternative to the multipole expansion[END_REF] One main obstacle in the implementation of these integrals is their frequency dependency. Due to the exponential shape of the interaction operator, this dependency cannot be factorized outside of the integral, thus requiring the output of a response calculation to define these integrals (see Fig. 4.1). Lindh et al. avoided the necessity to calculate all the frequencies by employing an interpolation scheme. [START_REF] Mickaël G Delcey | Exact semi-classical light-matter interaction operator applied to two-photon processes with strong relativistic effects[END_REF] In the next step of the algorithm, the property gradient

g Tp;ai = φ L a |cσ p e ik•r |φ S i + φ S a |cσ p e ik•r |φ L i , (4.9)
is obtained by transforming the property integrals to MO basis

g Tp;ai = c L * µa χ L µ |cσ p e ik•r |χ S ν c S νi + c S * µa χ S µ |cσ p e ik•r |χ L ν c L νi , (4.10)
where the implicit summation convention has been applied and the small component basis is generated according to kinetic balance (see Section 2.4.2). From the property gradient, transition moments can be computed according to Eqn. (1.121). Depending on the user input, these transition moments are then inserted in Eqn. (3.20), Eqn. (3.22) or both, yielding the anisotropic (differential) oscillator strength corresponding to the specific grid point. This result is accumulated with the grid weight, such that we end up with the isotropic (differential) oscillator strength once the loop terminates.

Truncated Interaction

In Figure 4.2, the flowchart of the truncated BED implementation is depicted. Similar to previous case, the starting point of this chart is the generation of the property integrals. Regardless of its type, the multipole moments generally require the following integrals

d 3 rG α 1 i 1 j 1 k 1 x u y v z w G α 2 i 2 j 2 k 2 = N α 1 i 1 j 1 k 1 N α 2 i 2 j 2 k 2 (4.11) × dxx i 1 +i 2 +u e -(α 1 +α 2 )x 2 dyy j 1 +j 2 +v e -(α 1 +α 2 )y 2 dzz k 1 +k 2 +w e -(α 1 +α 2 )z 2 ,
which essentially boils down to solving integrals of the form

∞ -∞ dxx i e -ar 2 = 1 + (-1) i ∞ 0 dxx i e -ar 2 , (4.12)
where the right-hand-side can be related to the gamma function, introduced in previous section (Eqn. (3.146)). Therefore, the multipole integrals can be expressed as

d 3 rx i y j z k e -ar 2 = 1 8 (1 + (-1) i )(1 + (-1) j )(1 + (-1) k )a -3+i+j+k 2 Γ 1 + i 2 Γ 1 + j 2 Γ 1 + k 2 , (4.13
) where I have left out the normalization constants and relabelled the powers and the exponents. In practice, however, this relation is seldom used, as these integrals can be obtained recursively using, for example, the McMurchie-Davidson scheme. Using a similar approach as in Eqn. (4.10), the property gradient can be obtained from the associated integrals, although the exact transformation depends on whether the operator in question is diagonal or off-diagonal with respect to the largeand small component. Contrary to the full interaction, these property integrals are independent of the frequency, although the response module is still needed to compute transition moments and provide the frequency-dependent prefactor of the oscillator strength.

These transition moments are all stored in the same one-dimensional array. A hierarchical ordering is applied to this array, which principally sorts the multipole moments based on their type, then their order and then the combinations of Cartesian powers. In the following, I will consider the simplest example of the generalized length representation electric multipole moments, although this storage scheme can straightforwardly be generalized to other types of multipole moments. For these purposes, I will index this multipole moment in an alternative manner

Q [L] ab = f |-ex L-a y a-b z b |i , (4.14)
which can be used to generate only the unique multipole moments of a given order. Furthermore, a and b can be understood as matrix indices that fill the lower triangle (see Fig. 4.3). Instead of explicitly storing a two-dimensional matrix, the multipole moments can be stored in a onedimensional array, where the element (a, b) in the matrix corresponds to element i = b + (a -1)(a -2)/2 in the array.

x 2 xy xz

y 2 yz z 2               x 2 xy xz y 2 yz z 2 Figure 4
.3: Storage of the 3-th order electric-length multipole moments. This scheme can be readily generalized for other types of multipole moments. For a given order, the unique multipole moments can be stored in a lower-triangular matrix (left), which can be represented as a one-dimensional array (right).

As can be inferred from Figure 4.2, the code proceeds differently if either isotropic or anisotropic conditions are imposed. Under anisotropic conditions, the transition moments are contracted with the wave-and polarization vectors, thus yielding Eqn. (3.64) or (3.87). The contracted transition moments are then used to construct the anisotropic oscillator strength. It should be noted that the subroutines written for these purposes hold for arbitrary order, thus implying that the anisotropic absorption cross-section can be calculated to arbitrary order.

To calculate the isotropic oscillator strength, however, a slightly more complicated scheme is required, because it involves the contraction between transition moments and the tensor, p q χ e k;j 1 e k;j 2 • • • e k;j 2n θ,φ (Eqn. (3.138)). A brute-force approach would involve a distinct subroutine for each order of σ [n] , whereas current approach uses a single subroutine for all orders. This feature is achieved by applying recursive subroutines, which can be defined in a self-referential manner. In Figure 4.4, a code snippet is given containing an example of a recursive loop.

At first sight, this subroutine does not seem to be all that special, since its main function is to fill up an array for a specific index value. However, upon closer inspection, it can be observed that this routine calls itself for negative outcomes of the if-statement. When the routine calls itself, the index of the array is shifted by one unit and the process is repeated until the final index is reached. What effectively happens, is that an array of a certain dimension is filled with all possible combinations of one, two and three. These values represent all possible index values of p q χ e k;j 1 e k;j 2 • • • e k;j 2n θ,φ . When the final index is reached, a specific element of this tensor is calculated, contracted with the transition moments and accumulated. At the end of the line, we are left with the isotropic truncated oscillator strength. As alluded to before, the generality of this subroutine allows the truncated isotropic oscillator strength to be accumulated to arbitrary order.

Convergence of the Multipole Expansion

Having a fully functional implementation at our disposal that can compute oscillator strengths beyond the electric-dipole approximation, we can proceed with our first calculations. For any newly developed functionality it is always good practice to proceed cautiously and start with calculations on simple test systems, which facilitates the identification of bugs and unanticipated results. Therefore, this section will mainly focus on the UV-Vis and X-ray absorption spectra of the radium atom, whose spherical symmetry will significantly simplify the problem at hand. The array FOSC contains the nth order oscillator strength for a given excitation and symmetry, ATMPFF contains the transition moments for a given excitation, operator and symmetry, while OMEGA denotes the frequencies. When the if statement is not met, a loop over 1,2 and 3 is initiated that fills the array INDICES at location m. In this loop the subroutine calls itself, while the index m is shifted by one, thus repeating the process for the next entry of INDICES. As a result, this subroutine will fill INDICES with all possible combinations of 1,2 and 3, until it has reached the final index of INDICES. At this point, the subroutine will call ROTFAC_BED, which creates the rotational tensor for that specific index combination and contracts it with the relevant transition moments. The recursive loop repeats this for all possible combinations in INDICES, thus accumulating the isotropic truncated oscillator strength iteratively.

Furthermore, this symmetry induces selection rules for multipole transitions. Presumably, the main characteristics of the spectra will be described by the electric-dipole oscillator strength (Eqn. (3.149)), whereas finer details are given by contributions depending on the magnetic-dipole and electric-quadrupole. In Section C.2, the relevant selection rules will be derived in detail. Here I will rather briefly summarize them:

• electric-dipole: ∆j = 0, ±1; ∆m = 0, ±1; (j = 0 j = 0)

• magnetic-dipole: ∆j = 0, ±1; ∆m = 0, ±1; (j = 0 j = 0)

• electric-quadrupole: ∆j = 0, ±1, ±2; ∆m = 0, ±1, ±2; (j = 0 j = 0)

• full interaction: (j = 0 j = 0).

In the following, I will put these rules to practice and calculate absorption intensities of the radium atom in both the core-and valence regions. The results from this section are taken from ref. 1.

Computational Details

Unless otherwise stated the data reported in this section have been obtained with a development version of the Dirac electronic structure code [START_REF] Dirac ; Jensen | a relativistic ab initio electronic structure program, Release DIRAC22[END_REF] (Figure 4.5: revision 52c65be; Table 4.1; Figure 4.6: revision 5a7d81c).

Using the full interaction, the valence-and core spectrum of the radium atom were calculated at the TDDFT [START_REF] Bast | Relativistic adiabatic timedependent density functional theory using hybrid functionals and noncollinear spin magnetization[END_REF] level of theory using the PBE0 [START_REF] John P Perdew | Generalized gradient approximation made simple[END_REF][START_REF] Adamo | Toward reliable density functional methods without adjustable parameters: The PBE0 model[END_REF] exchange-correlation functional, based on the Dirac-Coulomb Hamiltonian and within the restricted excitation window (REW) approach (see Section C.4). In these calculations the (SS|SS) integrals are replaced by an interatomic SS energy correction. [START_REF] Visscher | Approximate molecular relativistic Dirac-Coulomb calculations using a simple Coulombic correction[END_REF] For technical reasons that become clearer later, the calculations involving truncated interactions were restricted to the ns 1/2 → 7p 1/2 ; n = 1, 2, • • • , 7 excitations of the radium atom and carried out at the TD-HF level of theory. Furthermore, in these calculations integral screening was turned off and the (SS|SS) integrals included. For both series of calculations, the dyall.ae3z basis set is used. [START_REF] Kenneth | Relativistic double-zeta, triple-zeta, and quadruple-zeta basis sets for the 4s, 5s, 6s, and 7s elements[END_REF][START_REF] Kenneth | Core correlating basis functions for elements 31-118[END_REF] The labels indicate the character of the receiving orbital. Note the differences in scales on the axes in the valence and X-ray region. Oscillator strengths are summed over contributions from transitions within each degenerate (same ∆J components) and near-degenerate (different ∆J components) set, and the sticks have been convoluted with a Lorentzian lifetime broadening of 1000 cm -1 . The experimental 1s ionization energy is 103922 ± 7.2 eV. [START_REF] Bearden | Reevaluation of X-ray atomic energy levels[END_REF] Figure 4.5 shows the valence and K -edge spectra of Ra within and beyond the ED approximation, the latter computed with the full light-matter interaction operator. The energy range of the Kedge spectrum well exceeds the 1s 1/2 ionization energy (see caption Fig. 4.5), which presumably can be attributed to the lack of core-hole relaxation effects in these calculations. Expectedly, all ED forbidden transitions, except for excitations associated with ∆J = 0 change in total angular momentum quantum number, gain intensity upon going beyond the ED approximation. In the valence region, however, they remain several orders of magnitude smaller than the ED counterparts, such that ED and BED spectra are essentially identical. In the X-ray region, the main contributions from the 1s 1/2 → 6d manifold corresponds to ∆J = 2 transitions, while the ED allowed ∆J = 1 transitions dominate for the 1s 1/2 → 7/8p manifold. Note that the small energy differences between different ∆J components in a given set makes them indiscernible in the spectrum, and their oscillator strengths were thus combined in Figure 4.5. Upon inclusion of non-dipolar effects, intensity is primarily redistributed from the 1s 1/2 → 7/8p 3/2 sets (a ∼20% reduction compared to ∼13% for the 1s 1/2 → 7/8p 1/2 excitations) to the 6d transitions.

Full Light-Matter Interaction

However, even if the spectral broadness is sufficiently small to fully distinguish the peaks, the corrections to the dipole-allowed transitions are rather modest. Non-dipolar effects are expected to gain importance if either the wave length of light is small, or the spatial extent of the system is large. For the radium atom, these effects are suppressed due to the compactness of the 1s 1/2 orbital. In the Section 4.4, I will discuss an example of a system with symmetry equivalent centers, where the core transition is delocalized over the centers, thus enhancing non-dipolar effects.

Truncated Light-Matter Interaction

When carrying out equivalent calculations using the truncated light-matter interaction formulations, both in the velocity and the length representation, nonsensical results were obtained for core excitations. Rather than reporting these numbers, I will illustrate and analyze this behavior using a simpler computational setup. Table 4.1 reports anisotropic oscillator strengths for radium ns 1/2 → 7p 1/2 (n = 1, .., 7) excitations at various orders in the generalized velocity representation as well as obtained using the full light-matter interaction. The orbital rotation operator, Eqn. (1.85), is restricted to the ns 1/2 and the 7p 1/2 orbitals of the selected excitation, and I only report results for the B 1u irreducible representation of the D 2h point group. To avoid issues of numerical integration I have performed TD-HF rather than TDDFT calculations. Furthermore, to avoid possible numerical noise due to rotational averaging, I have chosen an oriented experiment, with the wave and polarization vectors oriented along the y-and z-axes, respectively.

It is observed that for the 7s 1/2 → 7p 1/2 excitation, the electric-dipole approximation holds since the zeroth-order oscillator strength f [0] reproduces the oscillator strength f full , using the full interaction, to within the reported digits. For other excitations, the second-order oscillator strength f [2] has to be included in order to get reasonable agreement with the full interaction. For the 1s 1/2 → 7p 1/2 transition, however, higher-order contributions to the oscillator strength blow up. A similar behavior, but to a lesser degree, is observed for the 2s 1/2 → 7p 1/2 transition. Also, it should be noted that the oscillator strength for the 3s 1/2 → 7p 1/2 transition, accumulated to 12th order, is negative. Very similar behavior is observed for multipolar gauge (data not shown). In Table 4.1 the corresponding norm k = ω/c of the wave vector is listed for each excitation. Interestingly, the apparent divergence in the expansion of the full light-matter interaction occurs when k ≈ 1 a -1 0 . Indeed, if it is not assumed that ω = ω f i , where ω f i is the excitation energy, and instead ω is treated as a variable, so as to artificially vary k appearing in the interaction operator, it is found that the oscillator strengths for all excitations blow up around k = 1 a -1 0 (Figure 4.6). It seems reasonable that the convergence behaviour of an expansion of oscillator strengths in orders of the norm of the wave vector should change when k ≈ 1 a -1 0 . However, this conclusion requires some caution, since k is not a dimensionless quantity. The proper expansion parameter is rather the dimensionless quantity kr and the above observations suggest that the effective radius r ≈ 1 a 0 . For the valence 7s 1/2 → 7p 1/2 excitation the effective radius r is more diffuse, which explains why the apparent divergence sets in for k < 1 a -1 0 , as seen in Figure 4.6. The oscillator strengths of given (even) order are calculated according to Eqn. (3.64) or (3.87).

I have also investigated to what extent transition moments over effective interaction operators T [n]

full of order n in the wave vector, Eqn. (3.30), sum up to transition moments over the full interaction operator and again find apparent divergences for core excitations. Again, when treating ω as a variable and not setting it equal to ω f i , we find that these apparent divergences occur for all ns 1/2 → 7p 1/2 excitations when k > 1 a -1 0 . Going deeper in our analysis, I note that transition moments are obtained by contracting the property gradient of the selected operator with the solution vector for the selected excitation, Eqn. (1.121). Due to the restrictions on the orbital rotation operator in our particular case, the scalar product is reduced to the multiplication of two numbers. It can be found that an expansion of the property gradient of the full interaction in orders of the wave vector displays the same apparent divergence for core excitations as was observed for both oscillator strengths and transition moments. Again, by artificially varying k, we find that these apparent divergences occur when k > 1 a -1 0 for all excitations. With our particular orientation of the experiment, the full and truncated effective interaction operators at order n are given by

T full (ω) = e ω cα z e +iky ; T [n] (ω) = e ω i n n! cα z (ky) n . (4.15)
Elements of the property gradient of the truncated effective interaction operator are accordingly given by

g T [n] ;ai = - e ω (ik) n ϕ L a |cσ z y n n! |ϕ S i + ϕ S a |cσ z y n n! |ϕ L i (4.16)
where superscripts L and S refer to the large and small components of molecular orbital ϕ p , respectively. As discussed in Section 4.1.1, the property gradient is compounded from products of Cartesian Gaussian-type orbitals (see Eqn. (4.1)) with two expansion coefficients on the form

c * µa χ µ |c y n n! |χ ν c νi , (4.17)
with the factor outside the curly brackets in Eqn. (4.16) multiplied on at the end. In the present case, the coefficients are real due to symmetry. [START_REF] Saue | Quaternion symmetry in relativistic molecular calculations: The Dirac-Hartree-Fock method[END_REF] For n = 12 it is found that the largest contribution, in terms of magnitude, to the property gradient comes from a small component p y function with exponent α 1 = 1.56556662(-02) a -2 0 combined with a large component p y function with exponent α 2 = 1.24964369(-02) a -2 0 . These are the most diffuse s and p functions, respectively, of the large component dyall.ae3z basis set. The resulting AO-integral has a value -1.19437467(+6) a.u. and is multiplied with a coefficient c 1 = -4.55940113(-8) from 1s 1/2 and a coefficient c 2 = -0.844080786 from 7p 1/2 . By calculating AO-integrals with high precision using Mathematica, [199] the above AO-integrals, provided by the HERMIT integral package, are found to be very stable. On the other hand, the very small c 1 coefficient is at the limits of the precision one can expect from the diagonalization of the Fock matrix, in particular given its ill-conditioning due to the presence of negative-energy solutions. I have, however, investigated the sensitivity of our results with respect to the HF convergence (in terms of the gradient) and find that they are quite stable at tight thresholds.

In the test case analyzed in Section 4.2.3, the wave k and polarization vectors are oriented along the y-and z-axes, respectively, such that the full and truncated effective interaction operator at order n are given by Eqn. (4.15). In the following, the convergence will be studied of the underlying AO-integrals over the truncated interaction towards the corresponding AO-integral over the full interaction. These involve only the scalar parts of the operators, so in practice I will work with the expression

χ µ |e +iky |χ ν = ∞ n=0 (ik) n n! χ µ |y n |χ ν , (4.18)
where χ µ and χ ν are scalar basis functions. I shall limit attention to p y functions (G α 010 ) since the largest integrals in this study involved such basis functions with diffuse exponents.

The final step of our analysis is to study the convergence of the AO-integrals over the truncated interaction towards the corresponding integral over the full interaction operator. Restricting attention to our particular case in Eqn. (4.15) and Gaussian p y functions we have

G α 1 010 | T full (ω)|G α 2 010 = ∞ m=0 G α 1 010 | T [2m] (ω)|G α 2 010 . (4.19)
After eliminating the factors on both sides related to the x-and z-integration, it can be shown that

G α 1 1 |e +iky |G α 2 1 y = ∞ n=0 (ik) n n! G α 1 1 |y n |G α 2 1 y ; G α j = N α j y j e -αy 2 , (4.20)
where the left-hand-side can be evaluated using the results from previous section (Eqn. (4.5))

G α 1 1 |e +iky |G α 2 1 y = N α 1 1 N α 2 1 π α 1 + α 2 i 2 √ α 1 + α 2 2 e -Q 2 H 2 (Q) . (4.21)
Using Eqn. (4.13), we obtain for the right-hand-side integral

G α 1 1 |y n |G α 2 1 y = N α 1 1 N α 2 1 1 2 1 + (-1) (2+n) (α 1 + α 2 ) -(n+3)/2 Γ n + 3 2 . (4.22)
It thus follows that this integral vanishes unless it is even-valued, i.e. n = 2m. Again, eliminating common factors yields the following expression

-(4Q 2 -2)e -Q 2 = ∞ m=0 (-1) m a m ; a m = Q 2m (2m + 2) (2m + 1) (m + 1)! (4.
a m+1 a m = Q 2 (2m + 3) (m + 1)(2m + 1) , (4.24)
which, depending on the specific order and the value of Q, is not always smaller than one. Eventually at elevated orders, this fraction necessarily becomes smaller than one, implying monotonic decrease after that order. For that reason, I will split this series in two parts

mc m=0 (-1) m a m + ∞ n=mc (-1) n a n , (4.25)
such that the second summation is monotonically decreasing. The first of these summation converges because it consists of a finite amount of terms, whereas the second summation obeys the Leibniz criterion. Therefore, the total summation should converge as well. For the 1s 1/2 → 7p 1/2 excitation and the above choice of exponents we find that m c ≈ Q 2 = 6998.7. For this value of Q, the left-hand-side of Eqn. (4.23) is essentially zero, whereas the right-hand side converges extremely slowly towards this value. In fact, using Mathematica, [199] no convergence was observed even after summing 10000 terms. Considering instead the 2s 1/2 → 7p 1/2 excitation, for which m c ≈ 240, reasonable convergence is found after summing 282 terms. The sheer amounts appearing in this expansion thus renders the schemes based on truncated interactions impractical at high excitation energy.

Conclusion

In summary it was found that for increasing excitation energies, the use of truncated light-matter interaction becomes increasingly problematic because of the slow convergence of such expansions. This is not a basis set problem which can be alleviated by increasing the basis set, since this slow convergence was observed at the level of the individual underlying AO-integrals. In particular, for core excitations, extremely slow convergence was observed for integrals involving Cartesian Gaussian-type orbitals with diffuse exponents. This can be understood, since such diffuse functions will be less efficient than tight ones in damping the increasing Cartesian powers appearing in an expansion of the full light-matter interaction in orders of the norm of the wave vector (Eqn. (4.18)). This in turn suggests that the use of Slater-type orbitals, which have slower decay than Gaussiantype orbitals, will be even more problematic. This is indeed the case, as shown in the appendix of ref. 1.

Basis Set Convergence

A proper choice of basis set is key to reach quantitative accuracy in quantum chemical calculations. Especially in the current context, where unconventional interaction operators are being used, it is important to give special considerations to the choice of basis set. Therefore, in this section, it will be assessed whether conventional basis sets are flexible enough to satisfy the demands of these interaction operators. Conventional basis sets are constructed by choosing a set of functions that minimize the energy. Typically, more energy is gained by adding tight functions than diffuse functions. [START_REF] Jensen | Introduction to computational chemistry[END_REF] Because of this, energy-optimized basis sets saturate the core region faster than the tail-region. The full and truncated light-matter interaction may require addition of further basis functions. A problem in this respect is the lack of a variational principle for transition moments. Alternatively, basis sets for use beyond energies can be constructed using the equivalent core approximation, [START_REF] William | Thermodynamic interpretation of chemical shifts in core-electron binding energies[END_REF][START_REF] Hans Ågren | On the validity of the equivalent core approximation in Born-Haber analyses of liquids and solutions[END_REF] Slater's rule [START_REF] John | Atomic shielding constants[END_REF][START_REF] Qian | Efficient basis sets for core-excited states motivated by Slater's rules[END_REF] or the completeness-optimization scheme. [START_REF] Delano | Completeness profiles of one-electron basis sets[END_REF][START_REF] Manninen | Systematic Gaussian basis-set limit using completenessoptimized primitive sets. A case for magnetic properties[END_REF] Another strategy is to systematically increase the basis set size, until the property of interest has become stable.

Sørensen et al. performed such a series in a non-relativistic framework using the full and truncated interaction operator, the latter limited to second order, expressed with the ANO-RCC basis sets. [START_REF] Kragh Sørensen | Applications to metal K pre-edges of transition metal dimers illustrate the approximate origin independence for the intensities in the length representation[END_REF][START_REF] Kragh Sørensen | Implementation of a semiclassical light-matter interaction using the Gauss-Hermite quadrature: A simple alternative to the multipole expansion[END_REF] Even though the series did not fully converge for the larger basis sets, their results indicated that the full interaction operator is more stable with respect to basis set choice. In addition, Sørensen et al. argued that each multipole moment requires different basis sets, making it exceedingly difficult to construct basis sets that can describe the truncated interaction at higher orders. They concluded that the full interaction operator should be the standard for calculations involving non-dipolar effects. For zeroth-and first-order interactions, the length representation seems to be the preferred choice in the literature, although there is not a concise answer as to which representation has a superior basis set convergence. [START_REF] Pecul | Density functional theory calculation of electronic circular dichroism using London orbitals[END_REF][START_REF] Šindelka | Floquet perturbation theory: Applicability of the finite level approximation in different gauges[END_REF][START_REF] Khamesian | Spectroscopy of linear and circular polarized light with the exact semiclassical light-matter interaction[END_REF][START_REF] Crossley | The Calculation of Atomic Transition Probabilities[END_REF] Another strategy to gauge the quality of a basis set is to compare with a reference value. In this section, the basis set requirements for non-dipolar effects are assessed by comparing calculations using Gaussian basis sets and the molecular code Dirac [START_REF] Saue | The DIRAC code for relativistic molecular calculations[END_REF] to equivalent finite-difference calculations using the atomic code GRASP. [START_REF] Kg Dyall | GRASP: A general-purpose relativistic atomic structure program[END_REF] If the grid size of the latter is chosen sufficiently large, the results effectively correspond to the complete basis set limit, thus forming a suitable reference to the basis set calculations. For these purposes, I will study the 1s 1/2 → 7p 1/2 and the 7s 1/2 → 7p 1/2 transitions in the radium atom, representative of core and valence excitations, respectively. This section is based on ref. [START_REF] Martin Van Horn | Transition moments beyond the electric-dipole approximation: Visualization and basis set requirements[END_REF], so the reader is referred to this article for more details.

Radial Transition Moment Distributions

Due to the spherical symmetry of the radium atom, its orbitals assume the general structure given in Eqn. (2.100), where the functions P (r) and Q(r) can either be calculated on a numerical grid, or from basis set calculations. Inserting these orbitals into the transition moments thus enables us to compare basis set calculations with a numerical reference. However, the alert reader may note that the transition moments from Eqn. (1.121) contain contributions from many orbitals, whereas the numerical approach gives single orbitals (see Section 2.4.1). For that reason, the restricted excitation window approach is applied to reduce Eqn. (1.121) to an orbital-orbital transition, i.e. 1s 1/2 → 7p 1/2 and 7s 1/2 → 7p 1/2 .

Since the s 1/2 and p 1/2 orbitals in the finite basis approximation are constructed exclusively from s-and p-exponents, the basis set considerations boil down to finding what additional diffuse or tight s/p-functions are required to converge intensities. In general, diffuse and tight functions improve the description of the outer and inner regions of the wave function. To examine which regions need improvement, we visualize the transition moment densities as radial distributions

R Ω (r) = 2π 0 dφ π 0 dθ r 2 sin θ Ω f i (r); Ω f i (r) = φ † n i ,κ i ,m i (r) Ωφ n f ,κ f ,m f (r) (4.26)
where Ω f i (r) is the density that integrates to the transition moment, T Ω . For the specific transition studied in this chapter, the quantum numbers of the two orbitals read:

n i = 1, 7,κ i = -1, m i = m f = 1
2 , n f = 7 and κ f = 1, whereas the relevant operators are Ω = T, Q, m, Q (see Eqs. (3.16) (3.57) (3.85)). From parity it follows that only odd electric and even magnetic multipoles contribute. We have chosen a frame in which the wave vector k is aligned with the z-axis and the polarization vector with the y-axis. The radial distribution for the electric-length multipoles then becomes

R [2n+1] Q (r) = er 2n+1 (2n + 1)(2n + 3) (P 7,1 P n i ,-1 + Q 7,1 Q n i ,-1 ), (4.27)
the electric-velocity

R [2n+1] Q (r) = -ecr 2n ω(2n + 1)(2n + 3) ((2n -1)P 7,1 Q n i ,-1 -(2n + 3)Q 7,1 P n i ,-1 ) (4.28)
and the magnetic multipole distribution

R [2n] M = iec2n(2n -1)r 2n+1 √ 2(2n + 1) 2 (2n + 3) (P 7,1 Q n i ,-1 + Q 7,1 P n i ,-1 ) , (4.29)
where the r-dependence of the radial functions P and Q have been suppressed. The radial distribution associated with the full interaction reads

R T (r) = iec 3ω j 0 (kr) -j 2 (kr) P 7,1 Q n i ,-1 + 3j 0 (kr)Q 7,1 P n i ,-1 , (4.30)
where j 0 (kr) and j 2 (kr) are the zeroth and second order spherical Bessel functions, given by j 0 (kr) = sin kr kr ; j 2 (kr) = 3 (kr) This relation implies that all Bessel functions except j 0 are zero at the origin. We can use this result to obtain

lim kr→0 R T (r) = iec 3ω P 7,1 Q n i ,-1 + 3Q 7,1 P n i ,-1 = iR [1] Q (r). (4.33)
Therefore, the electric-dipole approximation is correctly retrieved in the long wavelength limit of Eqn. (4.30). Without performing any calculation, the shape of Eqs. (4.27)-(4.30) already tells us useful information about the basis set requirements of the interaction operators. For example, the radial distributions corresponding to the multipole moments (Eqs. (4.27)-(4.29)) all contain a prefactor depending on a power of the radial distance. This prefactor is in competition with the terms associated with the atomic radial functions: the former tends towards infinity for large distances, whereas the latter decays to zero. Because the atomic functions decay exponentially, the transition moments are finite. From these considerations it follows that the multipole moments with increasing order sample regions farther and farther away from the nucleus and thus we expect that they may require additional diffuse functions for their proper description. This is in sharp contrast with the full interaction, Eqn. (4.30). Instead of the radial powers, this distribution depends on the spherical Bessel functions, which assume the following asymptotic form [START_REF] Frank Wj Olver | NIST handbook of mathematical functions hardback and CD-ROM[END_REF]

lim kr→∞ j (kr) ∼ (kr) -1 sin kr - 1 2 π . (4.34)
Therefore, the spherical Bessel functions decay to zero for kr → ∞, which implies that the radial distribution of the full interaction will decay much faster than the multipole distributions. Consequently, the full interaction is expected to require fewer diffuse basis functions than the multipole moment operators.

Computational Details

To compare basis set effects in both the valence and core regions, the 7s 1/2 → 7p 1/2 and the 1s 1/2 → 7p 1/2 transition of the radium atom were studied. The transitions were constrained to occur between single orbitals to facilitate the comparison between the Gaussian-type-orbital calculations in Dirac [START_REF] Dirac ; Jensen | a relativistic ab initio electronic structure program, Release DIRAC22[END_REF][START_REF] Saue | The DIRAC code for relativistic molecular calculations[END_REF] and the numerical reference calculations in GRASP. [START_REF] Kg Dyall | GRASP: A general-purpose relativistic atomic structure program[END_REF] Since GRASP only computes occupied orbitals, the orbitals were optimized, using average-of-configurations (AOC) Hartree-Fock, [START_REF] Thyssen | Development and Applications of Methods for Correlated Relativistic Calculations of Molecular Properties[END_REF] with respect to excited-state determinants, i.e., [Rn]7s 1 1/2 7p 1 1/2 for the valence transition and [Ra]1s -1 1/2 7p 1 1/2 for the core transition. Even though the 7p 1/2 orbital is formally occupied in these configurations, I treat it as a virtual orbital to construct transition moments. In the case of GRASP these were obtained by inserting the P nκ and Q nκ radial functions into Eqs. (4.27)-(4.30). Since transition moments are only determined up to a complex phase, I defined them such that all radial distributions are real and have a positive maximum value. Since the radial functions from GRASP are calculated on a numerical grid and thus effectively correspond to the complete basis set limit, the GRASP transition moments were used as reference. In the case of Dirac, I applied overlap selection to prevent the core-excited state to collapse during the SCF cycles. [START_REF] Lefebvre-Brion | CECAM-ORSAY Documentation for Molecular Calculations[END_REF][START_REF] Paul | Direct Near-Hartree-Fock Calculations on the 1s Hole States of NO +[END_REF][START_REF] Andrew Tb Gilbert | Self-consistent field calculations of excited states using the maximum overlap method (MOM)[END_REF] I then used the orbitals of the excited state calculations in a ground state four-component time-dependent Hartree-Fock (4c-TD-HF) calculation within the linear response regime. [START_REF] Bast | Relativistic adiabatic timedependent density functional theory using hybrid functionals and noncollinear spin magnetization[END_REF] The restricted excitation window method [START_REF] Stener | Time dependent density functional theory of core electrons excitations[END_REF][START_REF] South | 4-Component relativistic calculations of L 3 ionization and excitations for the isoelectronic species UO 2 2+ , OUN + and UN 2[END_REF] was invoked to only consider one amplitude in Eqn. (1.85). The radial distributions were then divided by this remaining amplitude to compare the results of the basis set calculations with the numerical reference values. In Dirac the radial distributions were calculated on a radial grid with the visualization module, which performs the angular integration on a Lebedev grid (L max = 64). Due to the quaternion symmetry scheme [START_REF] Saue | Quaternion symmetry in relativistic molecular calculations: The Dirac-Hartree-Fock method[END_REF] in Dirac, all transition moments are real by default (see Section 2.4.3).

The basis set calculations were carried out at the Hartree-Fock level of theory using the Dirac-Coulomb Hamiltonian and the dyall.aeXz (X=2,3,4) basis sets. [START_REF] Kenneth | Relativistic double-zeta, triple-zeta, and quadruple-zeta basis sets for the 4s, 5s, 6s, and 7s elements[END_REF][START_REF] Kenneth | Dyall double-zeta, triple-zeta, and quadruple-zeta basis set archive files[END_REF] These basis sets are constructed from uncontracted Cartesian Gaussian basis functions which are designed for correlated calculations and hence contain basis functions of high orbital angular momentum. I trimmed off the g, h and i functions because for atomic systems they will only contribute to the space of virtual orbitals. The small component basis sets were generated according to the condition of restricted kinetic balance [START_REF] Grant | Conditions for convergence of variational solutions of Dirac's equation in a finite basis[END_REF][START_REF] Schwarz | Basis set expansions of relativistic molecular wave equations[END_REF][START_REF] Stanton | Kinetic balance: A partial solution to the problem of variational safety in Dirac calculations[END_REF][START_REF] Kenneth | Kinetic balance and variational bounds failure in the solution of the Dirac equation in a finite Gaussian basis set[END_REF] and the (SS|SS) integrals were treated exactly, for consistency with the numerical calculations. Furthermore, I found that with the default linear dependence threshold (10 -6 and 10 -8 for the large and small component space, respectively), several basis functions were deleted. Especially the small component of the 1s 1/2 was susceptible to the deletion of functions. This deteriorates the quality of the results, so for all calculations, the linear dependence threshold was set to 10 -9 and 10 -10 for the large and small component space, respectively. In Section D.1, the problems associated with linear dependence are further discussed.

Results and Discussion

We start by comparing the radial distributions of the core and valence excitations within the electricdipole approximation (velocity representation; Figure 4.7). To illustrate their radial extent, the expectation value of the radial distance for different s orbitals are indicated. A striking observation is that the radial distribution for the core transition is much more localized around the nucleus than that of the valence transition. This follows from the locality of the core orbital. Furthermore, the valence distribution oscillates considerably more than the core distribution, and one can see that the oscillations follow the shell structure of the atom. The electric transition dipole distribution is stable with respect to the choice of basis set, since all basis set calculations overlap with the numerical reference. The relative deviations (depicted in the right corner of Therefore, for both transitions, the dyall.ae2z basis set is sufficient to properly describe electric transition dipole moments.

To gauge the importance of non-dipolar effects, the electric-velocity dipole distributions are compared to those of the full interaction in Figure 4.8. The basis set convergence of the full interaction appears to be similar as the electric-dipole moment, since the full interaction is unaffected by the choice of basis set. For the valence transition, the electric-dipole distributions basically coincide with the full interaction, which confirms the validity of the electric-dipole approximation in this energy regime. Even, for the core transition, the difference between the distributions of the electric-dipole and full interaction is modest. The electric-dipole approximation breaks down if the wavelength of the electromagnetic field is small compared to the spatial extent of the transition. Though the wavelength for the core transition falls in the hard X-ray regime, the compactness of the radium 1s 1/2 orbital reduces the importance of non-dipolar effects. In the next two sections, I will discuss systems with symmetry equivalent centers, for which the core transitions are delocalized, thus increasing the importance of non-dipolar effects.

The oscillatory behavior in Figures 4. 4.9 and 4.10, respectively). Note that the 7p 1/2 orbitals are, as expected, not identical for the two transitions, since the orbitals have been optimized in different excited configurations. For all orbitals, the basis set calculations nearly coincide with the numerical reference, which makes it difficult to assess the performance of each basis set. Therefore, the error curves in Figures 4.9 and 4.10 need to be analyzed to properly assess the basis set convergence. From the analytical formulas in Eqs. (4.27)-(4.29), it is expected that the basis set error is most relevant at large radial distances. For that reason, I will investigate the error at much larger distances than the range of the radial functions (bottom row subfigures).

To some extent, all error curves in these plots are oscillatory, reflecting the incompleteness of the basis set. In general, the amplitude of the oscillations diminish with larger basis sets, although the frequency increases. This trend can clearly be observed in Figures 4.9 and 4.10: the small error associated with the dyall.ae2z basis is even further reduced by using the dyall.ae3z basis set, at the expense of a higher frequency of the oscillations, while the dyall.ae4z set flattens most of the error curves. However, the additional improvement introduced by the dyall.ae4z basis set is marginal for orbital (top row) and their deviations from the numerical reference (bottom row). These orbitals were taken from the radium atom with [Rn]7s 1 1/2 7p 1 1/2 configuration. The basis set orbitals were calculated at the 4c-HF level using Dirac, while the numerical reference was calculated with GRASP. Note that the scaling is different for each individual box and that the error curves are plotted in a different range than the radial functions.

the Q 1,-1 function, and the dyall.ae4z curve seems to oscillate more compared to the dyall.ae3z curve.

It has proven to be challenging to completely eliminate the error of the Q 1,-1 function, as it is highly susceptible to problems associated with linear dependence. As explained in Section D.1 of the appendix, the first attempts to calculate the radial distributions of the magnetic multipole moments were met with difficulties. In these calculations, the default linear dependence threshold was in place, further accentuating the oscillations of the Q 1,-1 function, which in turn caused severe oscillations in the radial distributions of the magnetic multipole moments. Although it may seem straightforward in hindsight, finding the reason for these oscillations has been a problem that significantly halted the progress of this project. Only after extensive experimentation, I found almost accidentally that changing the linear dependence threshold ameliorated the problem. By increasing the linear dependence threshold, I could significantly reduce the deviations from the 1/2 7p 1 1/2 configuration. The basis set orbitals were calculated at the 4c-HF level using Dirac, while the numerical reference was calculated with GRASP. Note that the scaling is different for each individual box and that the error curves are plotted in a different range than the radial functions.

reference, although the basis set convergence of the Q 1,-1 function is still not completely smooth. The minute differences between numerical and basis set orbitals suggest the radial distributions of the multipole moments should be stable across basis sets series.

In practice, this is, however, not completely true. Figures 4.11 and 4.12 depict the radial distributions of the electric-length multipoles, Q[2n+1] , for the valence and core transitions, respectively. In both figures, the peak of the reference curve appears close to the origin and moves farther away at higher orders. Additionally, the peak height and the integrated value of these curves seems to rise upon increase of the order. The same trend can be observed in the error curves, which are most pronounced in the regions far from the nucleus. In the following, I will define an error below 1% as acceptable convergence. Therefore, the dyall.ae2z basis set is an acceptable choice for multipole moments below order 2. For 2 < n < 5, the dyall.ae3z is needed to reduce the error below 1%. Even with the largest basis set, the accuracy goal is not reached for the highest two orders. These trends are even more pronounced in the case of the core transition (Figure 4.12). For this transition, the largest dyall.ae4z basis set is already needed at n > 3. In general, the overall error increases with the order of the multipole moments. These findings can be understood by considering the form of the radial distribution in Eqn. (4.27). The multiplication by the power of the radial distance of increasing orders blows up the moderate deviations in radial densities (Figures 4.9 and 4.10).

0 1 2 3 R Q (r) ×10 1 0.22% 0.00% 0.00% n=0 0 6 12 18
r (a 0 ) 4.12 for higher orders. Likewise, the peaks of the radial distributions move away from the origin and become higher upon increasing the order. The overall errors indicate that the dyall.ae3z basis set is preferable for 2 < n < 5, whereas the dyall.ae4z basis set should be used at higher orders. However, for n = 6 and n = 7, the dyall.ae4z basis does not reduce the overall error below 1%. The magnetic multipole moments for the core transition (Figure 4.14) follow the same trend, although the higher order multipole moments tend to converge with more difficulties towards the numerical reference. Already at n = 4, the dyall.ae4z basis set is not enough to reduce the error below 1%. In general, the basis set convergence of the magnetic multipole moments seem to follow the same trends as the electric-length multipole moments, although the convergence of the former is much more demanding. Furthermore, note that the dyall.ae4z curve has two peaks for n = 5, whereas the reference curve 4.16 for valence and core transitions, respectively). In both cases, the reference curve follows the same pattern as the other multipole moment. For the valence transition, the dyall.ae2z basis set is sufficient to converge the lowest three orders, whereas the dyall.ae3z basis set should be used at higher orders. For the core transition, the radial distributions are extremely problematic to converge. At n > 3 the dyall.ae4z is still not enough to converge towards the reference. Furthermore, at these orders, the dyall.ae4z curves contain additional peaks that are not present in the reference curves. Due to these artifacts, the overall error for n > 4 follows a counter-intuitive trend: the dyall.ae2z basis set performs better then the larger bases.

Interestingly, the artificial peaks only appear for the magnetic and electric-velocity distributions, whereas the electric-length distributions converge smoothly towards the numerical reference. This observation can be understood from the problematic basis set convergence of the Q 1,-1 function (Figure 4.10). After all, the magnetic and electric-velocity distributions depend on the α-matrices, which couple the small and large components, whereas the electric-length distribution is diagonal in these two components (see for example Eqs. (4.27)-(4.30)). In Figure 4.10, it was observed that the larger basis sets introduce small oscillations in the error curve of the Q 1,-1 function. These oscillations in the electric-length multipoles are contained in the small-small contribution to the transition density and are thus comparatively small. For the two other types of multipole moments, the oscillations reside in the large-small contribution and are amplified at large distances by the power of the radial distance, thus creating the artificial peaks in Figures 4.14 and 4.16. Similar artifacts are avoided when applying the full interaction operator. As can be inferred from Eqn. (4.30), the error associated with the tail region of the orbitals is not blown out of proportion when applying the full interaction operator. Consequently, this renders the full interaction stable with respect to basis set choice, essentially being converged already with dyall.ae2z.

Conversely, for the core transition, the low-order multipole moments require the dyall.ae3z basis set to obtain reasonable accuracy, whereas even the dyall.ae4z basis set is not enough for the highest orders. However, it might be possible that similar accuracy can be achieved with smaller basis sets. Compared to the dyall.ae2z, the dyall.ae4z basis set contains additional tight functions and highangular momentum functions that might be irrelevant for the construction of multipole moments. I attempted to augment the dyall.ae2z basis set in an even-tempered fashion with diffuse functions to obtain a better balance between computational costs and performance, but all of these basis sets introduced unphysical oscillations, as shown in Section D. be constructed by applying a similar scheme as Jensen and coworkers. [START_REF] Ambroise | Probing basis set requirements for calculating core ionization and core excitation spectra using correlated wave function methods[END_REF] They augmented the cc-pVnZ basis set with the core polarization functions of the cc-pCV(n+1)Z basis set. I tried a similar procedure to add diffuse functions to the dyall.ae2z basis, but this also led to unphysical oscillations. These findings suggest that the improved convergence observed with increased cardinal number X of the dyall.aeXz series is not only due to the increased range of exponents, but also their distribution within that range.

What remains is to apply the basis sets to the calculation of oscillator strengths. By inspection of , it seems that the value of the transition moments become increasingly larger at higher orders, suggesting that the oscillator strengths of both the valence and core transitions have convergence problems with respect to the multipole expansion. To validate this proposition, I report the anisotropic oscillator strengths for the single orbital 1s 1/2 and 7s 1/2 → 7p 1/2 transitions for both the full and truncated interactions in Table 4.2. However, upon inspection of the values in Table 4.2, it becomes clear that the multipole expansion already converges at zeroth order for the valence transition. This appears to be in contradiction with the steady growth of transition moments seen in Figures 4.11,4.13 and 4.15. This can be understood from the damping of the transition moments by increasing powers of the wave vector k, Eqs. (3.64) (3.87). This is not the case for the 1s 1/2 → 7p 1/2 transition. The oscillator strength corresponding to this transition seems

4 0 4 8 R (r) ×10 2 0.04% 0.00% 0.00% n=0 0 4 8 12
r (a 0 ) to diverge at higher orders. In fact, these results are in line with Section 4.2, where it was found that the multipole expansion converges extremely slowly at higher excitation energies (ω = c (∼3728 eV)). [START_REF] Holmgaard List | Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory[END_REF] The basis set series from Table 4.2 follows the same trends as the radial distributions.

A striking observation in Table 4.2 is that even with the largest basis set the oscillator strengths in the length-and velocity representation differ by orders of magnitude for both core and valence transitions. More precisely, the ratio lr:vr at zeroth order in k (electric-dipole approximation) is about 0.09 for the valence transition and 8.10 for the core transition, clearly in favor of the latter (vr) when comparing with the full interaction. However, before drawing conclusions on the relative merits of these two representations, one should keep in mind that in Section 3.6.2 it is demonstrated that one can only expect equivalence between these two representations when three conditions are met: i) complete basis set limit, ii) variational conditions satisfied and iii) no restrictions on the excitation window. None of these conditions are met for the results presented in Table 4.2. In fact, in order to compare directly with the GRASP reference I have imposed a severely restricted excitation window and I am using orbitals of a singly-excited state and not the ground state. I will now investigate what happens when we successively comply with conditions iii) and ii). r (a 0 )

3.0 1.5 0.0 1.5 3.0 ×10 5 maintained the restrictions on the occupied space. For the core excitation this is mandatory to assure convergence to the right excitation, and for the valence excitation it is found that it is in any case completely dominated by excitation of the 7s 1/2 orbital. For the valence transition, comparison with Table 4.2 shows that the oscillator strength using the full interaction is significantly reduced. Its value is very well reproduced within the dipole-velocity representation, whereas the discrepancy with the dipole-length representation increases dramatically. For the core excitation there is, on the other hand, only moderate change with respect to Table 4.2. It is only when we in addition use the proper ground-state orbitals that the dipole-length and dipole-velocity oscillator strengths come into agreement (Table 4.4). Table 4.2: Anisotropic oscillator oscillator strengths ( c ω k = e z ; = e y ) of the ns 1/2 → 7p 1/2 (n = 1, 7) transitions for the full semi-classical interaction operator and accumulated to various orders in the truncated interaction, as indicated by the superscripted numbers in parenthesis, within the multipolar gauge (lr: length representation) and Coulomb gauge (vr: velocity representation), computed at the 4c-TD-HF level of theory with different basis sets. Numbers in parentheses are exponents of 10.

Basis ∆E (eV) f full gauge f (→0) f (→2) f (→4) f (→6) f (→8) f (→10) f (→12) 7s 1/2 →
Table 4.3: Anisotropic oscillator oscillator strengths ( c ω k = e z ; = e y ) of the ns 1/2 → 7p 1/2 (n = 1, 7) transitions for the full semi-classical interaction operator and the dipole-length and velocity oscillator strengths, calculated with excited state orbitals and no restrictions on the space of virtual orbitals, within the multipolar gauge (lr: length representation) and Coulomb gauge (vr: velocity representation), computed at the 4c-TD-HF level of theory with different basis sets. Numbers in parentheses are exponents of 10. Table 4.4: Anisotropic oscillator oscillator strengths ( c ω k = e z ; = e y ) of the ns 1/2 → 7p 1/2 (n = 1, 7) transitions for the full semi-classical interaction operator and the dipole-length and velocity oscillator strengths, calculated with ground state orbitals and no restrictions on the space of virtual orbitals, within the multipolar gauge (lr: length representation) and Coulomb gauge (vr: velocity representation), computed at the 4c-TD-HF level of theory with different basis sets. Numbers in parentheses are exponents of 10.

Basis

Basis ∆E (eV) f full f (→0) lr f (→0) vr
ratio (lr:vr)

7s 1/2 →

Conclusion

In this section, I have assessed the basis set requirements for 4c-TD-HF linear absorption calculations with the full semi-classical light-matter interaction operator as well as the truncated interactions in the generalized length and velocity representations. To simplify the basis set considerations, I focused my attention on the underlying transition moments for the 7s 1/2 → 7p 1/2 and the 1s 1/2 → 7p 1/2 transition in the radium atom. I considered the relativistic dyall.aeXz basis set with cardinal numbers X=2,3,4 and compared to a numerical reference. The comparison was facilitated by the generation and visualization of radial distributions of the corresponding transition moment densities.

To lowest orders, the dyall.ae2z basis set was sufficient to converge the length representation electric multipole moments for both transitions. At higher orders, the dyall.ae4z basis was needed. It was commonly observed that the core transition tends to be more difficult to converge than the valence transition. The basis set convergence of the magnetic multipole moments follows these trends as well, although it seems to be even more difficult to converge for the core transition. Therefore, for the generalized length representation, the dyall.ae4z basis set suffices to properly describe high-order multipoles, although the linear dependence threshold needs to be adjusted.

From all types of multipoles, the velocity representation electric multipoles appear to be most problematic to converge. Even though the valence multipoles converge towards the numerical reference, the core multipoles contain deviations from the reference in the form of oscillations and additional peaks. Due to these artifacts, the basis set convergence seems to be inverted: larger basis sets deviate further from the reference. This suggests that the generalized velocity representation should be applied with caution for n > 3. For the generalized velocity representation, I recommend to use the dyall.ae3z basis set, which does not seem to suffer as much from the artifacts as the dyall.ae4z basis set. In addition, I recommend to go not further than n = 3, because after this order, the reliability of this scheme becomes questionable.

The full interaction operator does not suffer from these problems and is already converged with the dyall.ae2z basis. Considering that the truncated interaction also suffers from slow convergence with respect to the multipole expansion, I recommend to apply the full interaction when calculating X-ray transitions.

These findings are transferable to the non-relativistic limit, because in this limit, the full interaction and the length representation electric multipole have the same characteristics. Furthermore, the magnetic multipoles and velocity representation electric multipoles converge faster towards the numerical reference than equivalent 4c calculations, because the problems associated with linear dependence disappear upon taking this limit. Therefore, I expect that in the non-relativistic limit, the basis set requirements of multipole moments are less demanding.

Conclusions are possibly less obvious for the intermediate 2-component relativistic level. The present approach would formally require the generation of radial distributions using properly picturechanged interaction operators, and these are for instance not available for the eXact 2-Component Hamiltonian (X2C), for which in general only a matrix representation, and not a real-space one, is available (see for instance discussion in Ref.117). However, again the elimination of the small components suggests that basis set convergence should improve.

Systems with Symmetry Equivalent Centers

For the calculations on the radium atom (see Section 4.2 and 4.3), it was observed that the dipoleallowed transitions only gained modest corrections when including non-dipolar effects, the explanation being that the compactness of the core-orbitals negate these effects. However, if the system under consideration contains symmetry equivalent centers, the core transitions that arise from them are delocalized, which, in principle, should enhance non-dipolar effects. For those purposes, I will consider the Cl K -edge in TiCl 4 . In addition, this system represents a case where there is no natural choice of gauge origin, thus forming a suitable system to test origin-(in)dependence of the two representations. This system has previously been studied in the context of non-dipolar effects in linear X-ray absorption using low-order multipole expansions. In particular, it was used to demonstrate the appearance of negative oscillator strengths [START_REF] Lestrange | The consequences of improperly describing oscillator strengths beyond the electric dipole approximation[END_REF] upon truncation of the light-matter interaction in the generalized velocity representation in a non-relativistic framework. [START_REF] Bernadotte | Origin-independent calculation of quadrupole intensities in X-ray spectroscopy[END_REF] Below, I will revisit this case and assess the necessity to go beyond the electric-dipole approximation. I further study numerically the gauge-origin dependence of the three schemes in the case of soft X-ray absorption. Ligand K -edge absorption spectroscopy supposedly provides direct information on the covalency of metalligand bonds due to the admixture of the ligand p-orbitals with the metal d-orbitals. [START_REF] Glaser | Ligand k-edge x-ray absorption spectroscopy: a direct probe of ligand-metal covalency[END_REF][START_REF] Edward I Solomon | Ligand k-edge x-ray absorption spectroscopy: covalency of ligand-metal bonds[END_REF] The Cl K -edge absorption of TiCl 4 has been studied both experimentally and also theoretically within and beyond the ED approximation using truncated multipole-expanded expressions. Its experimental spectrum features a broad pre-edge peak that require a two-peak fit (in toluene: at 2821.58 and 2822.32 eV with an approximate intensity ratio of 0.84). [START_REF] Debeer | Metal and ligand k-edge xas of organotitanium complexes: Metal 4p and 3d contributions to pre-edge intensity and their contributions to bonding[END_REF] In T d symmetry, the five 3d-orbitals of Ti belong to the e and t 2 irreducible representations, and the pre-edge bands can be assigned to excitations from the a 1 and t 2 Cl 1s-orbitals into the e and t 2 sets of 3d-orbitals on Ti, respectively. Here, I focus on the eight lowest-lying transitions (a 1 , t 2 → e) which give rise to three degenerate sets (E, T 1 and T 2 ) of which the latter is ED allowed. The results in this section are mainly based on the findings from ref. 1.

Computational Details

The data reported in this section have been obtained with a development version of the Dirac electronic structure code [START_REF] Dirac ; Jensen | a relativistic ab initio electronic structure program, Release DIRAC22[END_REF] (Tables 4.5-4.7: revision 52c65be).

The calculated results presented in this section have been obtained by time-dependent density functional theory (TD-DFT) calculations, based on the Dirac-Coulomb Hamiltonian and within the restricted excitation window (REW) approach [START_REF] Stener | Time dependent density functional theory of core electrons excitations[END_REF][START_REF] South | 4-Component relativistic calculations of L 3 ionization and excitations for the isoelectronic species UO 2 2+ , OUN + and UN 2[END_REF] using the PBE0 [START_REF] John P Perdew | Generalized gradient approximation made simple[END_REF][START_REF] Adamo | Toward reliable density functional methods without adjustable parameters: The PBE0 model[END_REF] exchangecorrelation functional and the dyall.ae3z basis sets. [START_REF] Kenneth | Relativistic double-zeta, triple-zeta, and quadruple-zeta basis sets for the 4s, 5s, 6s, and 7s elements[END_REF][START_REF] Kenneth | Core correlating basis functions for elements 31-118[END_REF] The small component basis sets were generated according to the condition of restricted kinetic balance, and the (SS|SS) integrals are replaced by an interatomic SS correction. [START_REF] Visscher | Approximate molecular relativistic Dirac-Coulomb calculations using a simple Coulombic correction[END_REF] A Gaussian model was employed for the nuclear charge distribution. [START_REF] Visscher | Dirac-Fock atomic electronic structure calculations using different nuclear charge distributions[END_REF] A 86-point Lebedev grid (L max = 12) was used for the isotropic averaging of the oscillator strengths based on the full light-matter interaction operator. The gauge origin was placed in the center-of-mass and spatial symmetry was invoked in all cases except for the gaugeorigin dependence calculations. The geometry of TiCl 4 was taken from Ref. [START_REF] Bernadotte | Origin-independent calculation of quadrupole intensities in X-ray spectroscopy[END_REF] where it was obtained using the BP86 exchange-correlation functional [START_REF] Becke | Density-functional exchange-energy approximation with correct asymptotic behavior[END_REF][START_REF] Perdew | Erratum: Density-functional approximation for the correlation energy of the inhomogeneous electron gas[END_REF] and the TZP basis set. [START_REF] Van Lenthe | Optimized slater-type basis sets for the elements 1-118[END_REF] 4.4.2 Full vs. Truncated Light-Matter Interaction Table 4.5 collects the isotropically averaged oscillator strengths for the pre-edge transitions computed in 4-component relativistic framework with the full light-matter interaction operator as well as accumulated to increasing orders (up to 12th order) in the wave vector within Coulomb gauge (velocity representation) and multipolar gauge (length representation). In line with the results of Lestrange et al., [START_REF] Lestrange | The consequences of improperly describing oscillator strengths beyond the electric dipole approximation[END_REF] negative oscillator strengths are found at second order for the 1 T 2 excitations in both length and velocity representation. The same issue appears for the 1 T 1 and 1 E sets, but at fourth order. As discussed previously, [START_REF] Lestrange | The consequences of improperly describing oscillator strengths beyond the electric dipole approximation[END_REF][START_REF] Kragh Sørensen | Applications to metal K pre-edges of transition metal dimers illustrate the approximate origin independence for the intensities in the length representation[END_REF] this behavior is expected when the cross terms involving the lower-order moments to f [n] dominate the diagonal contributions. As evident from the underlying contributions given in Table 4.6, the multipole expansions are alternating, and beyond fourth order, the correction is reduced at each order. Indeed, the expansions converge to the full expression at about 12th order. For the dipole-allowed 1 T 2 set, the correction introduced by non-dipolar effects is significant, reducing the oscillator strength by a factor of ∼5. As seen from the comparison of the ED and full (BED) oscillator strengths summed over the three sets of transitions, included in Table 4.5, the implication of going beyond the ED approximation is a redistribution of intensity among transitions. In particular, the ED forbidden 1 T 1 and 1 E transitions gain intensity beyond that of the T 2 set. It should be noted that this intensity redistribution cannot be observed in the spectrum due to broadening.

However, even in the hypothetical case where these transitions are well-separated, the validity of these results can be questioned. In the exact many-body wave function, the probability to find a core electron on either of the symmetry equivalent centers is strongly correlated with the motion of the other core electrons. For example, after a core electron is located on center A, the probability to find another electron on the same center decreases, whereas the probability to find it on other centers increases. This effect is poorly taken into account using SCF methods, were core electrons are described by orbitals delocalized over the centers and completely devoid of static correlation. Early accounts of this effects were made by Bagus et al. in their work involving core-ionized O + 2 , were a significant improvement of the ionization energy could be obtained by localizing the core orbital, albeit at the expense of losing the symmetry of the wave function. If we insist on using SCF methods to describe core excitations, we are left with a dilemma: either we localize the core orbital, which improves the energy values, but diminishes non-dipolar effects, or we use the delocalized orbitals, which induces the opposite effect.

However, this problem is a false dilemma, since there is a third option, as it was found that more complicated wave function methods (such as MCSCF [START_REF] Hans Agren | Symmetry adapted versus symmetry broken wavefunctions: the 1s core level ions of O +2[END_REF] or CCSD [START_REF] Stanton | On the choice of orbitals for symmetry breaking problems with application to NO 3[END_REF]) can both preserve symmetry while at the same time reproducing ionization energies. To properly assess the importance of non-dipolar effects in systems with symmetry equivalent centers, it is thus necessary to go beyond standard SCF methods. Extending the Dirac implementation to the MCSCF level of theory was one of the goals in this thesis, although it has not yet been completed. Therefore, the importance of non-dipolar effects for systems with symmetry equivalent centers remains partially unresolved. In the outlook (Sec. 5), it is further discussed what code has to be programmed to allow for MCSCF calculations beyond the electric-dipole approximation.

Table 4.5: Comparison of isotropically averaged oscillator strengths for Cl 1s →Ti 3d transitions of TiCl 4 for the full semi-classical interaction operator and accumulated to various orders, as indicated by the superscripted number in parenthesis, within multipolar gauge (lr: length representation) and Coulomb gauge (vr: velocity representation), computed at the 4c-TD-PBE0 level of theory with the dyall.ae3z basis set. Contributions from degenerate states have been summed. A 86-point (L max = 12) Lebedev grid was used to obtain the isotropically averaged full BED oscillator strengths. The gauge origin is placed on the Ti atom.

Final state ∆E (eV) gauge Lebedev grid was used to obtain the isotropically averaged full BED oscillator strengths. The gauge origin is placed on the Ti atom.

10 3 f (→0) 10 3 f (→2) 10 3 f (→4) 10 3 f (→6) 10 3 f (→8) 10 3 f (→10) 10 3 f (→12)
F.S. ∆E (eV) gauge f [0]
f [2] f [4] f [6] f [8] f [10] f [12] f full

Origin-Dependence

The above results were computed with the gauge-origin placed at the Ti atom. I now proceed to a numerical evaluation of their dependency on the gauge origin (O+a). The formulations based on the full semi-classical interaction operator and truncated interaction in the velocity representation are formally gauge invariant. In practical calculations, however, as discussed in Section 3.4.1, invariance in the latter case relies on the accurate cancellation of lower-order contributions multiplied with powers (k • a), where a is the displacement. In contrast, as discussed in Section 3.6.2, in the multipolar gauge formal gauge-origin invariance appears to only be achieved in the practically unreachable limit of the complete expansion of the fields. Table 4.7 collects the total isotropic oscillator strength for the dipole-allowed 1 T 2 set for each of the three schemes for going beyond the ED approximation using different choices for the gauge origin. As expected, the results for the full light-matter interaction operator remain unchanged, providing a numerical verification of its gauge-origin invariance. The same is true for the oscillator strengths in the generalized velocity representation. However, numerical noise from the cancellation of many terms in powers of the displacement becomes apparent at large displacements. For a displacement of 100 a 0 , instabilities start to appear at 10th order, and at 12th order, the oscillator strength exceeds the full result by one order of magnitude. The oscillator strengths in the multipolar gauge already at second order differ significantly upon shifting the origin from the Ti atom. In line with the discussion in Section 3.6.2, the two representations are roughly equivalent if the gauge origin coincides with the coordinate origin. Table 4.7: Gauge-origin dependency of the isotropically averaged oscillator strengths for the 1 T 2 set of Cl 1s →Ti 3d transitions of TiCl 4 for the full semi-classical light-matter interaction operator and accumulated to various orders within multipolar gauge (lr: length representation) and Coulomb gauge (vr: velocity representation), computed at the 4c-TD-PBE0 level of theory and the dyall.ae3z basis set. Numbers in parentheses are exponents of 10. At this level the excitation energy is calculated as 2773.351145 eV. Contributions from the degenerate set have been summed. An 86-point (L max = 12) Lebedev grid was used to obtain the isotropically averaged full BED oscillator strengths. The gauge origin is shifted along the x-axis (d x ) where d x = 0.0 a 0 corresponds to gauge-origin in the Ti atom.

dx (a 0 ) gauge f (→0) f (→2) f (→4) f (→6) f (→8) f (→10) f (→12) f full 0 lr 7.

Conclusion

Two important key conclusions follow from these calculations:

• the studied transitions stemming from the Cl K -edge in TiCl 4 have significant non-dipolar corrections, mainly induced by the delocalization of the Cl core orbitals. However, due to the closeness of the transitions, the accumulated value of these transitions is hardly affected. Even if this were not the case, it can be questioned whether the delocalization of the core orbital is a true physical effect. To properly assess this, it is required to go beyond SCF theory.

• Formal gauge-origin invariance of oscillator strengths in multipolar gauge hinges on commutator expressions that do not necessarily hold in a finite basis. This explains the notorious lack of order-by-order gauge-origin independence in practical calculations beyond the electricdipole approximation based on any truncated multipolar gauge formulation. [START_REF] Lestrange | The consequences of improperly describing oscillator strengths beyond the electric dipole approximation[END_REF][START_REF] Kragh Sørensen | Applications to metal K pre-edges of transition metal dimers illustrate the approximate origin independence for the intensities in the length representation[END_REF] As indicated in Section 3.6.2 these commutator relations, involving the Hamiltonian, correspond to a gauge transformation from the length to the velocity representation. In other words, gaugeorigin independence in multipolar gauge is shown by transforming to another gauge for which origin-independence holds. Thus far, I have not been able to show gauge-origin invariance while staying within multipolar gauge.

Electronic Circular Dichroism

I have not yet exploited the versatility of the BED implementation to its full extent, since all calculations until this point involved the absorption of linearly polarized light. In this section, I will thus focus on the isotropic and anisotropic ECD calculated using the full semi-classical lightmatter interaction, as well as the generalized velocity representation. This allows us to investigate the ECD response across the electromagnetic spectrum, from optical to X-ray regimes. So far, previous theoretical studies of the isotropic XNCD of molecules were based on a non-relativistic formulation while only considering the first-order truncated interaction, which is proportional to the dot-product of the electric-and magnetic-dipole moment (Eqn. (3.150)). [START_REF] Goulon | Detection limits for natural circular dichroism of chiral complexes in the x-ray range[END_REF][START_REF] Villaume | On circular dichroism and the separation between chromophore and chiral center: The near carbon k-edge x-ray absorption and circular dichroism spectra of noradrenaline and l-dopa[END_REF][START_REF] Zhang | Xray circular dichroism signals: a unique probe of local molecular chirality[END_REF][START_REF] Takahashi | Theoretical study of the x-ray natural circular dichroism of some crystalline amino acids[END_REF][START_REF] Kimberg | Calculation of K-edge circular dichroism of amino acids: Comparison of random phase approximation with other methods[END_REF][START_REF] Jiemchooroj | Near-edge x-ray absorption and natural circular dichroism spectra of L-alanine: A theoretical study based on the complex polarization propagator approach[END_REF][START_REF] Carravetta | Ordinary and rotatory intensities for X-ray absorption at the C-1s edge of organic chiral molecules: propylene oxide and trans-1,2-dimethylcyclopropane[END_REF][START_REF] Plashkevych | Theoretical study of x-ray circular dichroism of amino acids[END_REF][START_REF] Turchini | Core Electron Transitions as a Probe for Molecular Chirality: Natural Circular Dichroism at the Carbon K-edge of Methyloxirane[END_REF] A rough estimate of the first-order XNCD can be found by assuming that the transition in question is effectively atomic in nature, which is justified by the atom-like character of core orbitals. Under this assumption, atomic selection rules apply, implying that the electric-and magnetic-dipole moment transition rules dictate whether the XNCD vanishes. However, as demonstrated in Section C.2, the non-relativistic magnetic-dipole selection rules prescribe that atomic transitions can only occur between states that are split by spin-orbit coupling, thus suggesting that the XNCD is zero within this simplified scheme. In more realistic calculations, the core orbitals are polarized away from atomic symmetry, which renders the XNCD non-zero, an effect that was taken into consideration in the above mentioned studies. The current implementation accounts for two additional possible contributions: (i) effects of beyond first-order light-matter interactions and (ii) inclusion of relativistic effects, notably spin-orbit coupling that modifies selection rules (in particular, the magnetic transition-dipole selection rule, discussed in Section C.2). Consequently, this allows, for the first time, to realistically examine the ECD response of molecules across the valence and X-ray regimes.

However, unlike previous examples, it is not sufficient to use the same atomic target system, which has vanishing ECD due to its high symmetry. Therefore, as test systems, I consider the simplest disulfide chromophore models, dihydrogen disulfide H 2 S 2 and dimethyl disulfide (CH 3 S) 2 . Because of the low disulfide torsional barriers (∼6-11 kcal/mol [START_REF] Urban | The high-resolution rotational and torsional spectra of HSSH[END_REF][START_REF] Herbst | An experimentally derived torsional potential function for HSSH[END_REF][START_REF] Glauciete | A quantum chemical study of H 2 S 2 : Intramolecular torsional mode and intermolecular interactions with rare gases[END_REF][START_REF] Aquilanti | Intramolecular Dynamics of RS-SR' Systems (R, R' = H, F, Cl, CH 3 , C 2 H 5 ): Torsional Potentials, Energy Levels, Partition Functions[END_REF]), the two enantiomeric forms (P -and M -helix) cannot be resolved experimentally. However, the disulfide bridge is an important structural element in proteins, [START_REF] Thornton | Disulphide bridges in globular proteins[END_REF] where it preferentially occurs in non-planar, chiral conformations (C 2 symmetry) and hence displays structurally-induced axial chirality. An interesting perspective for complex systems (e.g., proteins) is the potential use of XNCD as a local probe of chirality. [START_REF] Plashkevych | Theoretical study of x-ray circular dichroism of amino acids[END_REF][START_REF] Zhang | Xray circular dichroism signals: a unique probe of local molecular chirality[END_REF] This could potentially complement the delocalized conformational information encoded in valence ECD. Because of its computational tractability, H 2 S 2 has been widely used to benchmark electronic structure methods for the calculation of chiroptical properties. [START_REF] Rauk | Chiroptical properties of disulfides. Ab initio studies of dihydrogen disulfide and dimethyl disulfide[END_REF][START_REF] Ha | Ab initio CI study of the optical rotatory strengths of HSSH[END_REF][START_REF] Pericou-Cayere | Theoretical treatment of the electronic circular dichroism spectrum and the optical rotatory power of H 2 S 2[END_REF][START_REF] Diedrich | Systematic Investigation of Modern Quantum Chemical Methods to Predict Electronic Circular Dichroism Spectra[END_REF][START_REF] Pecul | The ab initio calculation of optical rotation and electronic circular dichroism[END_REF][START_REF] Bednárová | Vibrational and electronic optical activity of the chiral disulphide group: Implications for disulphide bridge conformation[END_REF][START_REF] Scott | Electronic circular dichroism spectra using the algebraic diagrammatic construction schemes of the polarization propagator up to third order[END_REF] For the same reason, Goulon et al. also used it to estimate relative magnitudes of XNCD responses within the first-order truncated interaction and non-relativistic framework, reporting values below the experimental detection limits. [START_REF] Goulon | Detection limits for natural circular dichroism of chiral complexes in the x-ray range[END_REF] Here, I revisit the ECD of the disulfide chromophore across the valence, L-and K -edges, going beyond these approximations.

Computational Details

The geometries of H 2 S 2 and (CH 3 S) 2 were obtained using the B3LYP [START_REF] Stephens | Ab Initio Calculation of Vibrational Absorption and Circular Dichroism Spectra Using Density Functional Force Fields[END_REF][START_REF] Becke | Density-Functional Thermochemistry. III. The Role of Exact Exchange[END_REF][START_REF] Vosko | Accurate Spin-Dependent Electron Liquid Correlation Energies for Local Spin Density Calculations: A Critical Analysis[END_REF][START_REF] Lee | Development of the Colle-Salvetti Correlation-Energy Formula Into a Functional of the Electron Density[END_REF] exchange-correlation functional and the cc-pVTZ [START_REF] Dunning | Gaussian Basis Sets for Use in Correlated Molecular Calculations. I. The Atoms Boron through Neon and Hydrogen[END_REF][START_REF] Woon | Gaussian basis sets for use in correlated molecular calculations. iii. the atoms aluminum through argon[END_REF] basis set. Geometry optimizations were performed in Gaussian 16. [START_REF] Frisch | Gaussian16 Revision C.01[END_REF] To mimic the χ 3 disulfide angle typical for protein structures, [START_REF] Douglas | Disulfide by design 2.0: a web-based tool for disulfide engineering in proteins[END_REF] a constrained geometry optimization was performed for χ 3 = -87 • , corresponding to M -helical chirality. [START_REF] Pescitelli | Conformational aspects in the studies of organic compounds by electronic circular dichroism[END_REF] The restricted excitation window approach [START_REF] Stener | Time dependent density functional theory of core electrons excitations[END_REF][START_REF] South | 4-Component relativistic calculations of L 3 ionization and excitations for the isoelectronic species UO 2 2+ , OUN + and UN 2[END_REF] was used to selectively target the sulfur L-and K -edges. This also eliminates the issue of artificial transitions to quasicontinuum orbitals caused by finite basis set effects that otherwise often interferes simulations at the L-edge. [START_REF] Kadek | X-ray absorption resonances near L 2,3 -edges from real-time propagation of the Dirac-Kohn-Sham density matrix[END_REF][START_REF] Fransson | K-and L-edge X-ray absorption spectrum calculations of closed-shell carbon, silicon, germanium, and sulfur compounds using damped four-component density functional response theory[END_REF] A Gaussian model was employed for the nuclear charge distribution, [START_REF] Visscher | Dirac-Fock atomic electronic structure calculations using different nuclear charge distributions[END_REF] and an 86-point Lebedev grid (L max = 12) was used for the isotropic averaging of the differential linear absorption based on the full interaction operator. The gauge origin was placed in the center-ofmass (COM) and spatial symmetry was invoked in all cases except for the gauge-origin dependence calculations.

Excitation energies, linear and differential absorption cross sections for the full interaction operator as well as the multipole expansions within the generalized velocity gauge were computed using the PBE0 [START_REF] John P Perdew | Generalized gradient approximation made simple[END_REF][START_REF] Adamo | Toward reliable density functional methods without adjustable parameters: The PBE0 model[END_REF] exchange-correlation functional and the uncontracted aug-pcX-3 [START_REF] Ambroise | Probing Basis Set Requirements for Calculating Core Ionization and Core Excitation Spectroscopy by the ∆ Self-Consistent-Field Approach[END_REF] and augpc-3 [START_REF] Jensen | Polarization consistent basis sets: Principles[END_REF][START_REF] Jensen | Polarization consistent basis sets. II. Estimating the Kohn-Sham basis set limit[END_REF][START_REF] Jensen | Polarization consistent basis sets. III. The importance of diffuse functions[END_REF] basis sets for sulfur and hydrogen, respectively. The pcX-n basis set series was developed for describing core-excitation processes using the ∆SCF (Self-Consistent Field) approach at both the nonrelativistic and relativistic levels. The small component basis sets were generated within the condition of restricted kinetic balance. The relativistic calculations were performed using a Dirac-Coulomb Hamiltonian in which the (SS|SS) integrals are replaced by an interatomic SS energy correction. [START_REF] Visscher | Approximate molecular relativistic Dirac-Coulomb calculations using a simple Coulombic correction[END_REF] Gauge-origin invariance of the full semi-classical formulation of the isotropic and anisotropic rotatory strengths (Eqs. (3.20) (3.64) (3.87) (3.134) (3.138)) and its first-order truncated counterpart was confirmed numerically by shifting the gauge-origin (from 0 to 100 a 0 ) along the C 2 axis. This leads to a redistribution of the E1-E2 and E1-M1 contributions to R [START_REF] Holmgaard List | Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory[END_REF] xx and R [START_REF] Holmgaard List | Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory[END_REF] yy (Eqn. (3.153)) for transitions of B symmetry. As expected, the results remained unchanged for both the full and truncated formulations (data not shown). Simulated spectra were obtained by convolving the stick spectrum with Gaussian lineshape functions with full width at half maximum (FWHM) of 0.4 eV, and those for (CH 3 S) 2 were shifted by different offsets for each absorption edge to match their experimental counterparts.

To assess the effect of increased excitation energies on the differential oscillator strength, the L 1 -and K-edge spectra were calculated for the heavier analogues of H 2 S 2 : H 2 X 2 , X=Se and Te. In analogy with H 2 S 2 , the geometries for the two heavier analogous were obtained from a constrained optimization (fixing χ 3 = -87 • ) using the cc-pVTZ [START_REF] Wilson | Gaussian basis sets for use in correlated molecular calculations. IX. The atoms gallium through krypton[END_REF] and def2-VTZPP [START_REF] Weigend | Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy[END_REF] basis sets for X=Se and Te, respectively. Except for the choice of basis set, this series of calculations was carried out at the same level of theory as previously mentioned H 2 S 2 calculation. To reduce the computational cost for the heavier analogous, I used the dyall.av3z basis set [START_REF] Kenneth | Relativistic double-zeta, triple-zeta, and quadruple-zeta basis sets for the light elements H-Ar[END_REF] for all spectral calculations. To make the comparison between the structural analogues more precise, the H 2 S 2 calculations were repeated using this basis set.

Results and Discussion

Before considering the ECD response of the disulfides, I assign the linear absorption features across the valence and X-ray regions. I initially focus on (CH 3 S) 2 for which experimental gas-phase absorption spectra are available. [START_REF] Tokue | Photoexcitation of dimethyl sulfide and dimethyl disulfide in the vacuum ultraviolet region: Rydberg states and photofragment emissions[END_REF][START_REF] Hearn | The near U.V. absorption spectra of dimethyl sulfide, diethyl sulfide and dimethyl disulfide at T=300 K. Atmospheric Environment[END_REF][START_REF] Schnorr | Tracing the 267 nm-Induced Radical Formation in Dimethyl Disulfide Using Time-Resolved X-ray Absorption Spectroscopy[END_REF][START_REF] Ochmann | UV-Photochemistry of the Disulfide Bond: Evolution of Early Photoproducts from Picosecond X-ray Absorption Spectroscopy at the Sulfur K-Edge[END_REF][START_REF] Hitchcock | Sulfur and Chlorine K-shell spectra of gases[END_REF] Expectedly, and as shown in Figure D.7 in Section D.3, the spectral profiles for H 2 S 2 are similar, and because of its greater computational tractability, I consider this minimal disulfide in subsequent analyses.

Figure 4.17 displays the rotationally averaged linear and differential absorption spectra for valence, sulfur L-and K-edge transitions of (CH 3 S) 2 , computed using the full interaction operator (green shading) with corresponding oscillator strengths indicated as green sticks. Hereafter, I explicitly indicate the results of the full interaction with the superscript "full". For comparison, I also provide the lowest non-vanishing terms in the truncated generalized velocity representation (orange lines) for linear and differential absorption, i.e., zeroth-and first-order in the magnitude of the wave vector, respectively. Black sticks at the top indicate the location of the underlying electronic transitions and black lines (solid and dashed) the experimental absorption spectra. [START_REF] Tokue | Photoexcitation of dimethyl sulfide and dimethyl disulfide in the vacuum ultraviolet region: Rydberg states and photofragment emissions[END_REF][START_REF] Hitchcock | Sulfur and Chlorine K-shell spectra of gases[END_REF][START_REF] Hearn | The near U.V. absorption spectra of dimethyl sulfide, diethyl sulfide and dimethyl disulfide at T=300 K. Atmospheric Environment[END_REF][START_REF] Schnorr | Tracing the 267 nm-Induced Radical Formation in Dimethyl Disulfide Using Time-Resolved X-ray Absorption Spectroscopy[END_REF][START_REF] Ochmann | UV-Photochemistry of the Disulfide Bond: Evolution of Early Photoproducts from Picosecond X-ray Absorption Spectroscopy at the Sulfur K-Edge[END_REF] Apart from uniform shifts necessary to align the lowest-energy band to the respective experimental spectrum, the theoretical spectra capture well both relative intensities and peak splittings. [START_REF] Tokue | Photoexcitation of dimethyl sulfide and dimethyl disulfide in the vacuum ultraviolet region: Rydberg states and photofragment emissions[END_REF][START_REF] Hearn | The near U.V. absorption spectra of dimethyl sulfide, diethyl sulfide and dimethyl disulfide at T=300 K. Atmospheric Environment[END_REF][START_REF] Schnorr | Tracing the 267 nm-Induced Radical Formation in Dimethyl Disulfide Using Time-Resolved X-ray Absorption Spectroscopy[END_REF][START_REF] Ochmann | UV-Photochemistry of the Disulfide Bond: Evolution of Early Photoproducts from Picosecond X-ray Absorption Spectroscopy at the Sulfur K-Edge[END_REF][START_REF] Hitchcock | Sulfur and Chlorine K-shell spectra of gases[END_REF] The stick spectra were convolved with a Gaussian lineshape with FWHM of 0.4 eV. The theoretical absorption spectra have been uniformly shifted to align with the experimental counterparts (shift values indicated in the top panels). The same shifts were applied to the ECD spectra.

The first valence band (Figure 4.17a) is dominated by the two excitations from the symmetric and antisymmetric combinations of the non-bonding 3p orbital on each sulfur to the lowest unoccupied σ * SS orbital (b symmetry). This assignment is consistent with the analysis by Linderberg and Michl on H 2 S 2 . [START_REF] Linderberg | Inherent optical activity of organic disulfides[END_REF] Since the valence orbitals are found to have well-defined spin, I adopt a non-relativistic state notation for the valence states, i.e., 1 1 B and 2 1 A, respectively. They are separated by ∼0.15 eV. The second valence band originates from transitions into the σ * CS orbital (3 1 A and 2 1 B). Turning to the X-ray region, the first two bands at the sulfur L 2,3 -edge (Figure 4.17b) are dominated by transitions from the symmetric and antisymmetric combinations of the S 2p 3/2 and 2p 1/2 core orbitals to the σ * SS orbital. It should be noted that the A/B pair of associated transitions are essentially degenerate (less than 0.1 meV splitting) because of the limited overlap between the core orbitals on each of the sulfur atoms. The calculated spin-orbit splitting of ∼1.35 eV between the L 3 -and L 2 -branches is comparable to the experimental splitting reported for dihydrogen sulfide (∼1.2 eV [START_REF] Svensson | Observation of an energy shift in the S2p 3/2 -S2p 1/2 spin-orbit splitting between x-ray photoelectron and Auger-electron spectra for the H 2 S molecule[END_REF][START_REF] Svensson | Observation of an anomalous decay ratio between the molecular field split levels in the S 2p core photoelectron and LVV Auger spectrum of H 2 S[END_REF]). The L 3 /L 2 branching ratio of ∼1.4:1 (obtained by summing the underlying oscillator strengths) deviates significantly from its statistical value, which is obtained only in the limit of jj coupling. [START_REF] Thole | Linear relation between x-ray absorption branching ratio and valence-band spin-orbit expectation value[END_REF][START_REF] Bagus | A new analysis of Xray adsorption branching ratios: Use of Russell-Saunders coupling[END_REF] The second peak in each branch is dominated by excitations to the σ * CS orbital and is separated from the first peak by ∼1.4 -1.5 eV. Consequently, the second band in the spectrum contains contributions from both branches whereas the third band is associated with the second peak in the L 2 branch. These assignments agree with previous studies. [START_REF] Bernini | Core level (S 2p) excitation and fragmentation of the dimethyl sulfide and dimethyldisulfide molecules[END_REF][START_REF] Schnorr | Tracing the 267 nm-Induced Radical Formation in Dimethyl Disulfide Using Time-Resolved X-ray Absorption Spectroscopy[END_REF] The energy range considered as well as the basis set used in our calculations does not cover the fourth band in the experimental spectrum which, according to previous work, [START_REF] Schnorr | Tracing the 267 nm-Induced Radical Formation in Dimethyl Disulfide Using Time-Resolved X-ray Absorption Spectroscopy[END_REF] originates from excitations to higher-lying orbitals of mixed σ * CS /Rydberg character. Not surprisingly, the L 1and K-edge spectra bear strong resemblance (Figure 4.17c-d): they display two pre-edge features, separated by ∼1.5 eV, which originate from pairs of near-degenerate excitations from the bonding and antibonding combinations of sulfur s-orbitals into the σ * SS and σ * CS orbitals, respectively. [START_REF] Baba | Site-specific fragmentation in condensed (CH 3 S) 2 by sulfur K-edge photoexcitation[END_REF][START_REF] Pickering | Chemical Sensitivity of the Sulfur K-Edge X-ray Absorption Spectra of Organic Disulfides[END_REF][START_REF] Ochmann | UV-Photochemistry of the Disulfide Bond: Evolution of Early Photoproducts from Picosecond X-ray Absorption Spectroscopy at the Sulfur K-Edge[END_REF] A non-vanishing ECD response in these minimal disulfides results from axial chirality caused by trapping the disulfide bridge in a non-planar (i.e., C 2 ) conformation. As described above, the two lowest-energy transitions in each spectral domain are dominated by an excitation from the bonding or antibonding combinations of the relevant atomic orbitals on the sulfurs into the σ * SS orbital. This pairing of transitions manifests as bisignate features in the low-energy region of the ECD spectra. On the basis of the simple Bergson model for the low-energy transitions in the disulfide chromophore, [START_REF] Bergson | Molecular orbital treatment of the 3pπ interaction in five-membered cyclic disulphides[END_REF][START_REF] Bergson | A semi-empirical study of the interaction between lone-pair electrons with special reference to the problem of hybridization and the theory of restricted rotation about single bonds[END_REF] Linderberg and Michl [START_REF] Linderberg | Inherent optical activity of organic disulfides[END_REF] formulated a quadrant rule for the optical activity of the two low-energy valence transitions (dominated by excitations from the symmetric and antisymmetric combinations of non-bonding 3p orbitals on sulfurs to the σ * SS orbital) in organic disulfides. This rule relates the sign of the long-wavelength Cotton effect across the four dihedral quadrants and is a specific case of the C 2 -rule for general chromophores of effective C 2 symmetry. [START_REF] Wagnière | Polarization and the sign of the long-wavelength cottoneffects in chromophores of symmetry C 2[END_REF] Woody extended the theoretical analysis to also include the absolute sign of the lowest ECD band within each quadrant, [START_REF] Woody | Application of the Bergson model to the optical properties of chiral disulfides[END_REF] providing predictions in agreement with experimental results across different dihedral angles. [START_REF] Carmack | Circular dichroism and the absolute configuration of the chiral disulfide group[END_REF][START_REF] Dodson | Correlation of the chirality of the disulfide group with its molecular ellipticity[END_REF][START_REF] Göran Claeson | Temperature-Dependent Optical Rotatory Dispersion Measurements of Methyl (R)(-)-1,2-Dithiane-4-carboxylate. The Absolute Configuration of the Disulphide Group[END_REF][START_REF] Ludescher | On the chirality of the cystine disulfide group: Assignment of helical sense in a model compound with a dihedral angle greater than ninety degrees using NMR. and CD[END_REF] For the M -helical form considered here, a negative-first Cotton effect is observed, consistent with the quadrant rule. [START_REF] Woody | Application of the Bergson model to the optical properties of chiral disulfides[END_REF] The intensity asymmetry of the lowest-energy valence couplet is attributed to different intrafragment (i.e., CH 3 S-) contributions to the ECD signal of each transition. At higher energies, the electronic coupling between the (core) orbitals decreases, reducing both their energetic splitting and the intrafragment ECD contributions which become increasingly atomic-like. As a consequence, the paired core transitions become near-degenerate (energy splitting of a few meV or less) with rotational strengths of almost equal magnitudes but opposite signs (see Table 4.8). Hence, after additionally accounting for sources of broadening, including finite core-hole lifetimes (∼0.1 and ∼0.5 eV at the sulfur L-and K -edges, [START_REF] Campbell | Widths of the atomic K-N7 levels[END_REF][START_REF] Nicolas | Lifetime broadening of core-excited and -ionized states[END_REF] respectively), it may realistically only be possible to resolve the differential contributions with the absolute sign given by the most intense of the transitions. Accordingly, the effective signals are therefore reduced by orders of magnitude in the X-ray region (see accumulated values in Table 4.8).

The linear absorption profiles with the full interaction and the electric-dipole approximation essentially coincide across the four spectral regions. However, as shown by the underlying oscillator Table 4.8: Beyond first-order effects in the isotropically averaged and anisotropic rotatory strengths for the lowest intense valence, L-and K-edge transitions of (CH Transition ∆E (eV) strengths in Table 4.8, non-dipolar effects at the K-edge lead to intensity redistribution among the underlying near-degenerate transitions (i.e., unrelated to the arbitrary mixing allowed for degenerate states). Nonetheless, the overall spectral profiles within and beyond the electric-dipole approximation are essentially identical because of the nearly overlapping transitions. This is consistent with our previous findings from Section 4.4. In contrast, the beyond-first-order effects become evident in the differential K-edge absorption profile because of the signed nature of the underlying quantities. This leads to a factor-of-two overestimation of the ECD within the conventional first-order treatment. Introducing third-order contributions largely corrects this discrepancy at the sulfur K-edge but going to higher orders in the expansion is not a general remedy, as will be discussed below.

f full iso f [0] iso ∆f full iso ∆f [1] iso ∆f full k ∆f [1] k valence 1 1 B (n S → σ * SS ) 4
To better understand the nature of the chiral response across spectral regions, I computed the underlying anisotropic differential oscillator strength distributions, considering now the smaller H 2 S 2 . Figure 4.18 shows the full ECD distributions (points), compared with the first-order truncated counterparts (surfaces). The solid angle represents the propagation direction, the distance from the origin (COM) indicates the magnitude of the associated signal and the color its sign. Note that different scaling factors have been applied across the transitions (see upper right corner of each subfigure). The C 2 -rotation axis coincides with the z-axis, whereas the disulfide bond is along the x-axis. The shapes of the anisotropic distributions can be understood by decomposing the firstorder signals into isotropic and d-orbital contributions (Eqn. (3.153)). The resulting orbital weights of the excitations plotted in Figure 4.18 are reported in Table 4.9.

From symmetry considerations detailed in Section D.4, it is found that contributions from d xz and d yz vanish for excitations of both A and B symmetry. In A symmetry, R [START_REF] Holmgaard List | Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory[END_REF] zz is also zero by symmetry, such that the s-and d z 2 -contributions come in a fixed ratio, giving a toroid in the xyplane. For valence and L 3 -edge excitations of A symmetry this shape is modulated by the d x 2 -y 2contribution, giving the shape of a biconcave disc elongated along the y-and x-axis, respectively, depending on its relative sign. For the L 1 -and K-edge excitations the d xy -contribution completely dominates. This is also the case for the corresponding excitations in B symmetry. It may be noted that in A symmetry the E1-M1 contribution to the d xy term is zero by symmetry, whereas in B symmetry the E1-E2 and E1-M1 contributions are of similar magnitude and the same sign. Continuing to the L 3 excitation of B symmetry, the angular plot resembles that of its counterpart in A symmetry, albeit with opposite overall sign and rotated π/2 about the molecular axes. The latter can be understood from the relative weights of d z 2 -and d x 2 -y 2 -contributions, as seen in Table 4.9. Finally, the angular plot of the valence excitation in B symmetry is a biconcave disc elongated along the z-axis, arising from positive interference between s-and d z 2 -contributions contrary to the negative interference observed for its counterpart of A symmetry. Next, I compare these conventional first-order truncated ECD distributions with their full counterparts. For the valence, L 3 -and L 1 -edge transitions, the anisotropic distributions virtually coincide, thereby confirming the validity of the first-order truncated description also for the anisotropic signal in these energy regimes. On the other hand, the full and truncated ECD distribution for the K-edge transitions are seen to have the same overall shape, but markedly different size, such that the factor-of-two overestimation at first order of the isotropic response (Table 4.8) arises largely from an overall scaling. Closer inspection of the angular distribution of the full ECD distribution reveals that the lobes are not strictly perpendicular as in a d xy -orbital, hence indicating the contributions from solid harmonics of higher even angular momentum. To investigate this further, the order-by-order contributions are provided together with the full anisotropic ECD distribution in Figure 4.19. Although ∆f [3] ( = 0, 2, 4) resembles ∆f [1] , the inclusion of higher-order solid harmonics in the former leads to non-orthogonal lobes. Furthermore, the two distributions differ by an overall sign, such that the inclusion of ∆f [3] decreases the ECD signal. A possible issue is the rate of convergence of the truncated interaction towards the full one. The BED implementation of truncated interaction for linear absorption allows us to go to arbitrary order, a unique functionality of the Dirac code. In Section 4.2, it was demonstrated that the truncated treatment converges to the full interaction upon inclusion of higher-order terms, but for higher-energy transitions (photon energies beyond ∼3728 eV) the convergence behavior was too slow for practical applications. In the present case, I assessed the convergence of the isotropic differential oscillator strength expansion at the sulfur K-edge and found that the relative error is below the threshold of 1% at 7th order (Table 4.10). Indeed, as shown in Figure 4.19 the ∆f [5] distribution ( = 0, 2, 4, 6) is minute. A similar convergence rate is found for the isotropic linear oscillator strength. From these data, it follows that the rate of convergence is acceptable for the application to the sulfur K-edge, although this conclusion may differ when performing calculations at different energy scales.

To further explore the energy window at which the truncated formalism is functional, I have performed an additional series of the isotropic differential oscillator strength for the heavier H 2 X 2 analogues (X=Se and Te). A comparison of the two pre-edge features in the L 1 -and K-edge linear and differential absorption spectra with the full and lowest-order truncated interaction across the H 2 X 2 (X=S, Se, Te) series is depicted in Figure 4.20. As a consequence of the deeper core levels of the heavier elements, the deficiency of the first-order treatment of the ECD signal now appears also at the L-edge. The discrepancy increases with the transition energy (i.e., Se L 1 -edge < S K-edge < Te L 1 -edge < Se K-edge < Te K-edge), with the Te K-edge having a vast discrepancy between the full and first-order differential oscillator strength. Similar to the findings of Section 4.2, the convergence of the multipole expansion is met with difficulties for the heavier analogues. Indeed, when considering these structural analogues, only the selenide L-edge converges at a sufficient rate, whereas it is too slow to practically converge for all other edges. In both cases, the first-order treatment not only overestimates the rotatory strengths of individual transitions but most critically it incorrectly predicts the signs of (the intense) pairs of near-degenerate transitions. However, since the relative magnitude of these oppositely signed contributions is preserved, the first-order spectral profile yields the correct overall sign (given sufficiently large Gaussian broadening). By comparing the series of H 2 S 2 and its analogues, a trend can be observed between the rate of convergence and excitation energy. Therefore, the breakdown of the first-order description at the sulfur K-edge, and not at the sulfur L-edges, is a consequence of its order-of-magnitude higher transition energy (∼215 and 2427 eV at the L 1 and K-edges).

Conclusion

In this section, I have reported the application of the anisotropic and isotropic ECD signal using the full -and truncated semi-classical light-matter interaction operator within a four-component relativistic framework. This simultaneous account of beyond-first-order light-matter interactions and relativistic effects provides two additional sources of ECD which become increasingly important at high photon energies. The linear form of the light-matter interaction operator in the relativistic domain further enabled straightforward extension to a multipole-based scheme in the velocity representation that allows for the traditional (albeit, in general, ambiguous) decomposition into electric and magnetic contributions while retaining order-by-order gauge-origin independence.

The presented approach was used to investigate the ECD response of two prototypical disulfides, H 2 S 2 and (CH 3 S) 2 , across the electromagnetic spectrum, from valence to core transitions. To quantify the implications of higher-order effects, I compared the results of the full interaction to those obtained within the traditional lowest-order non-vanishing (i.e., first-order) truncated generalized velocity representation. Going beyond the electric-dipole approximation at the sulfur K -edge leads to non-negligible intensity redistribution among near-degenerate transitions but with no visible implications on the linear absorption profile. On the other hand, the differential absorption profile is not affected by such redistribution, because of its signed nature. This leads to an overall factor-oftwo overestimation. By examining the shapes of the underlying anisotropic ECD distributions, this discrepancy is found to largely originate from an overall scaling that is corrected upon introducing third-order contributions.

Critically, the first-order treatment deteriorates at higher transition energies (beyond ω=c∼3728 eV), as can be inferred from the additional calculations involving the heavier analogues of H 2 S 2 . In these cases, this treatment may even fail to predict the sign of individual differential oscillator strengths, although this is partially alleviated if the peaks are broad enough. At such energies, going to higher orders is not a practical remedy because of the slow convergence of the truncated interaction -the full interaction is a must. iso ) at various orders for H 2 S 2 , n = 0, 1, 2, 3, compared to the result of the full interaction for the two 1s 1/2 → σ * SS transitions of A/B symmetry. The superscript notation (→m) indicates accumulated contributions up through mth order. In particular, the 'ac'-labeled row indicates the (differential) oscillator strengths accumulated through 8th (9th) order.The errors upon truncation are defined as %δ∆f (→2n+1) iso = |(∆f (→2n+1) iso -∆f full iso )/∆f full iso |×100%. The results were obtained using 4c-TD-PBE0 level of theory and the uncontracted aug-pcx-3/aug-pc3 basis set. Numbers in parentheses are exponents of 10.

4A (1s

Conclusion

The main goal of this thesis is to assess the validity of the dipole approximation in the simulation of X-ray absorption spectroscopy. Here, the term dipole approximation should be understood as an umbrella term that denotes the first-order truncated multipole expansion corresponding to the absorption of linearly polarized light (electric-dipole approximation) and electronic circular dichroism (rotational strength). To answer this question, I have presented an applied and implementation at the four component relativistic level based on either the full-or truncated light-matter interaction.

In addition, it can describe the truncated interaction in two representations, the generalized lengthand velocity representation, which become equivalent in the complete basis set limit. In summary, the following conclusions were obtained in previous chapter:

1. At high excitation energies (∼ 3728 eV), the truncated interaction converges extremely slowly towards the full interaction, which renders this scheme impractical for simulations in the hard X-ray regime.

2. The truncated interaction is more difficult to converge with respect to the basis set choice than the full interaction. The basis set convergence of the generalized velocity representation is further complicated by problems associated with linear dependence of the small component function.

3. The compactness of the core orbital counteracts the magnitude of non-dipolar effects for transitions arising from a single center, even in the hard X-ray regime. In case of the radium atom, the full interaction provides a correction of approximately (∼ 13 -20%) compared to the electric-dipole approximation.

4. In general, it seems that non-dipolar effects redistribute the intensities in a near-degenerate manifold of transitions, the effects of which are typically not observable in an absorption spectrum of linearly polarized light due to spectral broadening.

5. In the presence of symmetry equivalent centers, the transitions stemming from these centers have significant non-dipolar corrections, although the validity of this effect can be questioned due to the shortcomings of SCF theory. Furthermore, for the absorption of linearly polarized light, the changes in signal are difficult to observe if spectral broadening is included as a result of the effect outlined in previous point.

6. In case of electronic circular dichroism, non-dipolar effects where already significant for H 2 S 2 , whereas for the heavier analogues these effects were even more pronounced. Furthermore, the changes in spectral features caused by non-dipolar effects are better preserved after spectral broadening due to the signed nature of circular dichroism.

Taking these points into consideration, we arrive at the general conclusion that at least for the absorption of linearly polarized light, the electric-dipole approximation is an excellent approximation for most systems. Even in the hard X-ray regime, the electric-dipole approximation can capture the overall characteristics of the spectrum. However, to describe the dipole-allowed transitions at high-accuracy and to even include dipole-forbidden transitions, it is mandatory to go beyond the electric-dipole approximation. If this level of accuracy is required, the full interaction operator provides more reliable results than schemes based on a truncated interaction, due to the difficult convergence of the latter with respect to basis set choice and the multipole expansion. Therefore, the full interaction should be the standard in calculations that demand high accuracy.

Another exception where the electric-dipole approximation might be insufficient is given by systems with symmetry equivalent centers, although spectral broadening makes it difficult to observe these effects. Hypothetically, the individual transitions can be resolved if spectral broadening is sufficiently small. However, to the best of my knowledge, the resolution that can be achieved with modern X-ray spectrometers is not enough. Perhaps future X-ray spectrometers will be accurate enough to measure these effects. It may be also worthwhile to simulate the absorption spectra of heavier analogues of TiCl 4 , e.g. TiBr 4 or TiI 4 , whose non-dipolar corrections may be so extreme that they can even be observed with significant spectral broadening. In addition, the spin orbit splitting of the Br and I L-edge, may induce interesting effects. However, to rigorously establish the validity of these effects, current implementation needs to be extended to the MCSCF or CC level of theory.

Non-dipolar effects seem to be most relevant for electronic circular dichroism, appearing already for H 2 S 2 and being less susceptible to the spectral broadening. For the heavier analogues of H 2 S 2 , a first-order treatment makes little sense due to the strong non-dipolar effects that are induced by the high excitation energies of the core electrons. Interestingly, non-dipolar effects seem to go handin-hand with the effects of relativity, as the heavy atoms in these species demand a fully relativistic treatment. Therefore, the implementation presented in this thesis is ideal to describe these effects. However, to exclude the possibility that these conclusions are system dependent, the ECD of other systems should be studied as well. A particularity of the H 2 X 2 series is the near degeneracy of the bonding and anti-bonding core orbitals, giving rise to bidentate signals that are so close that they mostly cancel. Future systems of interest may be chiral metal acetylacetonate complexes, or for that matter any chiral single-centered metal complex.

Chapter 5 Outlook

MCSCF

In Section 4.4, it is discussed how core transitions stemming from symmetry equivalent centers give rise to elevated non-dipolar effects because the core orbitals are delocalized over these centers. However, improved ionization energies can be obtained by localizing the core orbital, which in principle should decrease non-dipolar effects albeit at the expense of the symmetry of the wave function. At the end of this section, it is also mentioned that this problem is most rigorously solved using wave function methods. Therefore, as a part of this thesis, it was originally envisioned to extend the implementation in Dirac to the MCSCF level. However, this project has not been finished. In the current Dirac code, energy optimizations at the Kramers-restricted MCSCF level are possible, although only with the inclusion of symmetry. Being based on the quaternion scheme (see Section B.3), this implementation involves more complicated equations without symmetry, explaining why to this day this functionality is not in place. To calculate absorption intensities, two key features need to be implemented: the optimization of excited states and the calculation of transition moments from these states. The former feature can be realized by implementing the NEO algorithm, as described in ref. [START_REF] Hans Jørgen | A direct, restricted-step, second-order MC SCF program for large scale ab initio calculations[END_REF]. The implementation of this algorithm heavily benefits from the existing infrastructure for energy optimization. Transition moments at this level of theory are complicated by the fact that the MCSCF wave function contains contributions from multiple determinants. It can be shown that in the corresponding transition moments, the only non-vanishing contributions involve combinations of determinants that differ at most by one orbital. Transition moments at the SCF level do not require these considerations as these methods are based on single determinants.

Core-Hole Relaxation at the DFT Level

In the introduction of this thesis, it was argued that the description of the core-hole is important for X-ray spectroscopy. Typically, the relaxation effects arising from this hole, may significantly shift the energies in a spectrum. These effects can be taken into account using the static exchange approximation (STEX) [START_REF] Hans Ågren | Direct, atomic orbital, static exchange calculations of photoabsorption spectra of large molecules and clusters[END_REF][START_REF] Hans Ågren | Direct SCF direct static-exchange calculations of electronic spectra[END_REF]. STEX can be thought of as CI singles using a core-hole reference state. In the current state of Dirac, it is possible to perform calculation of this type using a reference state described at the HF level of theory. [START_REF] Ekström | Relativistic four-component staticexchange approximation for core-excitation processes in molecules[END_REF] However, within this reference, dynamic correlation is not taken into consideration. Furthermore, due to the choice of reference state, the excited state obtained with STEX is not necessarily orthogonal to the HF ground state, making the formalism somewhat more cumbersome.

Therefore, another side project in this thesis was to implement STEX at the DFT level of theory. To avoid the complications associated to non-orthogonality, a scheme was going to be implemented that uses the core-orbitals from the core-hole reference state. The main assumption of this scheme being that the core orbitals are not easily polarizable and thus relatively unchanged in the neutral and core-hole states. The implementation of this scheme heavily leans on the existing Tamm-Dancoff approximation code. However, this project never really took off due to an unresolved bug in this code.

X-ray Magnetic Circular Dichroism

Knowing that non-dipolar effects play an important role in electronic circular dichoism, it seems straightforward to extend the BED implementation to describe magnetic circular dichroism. This feature has already been implemented by Foglia et al., [START_REF] Nicolás | Going beyond the electric-dipole approximation in the calculation of absorption and (magnetic) circular dichroism spectra including scalar relativistic and spin-orbit coupling effects[END_REF] although in a non-relativistic framework. To generalize it to a relativistic framework, there are two possible routes: either quadratic response is applied, where the static magnetic field is an additional perturbation, or the magnetic field is included during the SCF cycles. Presumably, the isotropic averaging within the latter scheme cannot be carried out in a very elegant manner. The most straightforward method to perform this averaging is to put all possible directions of the magnetic field on a Lebedev grid. However, this requires a separate SCF calculation for every grid point, making the calculation rather expensive. Another challenge associated with this approach is that the magnetic field breaks time-reversal symmetry, implying that the formalism is more involved. Alternatively, this quantity is calculated using quadratic response, which in principle, is possible in Dirac. The remaining task would then be to identify which response function exactly has to be calculated, making this method the preferred one.

Full Interaction Operator in the Length Representation

Although the full interaction operator has many advantages, it does not explicitly distinguish between electric-and magnetic interactions. In multipolar gauge, this separation is inherent to the formulation, although it is formulated as a truncated expansion, which is accompanied by a plethora of problems (see Section 3.5). It is possible, however, to get the best of both worlds by utilizing the integral representation of multipolar gauge, where the gauge function is given by [START_REF] Brittin | Poincaré gauge in electrodynamics[END_REF][START_REF] Donald | Gauge transformations and the electric dipole approximation[END_REF]

χ a (r, t) = 1 0 dλδ a • Ã(λδ a + a, t) (5.1)
and corresponding potentials

φ a (r, t) = φ(a, t) -δ a • 1 0 dλE(λδ a + a, t) (5.2)
A a (r, t) = -1 0 dλλ δ a × B(λδ a + a, t) .

(5.

3)

The integral representations of these potentials are completely equivalent to Eqs. (3.71) and (3.77), which can be confirmed by expanding the position argument of above integrands around a. An advantageous feature of the integral representation is that it does not involve any truncation, thus being gauge-origin independent, as demonstrated in Section 3.5. The implementation of the absorption intensities derived from these potentials presumably shares some similarities with the implementation of the full interaction, although the integrals are different.

the successive Lorentz transformations, also referred to as Thomas precession. The electric field appearing in this expression can be rewritten in terms of the central potential

E = -∇φ(r) = - r r dφ(r) dr , (B.7)
which gives rise to the following expression for the magnetic field

B = - 1 2m e c 2 dφ(r) dr ; = r × p. (B.8)
The interaction between the magnetic field and the magnetic moment of the electron spin, μs = g e e 2me ŝ, are described by the operator where g e ≈ 2 is the electron g-factor. Typically, these effects can be captured by introducing above operator as a perturbation.

ĤSO = -μs • B = ξ(r) ˆ • ŝ; ξ(r) = g e e 4m 2
As alluded to before, the Dirac equation already has these effects build in its formulation. To understand this better, consider the time-dependent Dirac equation under the influence of the Coulomb potential from the nuclei. Transforming this equation from the molecular frame to a moving frame yields the following result

i γ µ ∂ µ -m e c + e c γ 0 φ nucl ψ = 0 Λ -→ i γ µ ∂ µ -m e c + eγ µ A µ ψ = 0, (B. 10
)
where ψ is the four-component wave function in the transformed frame. Although this transformation is far from the consecutive transformations as described by Thomas, the above example is sufficient to illustrate that a moving frame introduces magnetic interactions that are absent in the molecular frame. Furthermore, due to its relativistic character, the Dirac equation is equally valid in any inertial frame. Therefore, in the molecular frame, spin-orbit coupling is taken into consideration without introducing any additional perturbing operator.

B.2 Negative Energy Solutions

In this appendix, I will demonstrate that the action of the charge conjugation operator on the four component wave function yields the result provided by Eqn. where the following relation holds due to the non-commutativity b j = jb * . Returning to our eigenvalue problem, the following unitary quaternion matrix ought to be applied

U = 1 √ 2 I jI jI I , (B.23)
to block-diagonalize our matrix

U † ΩU = A + B j 0 0 -k(A + B j) k (B.24)
thereby reducing the eigenvalue problem to the half of its original dimension

A + B j c α -c β * j = c α -c β * j . (B.25)
By using the quaternion scheme, it becomes thus possible to benefit optimally from the reductions provided by time-reversal symmetry. In the quaternion representation the operator associated to this symmetry, can be represented by j [START_REF] Saue | Quaternion symmetry in relativistic molecular calculations: The Dirac-Hartree-Fock method[END_REF]

j c α -c β * j = c α * j + c β . (B .26)
In the following, it will also be illustrated how the quaternion scheme can be used to introduce point-group symmetry to relativistic calculations. However, before arriving at that point, let us see how these techniques carry over to the Dirac equation. For these purposes, it proves to be useful to reorder the four-component wave function

ψ L ψ S =     ψ Lα ψ Lβ ψ Sα ψ Sβ     →     ψ Lα ψ Sα ψ Lβ ψ Sβ     = ψ α ψ β . (B.27)
In this basis, the time-reversal operator can be represented as

K = -i σ y ⊗ I 2 K0 (B.28)
from which we obtain

Kψ =     -ψ Lβ * -ψ Sβ * ψ Lα * ψ Sα *     . (B.29)
If we assume the absence of external magnetic fields, the Dirac Hamiltonian in the reordered basis reads

ĥD =     V -ic∂ z 0 -ic∂ - -ic∂ z -2mc 2 + V -ic∂ - 0 0 -i∂ + V ic∂ z -ic∂ + 0 ic∂ z -2mc 2 + V     ; ∂ ± = ∂ x ± i∂ y (B.30)
particular Eqn. (B.3), it can be shown that the symmetry content of the large-and small component reads

Γ L = (Γ 0 , Γ Rz ) (Γ Ry , Γ Rx ) ⊗ Γ φ ; Γ S = (Γ xyz , Γ z ) (Γ y , Γ x ) ⊗ Γ φ = Γ L ⊗ Γ xyz , (B.35)
where Γ φ corresponds to a phase factor that depends on whether the spinor is gerade/ungerade Γ 0 /Γ xyz , or barred/unbarred Γ 0 /Γ Ry . This phase can also correspond to the irreps of the rotations or Cartesian coordinates, but in these cases the components in Eqn. (B.35) are merely reordered, thus preserving the overall symmetry structure. By applying this structure to the quaternion wave function, we obtain

Γ Q ψ = Γ Q ψ L Γ Q ψ S = Γ 0 , Γ Rr Γ xyz , Γ r ⊗ Γ φ . (B.36)
It can thus be concluded that in this scheme, the function of the quaternions is twofold: it preserves the structure of time-reversal symmetry, while simultaneously each quaternion unit points to a certain boson irrep. Within the quaternion scheme, the basis set expansion of an orbital can be expressed as

Q ψ i = χ L 0 0 χ S Q c L i Q c S i ; Q c X = c X 0 + ȋc X 1 -jc X 2 + kc X 3 (X = L, S), (B.37)
where the coefficients are now given by quaternion numbers. For a symmetry-adapted basis, our orbital is given by the expression

Q ψ k = i χ L i (Γ i ) c L 0;i (Γ 0 ) + c L 1;i (Γ Rz ) ȋ -c L 2;i (Γ Ry ) + c L 3;i (Γ Rx ) k j χ S j (Γ j ) c S 0;j (Γ xyz ) + c S 1;j (Γ z ) ȋ -c S 2;j (Γ y ) + c S 3;j (Γ x ) k , (B.38)
where the coefficients vanish if the symmetry of the basis function does not agree with the coefficient, e.g. c L 0;i (Γ 0 ) = 0 if Γ 0 = Γ i . For the C 2v , D 2 and D 2h point groups, this structure implies that the coefficient of each basis function only has contributions from one quaternion unit. Following a rather technical procedure that is described with more detail in ref. [START_REF] Saue | Quaternion symmetry in relativistic molecular calculations: The Dirac-Hartree-Fock method[END_REF], it can be shown that this single quaternion cancels with the quaternion units in the Dirac Hamiltonian, hence forming a single real matrix. Likewise, for point groups of lower symmetry, such as C 2 , C s and C 2h , each coefficient has contributions from two quaternion units. From a similar procedure, it follows that we are left with a complex matrix representation of the Dirac Hamiltonian. For the C 1 and C i point groups, however, the coefficients have contributions from all quaternion units, thus implying that the matrix representation keeps its quaternion structure. Therefore, point-groups of higher symmetries are accompanied by a reduction of the algebra, which simplifies the calculation significantly.

Using this scheme, it is possible to re-express the familiar quantities from chapter 1 in quaternion form. Some examples are: the one-body density matrix

Q D pq = D pq + D pq j, (B .39)
the Fock matrix

Q F pq = F pq + F pq j (B.40)
which generally can be understood as irreducible representations of the full rotation group. However, to fully exploit the restrictions imposed by spherical symmetry, we should also express the operator in the transition moment in terms of this symmetry. The irreducible tensor operators generally obey this condition. An irreducible tensor operator of rank k, comes with 2k + 1 components

T k = {T k q | q = k, k -1, • • • , -k}, (C.12)
where each component obeys the commutator relations

[ ĵz , T k q ] = mT k q ; [ ĵ± , T k q ] = k(k + 1) -q(q ± 1)T k q±1 . (C.13)
In this expression, I have introduced the ladder operators ĵ± = ĵx ± i ĵy , which deserve their name from their action on angular momentum eigenstates

ĵ± |jm = j(j + 1) -m(m ± 1)|jm ± 1 . (C.14)
The symmetry reductions arising from irreducible tensor operators become manifest in the Wigner-Eckart theorem

jm|T k q |j m = (-1) j-m j k j -m q m (j||T k ||j ), (C.15)
where the integral on the right-hand-side is independent on the orientation of the system, i.e. the values of m, m and k. Furthermore, the prefactor of this integral is given by the 3j-symbol, which can be related to the Clebsch-Gordan coefficients (Eqn. (2.93)) according to

j 1 m 1 j 2 m 2 |j 1 j 2 ; jm = (-1) j 1 -j 2 -m 2j + 1 j 1 j 2 j m 1 m 2 -m = C j 1 j 2 j m 1 m 2 m . (C.16)
The 3j-symbol mediates all selection rules, only being non-zero if the following conditions apply

m 1 + m 2 = m (C.17) |j 1 -j 2 | ≤ j ≤ j 1 + j 2 j 1 + j 2 + j ∈ Z.
Typically, the j-dependent integral is found by exploiting the fact that it is the same for all components of T k . For example, by expressing the angular momentum operator as a irreducible tensor operator, ĵ1 = {-1 In the following, I will be treating operators that are derived from cα • A, where cα acts on spin space, whereas A on position space. It will thus be useful to combine two irreducible tensor operators S k 1 and T k 2 , which may act on different spaces, into a new spherical tensor operator

[S k 1 ⊗ T k 2 ] k q = k 1 q 1 =-k 1 k 2 q 2 =-k 2 C k 1 k 2 k q 1 q 2 q S k 1 q 1 ⊗ T k 2 q 2 ; q = k, k -1, • • • , -k; |k 1 -k 2 | ≤ k ≤ k 1 + k 2 .
(C.20) As a first example, I will apply this machinery in the derivation of electric-dipole selection rules. For these purposes, I will conveniently express the electric-dipole moment operator in spherical basis Within this basis, we can rewrite this operator as an irreducible tensor operator of rank 1

Q [1] = - 4π 3
Q [1] = -erC 1 ; C 1 = { 4π 3 Y 1-1 , 4π 3 Y 10 , 4π
-jm|erC 1 q |j m = (-1) j-m j 1 j -m q m (j||C 1 ||j ). (C.23)
From Eqn. (C.17), the 3j-symbol yields non-zero results if

m + q = m (C.24) |j -1| ≤ j ≤ |j + 1| j + j + j ∈ Z.
Therefore, the selection rules of electric-dipole transitions are ∆j = ±1, 0; ∆m = ±1, 0, with the caveat that any transition of the type j = 0 → j = 0 are forbidden, as can be inferred from the 3j-symbol. Exactly the same selection rules apply to the velocity representation, as we can express the α matrices in irreducible tensor form as well

α 1 = {- 1 √ 2 α -, α z , 1 √ 2 α + }; α ± = α x ± iα y , (C.25)
which has the same rank as Eqn. (C.22), thus suggesting that the same selection rules apply. Deriving selection rules for magnetic-dipole transitions is more difficult, as this operator has a more complicated structure, being defined as the cross-product between the position and current vector. It is possible, however, to apply the same procedure as the one outlined before. Again, we have to express the relativistic magnetic-dipole moment operator as an irreducible tensor operator. In other words, we first have to express this operator in spherical basis, which can be achieved by evaluating the cross-product of Eqn. (C.22) and (C.25). From [START_REF] Grant | The Dirac Equation[END_REF]Equation B.3.149], it follows that the cross-product of two first rank spherical tensor can be expressed as

(a 1 × b 1 ) q = -i √ 2[a 1 ⊗ b 1 ] 1 q = -i √ 2 q 1 q 2 q C 111 q 1 q 2 q a 1 q 1 b 1 q 2 , (C.26)
where in the second equality, I have used the composition rule for two spherical tensors (Eqn. (C.20)). Expressed in spherical basis, the transition moment stemming from the magnetic-dipole moment can be expressed as jm| m [START_REF] Holmgaard List | Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory[END_REF] q

|j m = i √ 2 jm|er[C 1 ⊗cα 1 ] 1 q |j m = (-1) j-m i √ 2 j 1 j -m q m (j||er[C 1 ⊗cα 1 ] 1 ||j ).
(C.27) Therefore, the magnetic-dipole selection rules are the same as for the electric-dipole: ∆j = ±1, 0; ∆m = ±1, 0(j = 0 j = 0). Alternatively, selection rules can be derived in terms of the orbital angular momentum of the large-and small component (∆ = 0, 2). However, this approach will not be pursued here, as the orbital angular momentum is not a good quantum number, which makes these selection rules less useful in current context. For a full derivation, see ref. [START_REF] Kaniauskas | Relativistic investigation of electronic transitions in many-electron atoms[END_REF].

It should be noted, however, that these selection rules drastically differ from the ones derived in a non-relativistic framework, where the magnetic-dipole moment assumes the more simple form m [START_REF] Holmgaard List | Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory[END_REF] nr = e 2mc ˆ . Further recalling that the non-relativistic one-electron atomic states can be expressed as ψ n m (r) = 1 r R n (r)Y m (θ, φ), the selection rules follow from

ψ n i i m i | e 2mc ˆ |ψ n f f m f = ∞ 0 drR n i i R n f f π 0 dθ sin θ 2π 0 dφY * m i i e 2mc ˆ Y m f f , (C.28)
where I have used the fact that the non-relativistic magnetic-dipole moment operator is purely angular. By expressing the angular momentum operator in spherical basis, Eqn. (C.14) can be exploited. Orthogonality relations subsequently give the following selection rules ∆n = 0; ∆ = 0, which suggest that magnetic-dipole transitions can only occur inside the manifold of quasidegenerate states split by spin-orbit coupling. As noted in Section 4.5, this difference in selection rules can have severe consequences on the magnitude of the first-order isotropic differential oscillator strength. Although I will not show it explicitly here, the electric-quadrupole selection rules can be derived by expressing this operator in terms of the following irreducible tensor operator of rank 2:

C 2 = { 4π 5 Y 2-2 , 4π 5 Y 2-1 , 4π 5 Y 20 , 4π 5 Y 21 , 4π 5 Y 22 }. (C.29)
From the Wigner-Eckart theorem it can thus be concluded that its selection rules are given by ∆j = ±2, ±1, 0; ∆m = ±2, ±1, 0(j = 0 j = 0). Using the procedure outlined above, it is in principle possible to find selection rules for multipole moments to arbitrary order.

In the following, however I will use the machinery of irreducible tensor operator to assess the selection rules of the full interaction operator, which I will conveniently expand using a plane-wave expansion where the α matrix is expressed in spherical basis. Starting from the irreducible tensor operator

X k q = [α (1) ⊗ C (l) ] k q = k 1 q 1 =-k 1 k 2 q 2 =-k 2 C 1lk q 1 q 2 q α q 1 C (l) q 2 , (C.31)
it can be shown that the full interaction operator can be expressed in terms of this operator by multiplying above operator with C 1 k pmq and exploiting the orthogonality of the Clebsch-Gordan coefficients

|j 1 +j 2 | j 3 =|j 1 -j 2 | j 3 m 3 =-j 3 C j 1 j 2 j 3 m 1 m 2 m 3 C j 1 j 2 j 3 m 1 m 2 m 3 = δ m 1 m 1 δ m 2 m 2 , (C.32)
where the final result can be identified with the terms appearing in Eqn. (C.30)

| +1| k=| -1| k q=-k C 1 k pmq [α (1) ⊗ C ] k q = k,q k 1 q 1 =-k 1 k 2 q 2 =-k 2 C 1 k pmq C 1 k q 1 q 2 q α q 1 C q 2 (C.33) = k 1 q 1 =-k 1 k 2 q 2 =-k 2 δ pq 1 δ mq 2 α q 1 ⊗ C q 2 = α p C m .
Therefore, the full interaction operator can be expressed as

Tp = e ω cα p e ik•r = √ 4π e ω ∞ =0 m=- √ 2 + 1i j (kr)Y m * (e k ) | +1| k=| -1| k q=-k C 1 k pmq X k q . (C.34)
Due to the complexity of this expression, which has contributions from nearly all irreps, it may seem as if there are no selection rules associated to this operator. However, upon closer inspection, it follows that the index k assumes values between 1 and ∞, which in turn implies that a transitions between two states of zero angular momentum always vanishes. Therefore, the selection rules j = 0 j = 0 seems to be a universal rule that applies throughout each order of the multipole expansion.

C.3 Manipulations of Summation Indices

In this appendix, I will illustrate several useful techniques to rewrite summation indices. For these purposes, let A be a square matrix of dimension N and suppose that we want to calculate its sum over all elements Written in this form, the summation is carried out row-wise, whereas we could also have defined it column-wise Typically, once the transformation has been completed, the prime on the index m is dropped, yielding the relation

N n=0 N -l m=l A n,m-l (C.43)
Another useful example is the reversal of a summation, which can carried out using the substitution

m → m = N -m, (C.44)
and inserting the relation

m = N -m N n=0 N N -m =0 A n,N -m = N n=0 0 -m =-N A n,N -m = N n=0 0 m =N A n,N -m = N n=0 N m =0 A n,N -m , (C.45)
where the prime on the indices can be dropped afterwards.

C.4 Restricted Excitation-Window Approach

The aim of this appendix is to briefly discuss the restricted excitation-window approach. In Section 3.6.2, the conditions were given that indicate whether the generalized length-and velocity representation are equivalent at the SCF level of theory. As a side note, I argued that this equivalence is broken if the restricted excitation-window approach is applied. Since this approach takes such an important position in this work, I will devote this section to the discussion of its principles.

In Chapter 1, we have learned that solution vectors, and hence transition moments, are obtained by solving the generalized eigenvalue problem from Eqn. (1.114). It was also noted that the Hessian matrix appearing in this problem is typically much too large to store, thus requiring us to project it onto a set of trial vectors and solve it in this reduced basis. In the context of core transitions, such methods are cumbersome because they are generally bottom-up approaches. Therefore, before obtaining the core transition, all transitions of lower excitation energies need to be calculated first.

To solve this problem, the orbital rotation operator (Eqn. (1.85)) is restricted, such that the occupied part only contains contributions from the core orbitals. For example, if we want to study the K-edge of the radium atom, the orbital rotation operator assumes the form

κ(t) = a κ ai 1s â † a âi 1s -κ * ai 1s â † i 1s âa , (C.46)
where the Einstein summation convention is not applied and i 1s is the index corresponding to the 1s 1/2 orbital of the radium atom. This procedure is referred to as the restricted excitationwindow approach. [START_REF] Stener | Time dependent density functional theory of core electrons excitations[END_REF][START_REF] South | 4-Component relativistic calculations of L 3 ionization and excitations for the isoelectronic species UO 2 2+ , OUN + and UN 2[END_REF] Closely related is the core-valence separation scheme, which applies to EOM-CC. [START_REF] Cederbaum | Many-body theory of core holes[END_REF][START_REF] Halbert | Relativistic EOM-CCSD for Core-Excited and Core-Ionized State Energies Based on the Four-Component Dirac-Coulomb (-Gaunt) Hamiltonian[END_REF] Approximation schemes of this type are justified because core transitions typically do not contain significant contributions from occupied valence orbitals. These methods thus allow to selectively study transitions stemming from the core shell. However, due to the nature of the restricted excitation-window approach, the terms appearing in Eqn. (3.107) do not cover all occupied-virtual/virtual-orbital pairs, thus breaking equivalence of the generalized length-and velocity representation. the kinetic balance condition is not met (see Section 2.4.2). In radial form, a kinetically balanced basis assumes the form[90]

R S ∝ √ 2 + 3R α, L +1 -2 √ 2 + 1R α, L -1 for κ L = L > 0 R α, L +1 for κ L = -( L + 1) < 0, (D.3)
As an example, the small component S = 1 can be generated either from L = 0 or L = 2, where the latter function is to be used for p 1/2 and the former for p 3/2 . In the atomic case, the 2-component basis functions do not mix since they have different angular parts, corresponding to κ S = +1 and κ S = -2, respectively. However, in the Dirac code, a scalar basis is employed

G α m = R α (r)Y m (θ, φ), (D.4)
Small component basis functions with radial parts corresponding to both forms of Eqn. (D.3) are generated, hence adhering to kinetic balance. However, now, in the case of e.g. S = 1, basis functions can contribute to both p 1/2 and p 3/2 , hence possibly amplifying linear dependencies.

Figure D.2:

Radial functions of the large and small component of the 1s 1/2 ( r = 0.01454 a 0 )and 7p 1/2 orbital (top row) and their deviations from the numerical reference (bottom row). These orbitals were taken from the radium atom with [Ra]1s -1 1/2 7p 1 1/2 configuration. The basis set orbitals were calculated at the 4c-HF level using Dirac, while the numerical reference was calculated with GRASP. These calculations where carried out using the default linear dependence threshold: 10 -6 and 10 -8 for the large and small component space. Note that the scaling is different for each individual box and that the error curves are plotted in a different range than the radial functions.

D.2 Augmented Basis Sets

In Section 4.3, I have investigated the absolute basis set convergence of the full interaction operator and the various multipole moments. In general, we have observed that the high-order multipole moments require at least the dyall.ae4z basis set. However, compared to the dyall.ae2z basis set, the dyall.ae4z basis contains additional tight functions, in addition to functions of higher orbital angular momentum, that may not contribute to the high-order multipole moments. Therefore, I investigated the effects of augmenting the dyall.ae2z basis set, such that a better balance could be found between accuracy and basis set size. I have employed two schemes to construct a new basis set: I Adding diffuse functions to the dyall.ae2z basis set in an even-tempered fashion.

At the three highest orders, the augmented basis sets completely fail to capture the shape of the reference due to severe oscillations, despite the use of tight thresholds for linear dependence. It can thus be concluded that the basis sets from Scheme I are not a proper substitute for the dyall.ae3z and dyall.ae4z basis sets. Figures D.5 and D.6 depict the results from Scheme II for the length representation electric and magnetic transition multipole moments, respectively. The basis sets constructed with this scheme provide no obvious improvement over the dyall.ae2z basis sets. Therefore, Scheme II is not a suitable alternative to the dyall.ae3z and dyall.ae4z basis sets.

D.3 H

D.4 Analysis of the Rotational Strength Tensor

Elements of the rotational strength tensor of Eqn. (3.152) of the main text, using the Einstein summation convention, can be split into the electric-dipole/electric-quadrupole (E1-E2) and electricdipole/magnetic-dipole (E1-M1) contributions

R E1-E2 ij = 4m e ω 2 ce 2 ε ipq f | Q[2] j;p |i f | Q[1] q |i (D.5) R E1-M1 ij = 4m e ω 2 ce 2 δ ij f |- i ω m[1] q |i f | Q[1] q |i -f |- i ω m[1] i |i f | Q[1] j |i (D.6)
and analyzed in terms of symmetry. The target molecules, in the chosen geometries, have C 2 symmetry. Fixing the origin at the center of mass we find that for excitations of irrep A the rotational strength tensor has the structure

R E1-E2 A = 4m e ω 2 e 2 c    1 2 Q [2] xy Q [1] z 1 2 Q [2] yy Q [1] z 0 -1 2 Q [2] xx Q [1] z -1 2 Q [2] yx Q [1] z 0 0 0 0    (D.7) R E1-M1 A = 4m e ω 2 e 2 c    -i ω m [1] z Q [1] z 0 0 0 -i ω m [1] z Q [1] z 0 0 0 0    (D.8)
where a shorthand notation is used for transition moments, e.g. Q

-1 2 Q [2] xz Q[1] y -1 2 Q [2] yz Q [1] y 0 1 2 Q [2] xz Q [1] x 1 2 Q [2] yz Q [1] x 0 0 0 1 2 Q [2] zx Q [1] y -1 2 Q [2]
zy Q [START_REF] Holmgaard List | Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory[END_REF] x    (D.9)

R E1-M1 B = 4m e ω 2 e 2 c    -i ω m [1] y Q [1] y i ω m [1] x Q [1] y 0 i ω m [1] y Q[1]
x -i ω m

[1]

x Q

[1] x 0 0 0 -i ω m

[1]

x Q [START_REF] Holmgaard List | Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory[END_REF] x -i ω m

[1]

y Q It should be noted that in both irreps the elements R

[1] xz , R [1] yz , R [1] zx , R [1]
zy are strictly zero by symmetry; in the A irrep elements R [START_REF] Holmgaard List | Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory[END_REF] zz are zero as well. des seuils seuls, mais plutôt à partir de leurs régions voisines. Dans la région au-delà du seuil, on peut observer des oscillations subtiles, induits par des interférences de photoélectrons qui sont rétrodiffusés par leur environnement local. La technique de mesure de ces oscillations, également appelée structure fine d'absorption des rayons X étendue (EXAFS), nous fournit des informations sur la structure cristalline de l'échantillon. Cependant, cette thèse sera consacrée plutôt au spectroscopie de structure près du front d absorption de rayons X, qui peut être sondé à l'aide de la structure fine d'absorption des rayons X en bordure (NEXAFS). Dans cette plage d'énergie, les rayons X ont encore suffisamment d'énergie pour interagir avec les électrons internes, bien qu'ils soient excités vers des niveaux liés et vides. Étant donné que les orbitales internes tendent à être localisées autour des noyaux, le NEXAFS fournit des informations spécifiques au site sur le système à l'étude, formant ainsi un complément à la spectroscopie UV-Vis, qui implique des excitations de valence qui sont délocalisées sur toute la molécule. Dans sa forme la plus simple, le NEXAFS est basé sur la lumière polarisée linéairement, bien que cette procédure ait un point faible critique : elle ne distingue pas entre les membres d'une paire énantiomère. Pour faire cette distinction, il faut mesurer son dichroïsme circulaire électronique (ECD), qui est défini comme la différence d'absorption entre la lumière polarisée circulairement à gauche et à droite. En général, cette quantité est mesurée dans la plage UV-Vis, bien qu'elle se manifeste également dans la gamme des rayons X, où elle est appelée dichroïsme circulaire naturel des rayons X (XNCD) par convention. Le XNCD présente les mêmes avantages que le NEXAFS ordinaire en ce sens qu'il fournit plus d'informations locales par rapport à l'ECD dans la plage UV-Vis.

Cependant, la création de lumière polarisée circulairement à des fréquences élevées peut être assez difficile, car les rayons X sont notoirement difficiles à manipuler à l'aide d'éléments optiques conventionnels. De nos jours, cependant, cet exploit technique peut être réalisé de manière routinière grâce aux récents développements de la radiation synchrotron. Initialement produit en tant que sous-produit de la physique des particules, ces dispositifs sont des accélérateurs de particules circulaires qui utilisent des champs magnétiques pour contraindre les électrons à des orbites circulaires dans leur intérieur creux. Les rayons X peuvent être extraits de ces électrons en appliquant un champ magnétique alternatif supplémentaire qui est perpendiculaire à la trajectoire des électrons. Les oscillations induites des électrons donnent naissance à une source cohérente de rayons X qui permet un contrôle complet de la polarisation. Cela rend la radiation synchrotron particulièrement utile pour effectuer des expériences NEXAFS. D'autres techniques alternatives qui ont gagné en popularité sont basées sur les lasers à électrons libres ou la génération d'harmoniques élevées.

Même si de nombreuses informations utiles peuvent être extraites de ces expériences seules, l'attribution et l'interprétation des spectres nécessitent des apports de la théorie et de la simulation. Par exemple, la stéréochimie absolue des molécules chirales ne peut être déterminée que lorsque le XNCD est comparé à des résultats équivalents issus de simulations chimiques quantiques. De plus, en conjonction avec la théorie, le NEXAFS peut être utilisé pour déterminer l'état d'oxydation et l'environnement de coordination local des complexes métalliques. Par conséquent, les avancées dans les techniques expérimentales doivent aller de pair avec le développement de la théorie et de ses mises en oeuvre.

Pour simuler la spectroscopie des rayons X, il semble naturel d'appliquer la théorie bien établie développée pour le régime UV-Vis. Cependant, une grande disparité avec l'expérience est observée si l'on transfère sans réflexion cette théorie au domaine des rayons X. Pour corriger cette disparité, il est donc nécessaire d'inclure plusieurs effets au-delà de ce qui est généralement requis pour la spectroscopie UV-Vis. Tout d'abord, puisque les effets relativistes sont générés dans la région du coeur, ils doivent être pris en compte dans la simulation des excitations du coeur même pour des éléments assez légers. Deuxièmement, après l'excitation de l'électron du coeur, le système est laissé avec un trou qui réduit l écrántage de la charge nucléaire. En conséquence, la charge nucléaire effective augmente, ce qui équivaut à une stabilisation supplémentaire de l'état excité, appelée relaxation du trou du coeur. Troisièmement, dans le régime UV-Vis, la longueur d'onde est considérablement plus grande que l'étendue de la molécule, ce qui suggère que les champs semblent uniformes du point de vue de la molécule. En raison de cette différence d'échelle, l'interaction lumière-matière peut être effectivement décrite par les premiers termes d'une expansion multipolaire, ce qui donne l'approximation électrique dipolaire pour la lumière polarisée linéaire et la force de rotation pour le dichroïsme circulaire, cette dernière dépendant à la fois du moment dipolaire électrique et magnétique. Dans les deux cas, seules les contributions dipolaires doivent être prises en compte. Il semble donc naturel d'utiliser l'approximation dipolaire comme terme générique pour désigner ces deux schémas. Dans le régime des rayons X, cependant, l'approximation dipolaire est remise en question.

Dans cette thèse, j étudiais si l'approximation dipolaire est valable dans la simulation de la spectroscopie d'absorption des rayons X.

En général, il existe deux approches possibles pour inclure les effets non dipolaires : soit l'interaction lumière-matière semi-classique est traitée de manière exacte, soit l'expansion multipolaire est tronquée au-delà de l'ordre zéro. Cependant, les expansions tronquées introduisent inévitablement une dépendance à l'origine de jauge dans le calcul, un problème que Bernadotte et al. ont résolu en développant la section efficace d'absorption plutôt que l'interaction lumièrematière elle-même. De plus, des schémas de ce type peuvent également donner de sections efficaces d'absorption négatives et de divergences apparentes, qui sont, en principe, toutes deux atténuées si suffisamment de termes sont inclus dans l'expansion multipolaire, bien que le dernier problème puisse nécessiter une quantité déraisonnablement de termes. Il existe plusieurs autres exemples d'implémentations, en plus de celles des références susmentionnées, qui sont capables de calculer les intensités d'absorption en utilisant soit une interaction tronquée, soit l'interaction complète. Certains de ces schémas ont même été étendus pour modéliser des phénomènes plus complexes tels que le dichroïsme circulaire magnétique des rayons X, la diffusion des rayons X ou l'absorption transitoire. Cependant, toutes ces implémentations ont une plage spectrale d'application limitée, car elles sont dérivées d'un cadre non relativiste, ayant au plus des corrections relativistes perturbatives. Dans le régime des rayons X durs, où les effets non dipolaires sont les plus pertinents, la lumière excite généralement les orbitales du coeur des éléments lourds, ce qui nécessite un traitement entièrement relativiste. Pour répondre à la question principale d'une manière qui s'applique également au régime des rayons X durs, il est donc impératif de travailler dans un cadre relativiste.

Dans cette thèse, je présenterai une implémentation pour simuler les interactions lumière-matière en utilisant à la fois l'interaction complète et tronquée dans un cadre relativiste à quatre composants. Cette implémentation fait partie de Dirac, un code de chimie quantique spécialisé dans les calculs moléculaires relativistes à deux ou quatre composants. Cependant, il convient de noter que même ce niveau de théorie a une portée limitée. À des échelles d'énergie extrêmes, telles que celles rencontrées dans les cations fortement chargés, les effets de l'électrodynamique quantique peuvent jouer un rôle dans les spectres d'absorption des rayons X. De plus, je n'ai pas inclus les effets du trou du coeur, dont l'importance a été évoquée dans le paragraphe précédent. Cependant, l'inclusion de ces effets dépasse le cadre de cette thèse et ne sera pas poursuivie plus avant.

E.2 Chapitre 2

Les électrons de coeur sont généralement pas pris en compte dans la formation de liaisons, car ils ne sont pas facilement polarisés et conservent ainsi largement leur caractère atomique. Cependant, en spectroscopie des rayons X, les électrons de coeur jouent un rôle central, car ces expériences fonctionnent à des énergies suffisamment élevées pour sonder ces électrons. La simulation de la spectroscopie des rayons X est confrontée à des difficultés, car la théorie conventionnelle de la spectroscopie UV-Vis (voir le chapitre 1) ne peut pas reproduire avec précision les expériences des rayons X. Au moins certaines de ces lacunes sont expliqués dans les caractéristiques spéciales des électrons de coeur. En raison de leur proximité avec le noyau, les électrons de coeur subissent une attraction beaucoup plus forte que les électrons de valence, augmentant ainsi considérablement leur vitesse. Un effet analogue se produit dans le système solaire, où la planète Mercure orbite autour du soleil à des vitesses beaucoup plus élevées que les planètes plus éloignées. En général, la vitesse des électrons augmente avec un nombre atomique plus élevé. Déjà pour des éléments de taille modérée, les électrons de coeur se déplacent si rapidement que la mécanique quantique classique échoue à prendre correctement en compte leur mouvement. Cependant, pour les éléments dans la partie basse du tableau périodique, les influences des effets relativistes s'étendent au-delà du coeur, car les orbitales de tous les autres électrons doivent rester orthogonales aux orbitales de coeur, contractant ainsi efficacement la couche de valence. Pour obtenir ne serait-ce qu'une fonction d'onde qualitativement correcte des molécules contenant de tels éléments, les effets relativistes doivent être inclus.

Avec la mécanique quantique, la relativité restreinte a révolutionné le paradigme de la physique moderne. Cette dernière théorie exige une réévaluation drastique des concepts d'espace et de temps, bien qu'elle ne gagne de la pertinence qu'à des échelles d'énergie extrêmement élevées. Cependant, cette théorie est classique dans le sens où elle concerne le mouvement déterministe de particules ponctuelles, tandis que la mécanique quantique est fondamentalement de nature probabiliste. Pour inclure les effets de la relativité dans nos calculs, elle doit être étendue au domaine de la mécanique quantique. Heureusement, ce travail a déjà été fait pour nous par Paul Dirac, qui a formulé une extension relativiste de l'équation de Schrödinger. Les mérites de l'équation de Dirac résident dans ses valeurs d'énergie plus précises, sa description naturelle du spin et du couplage spin-orbite, ainsi que sa prédiction de l'antimatière. Comme mentionné précédemment, l'équation de Dirac est particulièrement utile pour décrire des propriétés qui dépendent des électrons de coeur, telles que la spectroscopie RMN, la spectroscopie Mössbauer ou la spectroscopie des rayons X. Dans ce travail, une approche relativiste sera suivie pour décrire correctement les électrons de coeur. Par conséquent, ce chapitre sera consacré aux méthodes de la chimie quantique relativiste.

E.3 Chapitre 3

Nous avons parcouru un long chemin depuis le début de notre discussion sur les principes de base de la théorie de réponse linéaire de SCF, suivie de la chimie quantique relativiste. Avec les outils théoriques des chapitres précédents en place, nous sommes presque prêts à discuter de quelques exemples pratiques. Cependant, avant de continuer, je dois aborder une hypothèse clé formulée dans le chapitre 1. Je rappelle qu'en régime UV-Vis, l'étendue spatiale de la molécule cible est généralement beaucoup plus petite que la longueur d'onde de la lumière. Pour cette raison, il est justifié d'appliquer l'approximation du dipôle électrique, selon laquelle le champ électromagnétique est approximé comme un champ électrique uniforme. Cependant, en régime X, cette approximation est remise en question car la longueur d'onde est de taille comparable par rapport à la molécule. La Figure 3.1 représente une situation similaire à celle de la Figure 1.1, bien que dans ce cas, la molécule de benzène interagisse avec des rayons X, ayant une longueur d'onde beaucoup plus courte (λ = 2, 6 nm).

Compte tenu du fait que l'approximation du dipôle électrique revient essentiellement à un développement multipolaire à l ordre zéro, il existe généralement deux méthodes pour corriger les lacunes de cette approximation. La première méthode, la plus directe, évite complètement un développement multipolaire et repose son interaction sur les champs électromagnétiques exacts, conservant ainsi leur caractère sinusoïdal dans l'opérateur d'interaction. En utilisant l'opérateur d'interaction complet, tous les effets non dipolaires sont inclus de manière exacte, bien que le formalisme soit quelque peu plus compliqué en raison de la dépendance en fréquence de cet opérateur.

Alternativement, le développement multipolaire peut être tronqué au-delà de l ordre zéro, prenant ainsi en compte les moments multipolaires d'ordre supérieur. Les moments multipolaires peuvent être compris comme des idéalisations des distributions de charge et de courant. Par exemple, le moment dipolaire électrique est une quantité vectorielle qui donne la direction globale dans laquelle la distribution de charge est polarisée. Le moment quadrupolaire électrique, en revanche, décrit la largeur globale d'une distribution de charge, semblable à la variance des distributions statistiques. Les inhomogénéités supplémentaires des distributions sont décrites par des moments multipolaires d'ordre supérieur, qui, sous forme cartésienne, sont définis comme suit :

Q [n] j 1 •••jn = d 3 rr j 1 r j 2 • • • r jn ρ(r, t); m [n] j 1 •••j n-1 ;i = n n + 1 d 3 rr j 1 r j 2 • • • r j n-1 r × j(r, t) i . (E.12)
En incluant les contributions de tous les moments multipolaires, les distributions de charge ou de courant peuvent être récupérées exactement. Cependant, en pratique, cela n'est souvent pas nécessaire, car généralement, les contributions les plus dominantes proviennent des moments multipolaires d'ordre le plus bas. Les distributions de charge ou de courant compliquées peuvent donc être simplifiées en ne considérant que ces contributions dominantes. Cette approche se révèle particulièrement utile lors de la description des interactions intermoléculaires. Dans ce contexte, cependant, j'utiliserai cette méthode pour approximer les interactions lumière-matière.

En utilisant soit l'interaction lumière-matière complète, soit l'interaction tronquée, je décrirai deux types de spectroscopie : près du front d absorption de rayons X (NEXAFS) et le dichroïsme circulaire naturel des rayons X (XNCD). La NEXAFS peut être calculée directement à partir de la section efficace d'absorption dérivée de la lumière polarisée linéairement. Le calcul du XNCD, cependant, est un peu plus compliqué, car il implique la différence entre la section efficace d'absorption de la lumière polarisée circulairement à gauche et à droite :

E.4 Chapitre 4

E.4.1 Implémentation

Dans la chimie quantique contemporaine, la formulation de la théorie n'est que la moitié du travail, car une implémentation efficace est cruciale pour rendre les calculs chimiques quantiques réalisables, ce qui impliquent souvent de nombreuses opérations mathématiques fastidieuses. Pour cette raison, cette section sera consacrée à l'implémentation de l'opérateur d'interaction lumière-matière complet et tronqué. Ces deux développements ont été réalisés dans DIRAC, qui est un code chimique quantique adapté aux calculs relativistes à quatre composantes. L'implémentation de ces interactions est encore en cours, donc ce qui suit est une invertoire de toutes les fonctionnalités disponibles. Il est possible d'effectuer des calculs d'intensité à l'aide des opérateurs d'interaction complets et tronqués, à la fois dans des conditions isotropes et anisotropes. Cependant, dans l'implémentation actuelle, les calculs ECD ne peuvent être effectués qu'à l'aide de l'opérateur d'interaction complet, tandis que l'ECD tronqué est encore en développement. L'ECD tronqué rapporté plus loin dans ce chapitre a été calculé à l'aide d'un script externe avec les moments de transition de DIRAC en entrée. Dans ce qui suit, je donnerai un bref aperçu de l'implémentation de l'interaction complète et tronquée, en mettant en évidence leurs caractéristiques uniques. Dans la section à venir et les suivantes, cette partie du code sera désignée sous le nom d'implémentation BED, l'abréviation signifiant au-delà de l'approximation électro-dipolaire. Bien que cette abréviation ne s'applique pas nécessairement à l'ECD, qui est déjà au-delà de cette approximation au premier ordre, elle sera quand même utilisée par souci de cohérence.

E.4.2 Convergence de l'Expansion Multipolaire

Avoir une implémentation entièrement fonctionnelle à notre disposition qui peut calculer des forces d'oscillateur au-delà de l'approximation électro-dipolaire nous permet de procéder à nos premiers calculs. Pour toute nouvelle fonctionnalité développée, il est toujours bon de procéder avec prudence et de commencer par des calculs sur des systèmes de test simples, ce qui facilite l'identification des erreurs et des résultats inattendus. Par conséquent, cette section se concentrera principalement sur les spectres d'absorption UV-Vis et absorption des rayons X de l'atome de radium, dont la symétrie sphérique simplifiera considérablement le problème à portée de main. De plus, cette symétrie induit des règles de sélection pour les transitions multipolaires. Il est probable que les principales caractéristiques des spectres seront décrites par la force d'oscillateur électrique-dipolaire, tandis que les détails plus fins seront donnés par des contributions dépendant du moment magnétique-dipolaire et du quadrupôle électrique. Les règles de sélection pertinentes sont les suivantes :

• dipolaire-électrique: ∆j = 0, ±1; ∆m = 0, ±1; (j = 0 j = 0)

• dipolaire-magnétique : ∆j = 0, ±1; ∆m = 0, ±1; (j = 0 j = 0)

• quadrupôle électrique : ∆j = 0, ±1, ±2; ∆m = 0, ±1, ±2; (j = 0 j = 0)

• interaction complète : (j = 0 j = 0).

Dans ce chapitre, je vais mettre en pratique ces règles et calculer les intensités d'absorption de l'atome de radium à la fois dans les régions centrale et de valence.

Sauf indication contraire, les données rapportées dans cette section ont été obtenues avec une version en développement du code de structure électronique DIRAC. En utilisant l'interaction complète, le spectre de valence et de coeur de l'atome de radium a été calculé au niveau de la théorie TDDFT en utilisant la fonction d'échange-corrélation PBE0, basée sur l'hamiltonien de Dirac-Coulomb et dans le cadre de la méthode de fenêtre d'excitation restreinte (REW). Dans ces calculs, les intégrales biéléctronique ont été remplacées par une correction d'énergie SS interatomique. Pour des raisons techniques qui deviennent plus claires par la suite, les calculs impliquant des interactions tronquées ont été limités aux excitations ns 1/2 → 7p 1/2 ; n = 1, 2, • • • , 7 de l'atome de radium et effectués au niveau de la théorie TD-HF. De plus, dans ces calculs, le criblage des intégrales (SS|SS) a été désactivé et les intégrales (SS|SS) ont été incluses. Pour les deux séries de calculs, l'ensemble de base dyall.ae3z a été utilisé.

Il a été constaté que pour des énergies d'excitation croissantes, l'utilisation de l'interaction tronquée lumière-matière devient de plus en plus problématique en raison de la convergence lente de ces expansions. Il ne s'agit pas d'un problème de base qui peut être atténué en augmentant la base, car cette lente convergence a été observée au niveau des intégrales atomiques sous-jacentes OA. En particulier, pour les excitations de coeur, une convergence extrêmement lente a été observée pour les intégrales impliquant des orbitales gaussiennes cartésiennes avec des exposants diffus. Cela peut s'expliquer par le fait que de telles fonctions diffuses sont moins efficaces que les fonctions serrées pour amortir les puissances cartésiennes croissantes apparaissant dans une expansion de l'interaction lumière-matière complète en fonction de la norme du vecteur d'onde. Cela suggère à son tour que l'utilisation d'orbitales de type Slater, qui ont une décroissance plus lente que les orbitales gaussiennes, sera encore plus problématique.

E.4.3 Convergence de la base

Le choix approprié de la base est essentiel pour atteindre une précision quantitative dans les calculs chimiques quantiques. Surtout dans le contexte actuel, où des opérateurs d'interaction non conventionnels sont utilisés, il est important de tenir compte spécialement du choix de la base. Par conséquent, dans cette section, il sera évalué si les bases conventionnelles sont suffisamment flexibles pour satisfaire les exigences de ces opérateurs d'interaction.

Les bases conventionnelles sont construites en choisissant un ensemble de fonctions qui minimisent l'énergie. En général, on gagne plus d'énergie en ajoutant des fonctions serrées qu'en ajoutant des fonctions diffuses. En raison de cela, les bases optimisées en termes d'énergie saturent plus rapidement la région du coeur que la région au-délà du valence. L'interaction lumière-matière complète et tronquée peut nécessiter l'ajout de fonctions de base supplémentaires. Un problème à cet égard est l'absence d'un principe variationnel pour les moments de transition. Alternativement, des bases pour une utilisation au-delà des énergies peuvent être construites en utilisant l'approximation du coeur équivalent, la règle de Slater ou le schéma d'optimisation de la complétude. Une autre stratégie consiste à augmenter systématiquement la taille de la base jusqu'à ce que la propriété d'intérêt soit stabilisée.

Sørensen et al. ont réalisé une série de calculs dans un cadre non relativiste en utilisant l'opérateur d'interaction complet et tronqué, ce dernier étant limité à l'ordre deux, exprimé avec les bases ANO-RCC. Bien que la série n'ait pas complètement convergé pour les bases plus grandes, leurs résultats indiquent que l'opérateur d'interaction complet est plus stable en ce qui concerne le choix de la base. De plus, Sørensen et al. ont fait valoir que chaque moment multipolaire nécessite des bases différentes, ce qui rend extrêmement difficile la construction de bases pouvant décrire l'interaction tronquée à des ordres plus élevés. Ils ont conclu que l'opérateur d'interaction complet devrait être la norme pour les calculs impliquant des effets non dipolaires. Pour les interactions d ordre zéro et de premier ordre, la représentation en longueur semble être le choix privilégié dans la littérature, bien qu'il n'y ait pas de réponse concise quant à la supériorité de la convergence de la base dans une représentation donnée.

Une autre stratégie pour évaluer la qualité d'une base consiste à la comparer à une valeur de référence. Dans cette section, les besoins en termes de base pour les effets non dipolaires sont évalués en comparant des calculs utilisant des bases gaussiennes avec le code moléculaire DIRAC à des calculs de différences finies équivalents avec le code atomique GRASP. Si la taille de la grille de ce dernier est suffisamment grande, les résultats correspondent effectivement à la limite de la base complète, ce qui en fait une référence appropriée pour les calculs de base. À cette fin, j'étudierai les transitions 1s 1/2 → 7p 1/2 et 7s 1/2 → 7p 1/2 dans l'atome de radium, qui sont représentatives des excitations de coeur et de valence, respectivement.

Pour comparer les effets de la base dans les régions de valence et de coeur, j'ai étudié les transitions 7s 1/2 → 7p 1/2 et 1s 1/2 → 7p 1/2 de l'atome de radium. Les transitions étaient contraintes à se produire entre des orbitales simples pour faciliter la comparaison entre les calculs utilisant des orbitales de type gaussien dans DIRAC et les calculs de référence numériques dans GRASP. Comme GRASP ne calcule que les orbitales occupées, les orbitales ont été optimisées en utilisant la méthode de Hartree-Fock avec moyenne des configurations (AOC) par rapport à des déterminants d'états excités, c'est-à-dire [Rn]7s 1 1/2 7p 1 1/2 pour la transition de valence et [Ra]1s -1 1/2 7p 1 1/2 pour la transition de coeur. Même si l'orbitale 7p 1/2 est formellement occupée dans ces configurations, je la traite comme une orbitale virtuelle pour construire les moments de transition. Dans le cas de GRASP, ceux-ci ont été obtenus en insérant les fonctions radiales P nκ et Q nκ dans les équations décrivant les distributions radiales. Comme les moments de transition sont déterminés jusqu'à une phase complexe, je les ai définis de manière à ce que toutes les distributions radiales soient réelles et aient une valeur maximale positive. Étant donné que les fonctions radiales de GRASP sont calculées sur une grille numérique et correspondent ainsi effectivement à la limite de la base complète, les moments de transition de GRASP ont été utilisés comme référence. Dans le cas de DIRAC, j'ai appliqué la sélection par recourrement pour éviter que l'état excité de coeur ne s'effondre pendant les cycles SCF. J'ai ensuite utilisé les orbitales des calculs de l'état excité dans un calcul d'état fondamental de Hartree-Fock dépendant du temps à quatre composantes (4c-TD-HF) dans le cadre de la réponse linéaire. La méthode de la fenêtre d'excitation restreinte a été invoquée pour ne considérer qu'une seule amplitude. Les distributions radiales ont ensuite été divisées par cette amplitude restante pour comparer les résultats des calculs de base avec les valeurs de référence numériques. Dans DIRAC, les distributions radiales ont été calculées sur une grille radiale avec le module de visualisation, qui effectue l'intégration angulaire sur une grille de Lebedev (L max = 64). En raison du schéma de symétrie quaternion dans DIRAC, tous les moments de transition sont par défaut réels.

Les calculs de base ont été effectués au niveau de la théorie de Hartree-Fock en utilisant le Hamiltonien de Dirac-Coulomb et les bases dyall.aeXz (X=2,3,4). Ces bases sont construites à partir de fonctions de base gaussiennes cartésiennes non contractées qui sont conçues pour les calculs corrélés et contiennent donc des fonctions de base de grand moment cinétique orbital. J'ai supprimé les fonctions g, h et i car, pour les systèmes atomiques, elles ne contribuent qu'à l'espace des orbitales virtuelles. Les bases des petites composantes ont été générées en respectant la condition de l'équilibre cinétique restreint et les intégrales (SS|SS) ont été traitées exactement, pour une cohérence avec les Les géométries de H 2 S 2 et (CH 3 S) 2 ont été obtenues en utilisant la fonctionnelle d'échangecorrélation B3LYP et l'ensemble de bases cc-pVTZ. Les optimisations de géométrie ont été effectuées dans Gaussian 16. Pour imiter l'angle de disulfure χ 3 typique des structures protéiques, une optimisation de géométrie contrainte a été réalisée pour χ 3 = -87 • , correspondant à la chiralté M -hélicoïdale. L'approche de fenêtre d'excitation restreinte a été utilisée pour cibler sélectivement les seuils du soufre L-et K. Cela élimine également le problème des transitions artificielles vers les orbitales quasi-continues causé par les effets du jeu de bases finies, qui interfèrent souvent avec les simulations au bord L. Un modèle gaussien a été utilisé pour la distribution de charges nucléaire, et une grille Lebedev à 86 points (L max = 12) a été utilisée pour la moyenne isotropique de l'absorption linéaire différentielle basée sur l'opérateur complet de l'interaction. L'origine du jauge a été placée au centre de masse (CM) et la symétrie spatiale est pris en compte dans tous les cas, sauf pour les calculs de dépendance à l'origine du jauge.

Les énergies d'excitation, les sections efficaces d'absorption linéaire et différentielle pour l'opérateur complet d'interaction ainsi que les expansions multipolaires dans le jauge de vitesse généralisée ont été calculées en utilisant la fonctionnelle d'échange-corrélation PBE0 et les ensembles de bases non contractés aug-pcX-3 et aug-pc-3 pour le soufre et l'hydrogène, respectivement. La série de jeux de bases pcX-n a été développée pour décrire les processus d'excitation du coeur en utilisant l'approche ∆SCF (Self-Consistent Field) aux niveaux non relativiste et relativiste. Les jeux de bases des petites composantes ont été générés dans le cadre de l'équilibre cinétique restreint. Les calculs relativistes ont été effectués en utilisant un Hamiltonien Dirac-Coulomb dans lequel les intégrales (SS|SS) ont été remplacées par une correction d'énergie (SS-SS). L'invariance de l'origine du jauge de la formulation semi-classique complète des intensités rotatoires isotropes et anisotropes et de sa forme tronquée du premier ordre a été confirmée numériquement en déplaçant l'origine du jauge (de 0 à 100 a 0 ) le long de l'axe C 2 . Cela conduit à une redistribution des contributions E1-E2 et E1-M1 à R [START_REF] Holmgaard List | Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory[END_REF] xx et R [START_REF] Holmgaard List | Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory[END_REF] yy pour les transitions de symétrie B. Comme prévu, les résultats sont restés inchangés à la fois pour les formulations complètes et tronquées (données non présentées). Les spectres simulés ont été obtenus en convoluant le spectre à bâtonnets avec des fonctions de forme gaussienne avec une largeur totale à mi-hauteur (FWHM) de 0,4 eV, et ceux de (CH 3 S) 2 ont été décalés par des décalages différents pour chaque bord d'absorption afin de correspondre à leurs homologues expérimentaux.

Pour évaluer l'effet de l'augmentation des énergies d'excitation sur la force oscillatrice différentielle, les spectres des seuils L 1 et K ont été calculés pour les analogues plus lourds de H 2 S 2 : H 2 X 2 , X=Se et Te. En analogie avec H 2 S 2 , les géométries des deux analogues plus lourds ont été obtenues à partir d'une optimisation contrainte (fixant χ 3 = -87 • ) en utilisant les ensembles de bases cc-pVTZ et def2-VTZPP pour X=Se et Te, respectivement. À l'exception du choix du jeu de bases, cette série de calculs a été réalisée au même niveau de théorie que les calculs précédemment mentionnés pour H 2 S 2 . Pour réduire le coût du calcul des analogues plus lourds, j'ai utilisé le jeu de bases dyall.av3z pour toutes les simulations spectrales. Pour rendre la comparaison entre les analogues structuraux plus précise, les calculs de H 2 S 2 ont été répétés en utilisant ce jeu de bases.

Aller au-delà de l'approximation du dipôle électrique au seuil K du soufre entraîne une redistribution non négligeable de l'intensité parmi les transitions presque dégénérées, mais sans implications visibles sur le profil d'absorption linéaire. En revanche, le profil d'absorption différentielle n'est pas affecté par une telle redistribution, en raison de sa nature signée. Cela entraîne une surestimation globale d'un facteur deux. En examinant les formes des distributions ECD anisotropes sous-jacentes, on constate que cette disparité provient en grande partie d'une mise à l'échelle globale corrigée en introduisant des contributions du troisième ordre.

iii 3

 3 Light-Matter Interactions Beyond the Dipole Approximation 62 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.2 General Expressions for Monochromatic Light . . . . . . . . . . . . . . . . . 64 3.3 Full Semi-Classical Light-Matter Interaction . . . . . . . . . . . . . . . . . . 66 3.4 Truncated Light-Matter Interactions: Generalized Velocity Representation . 70 3.4.1 Origin Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.4.2 Simplifications from Time-Reversal Symmetry . . . . . . . . . . . . . 72 3.4.3 Electric and Magnetic Contributions . . . . . . . . . . . . . . . . . . 75 3.5 Truncated Light-Matter Interactions: Generalized Length Representation . . 77 3.6 Connecting the Generalized Length-and Velocity Representation . . . .

127 4. 4 . 3

 43 Origin-Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.5 Electronic Circular Dichroism . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.5.1 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.5.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4.5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.2 Core-Hole Relaxation at the DFT Level . . . . . . . . . . . . . . . . . . . . . 145 5.3 X-ray Magnetic Circular Dichroism . . . . . . . . . . . . . . . . . . . . . . . 146 5.4 Full Interaction Operator in the Length Representation . . . . . . . . . . . . 146

  (t)e -iEnt )|n = n c n (t)e -iEnt Ĥ|n (V mn (t)c n (t); ω mn = 1 (E m -E n ) (1.16)

α

  Vα (ω y )|0 ω yω m0 + i e (-iωy+iω m0 + )t . (1.20)

  c

Figure 2 . 1 :

 21 Figure 2.1: schematic representation of the atomic shell model. Compared to the valence electrons (yellow) the core electrons (orange) move at an elevated speed, thus inducing relativistic effects.

Figure 2 . 2 :

 22 Figure 2.2: Left: ordinary rotation of axes, from unprimed (dashed) to primed (full) axes. Transformations of this type are a subgroup of all Lorentz transformations, where only the spatial coordinates are mixed, i.e. ∃i, j s.t. Λ j i = 0 . Here, the ct-axis is left out intentionally. Right: boost i.e. ∃µ s.t. Λ µ 0 = 0 and Λ 0 µ = 0, from unprimed (dashed) to primed (full) axes. Here, the z-axis is left out and the diagonal black line represents a light ray. Note that in the primed frame, the ct -and x -axis have tilted symmetrically towards the light line, thus leaving the speed of light unchanged in the transformed frame.

  89, Section 1.6][90, Chapter 5][110, Section 2.8.1]

Figure 3 . 1 :

 31 Figure 3.1: Schematic representation of benzene molecule interacting with X-rays (λ = 2.6 nm), consisting of an electric (violet) and magnetic (grey) component. Note that in this figure, the wave length is comparable in size to the benzene molecule, suggesting that it is affected by the curvature of the light. The mathematical expressions describing the electromagnetic fields are given at the bottom of the box. The propagation-and polarization direction are described by the unit vectors e k and , respectively, with the latter being defined by the relation e k = c ω k.

Figure 3 . 2 :

 32 Figure 3.2: Snapshot of linearly polarized light traced out over space, in terms of its electric (red) and magnetic (blue) component, at an angle θ.

  (a) Right handed circularly polarized light (b) Left handed circularly polarized light

Figure 3 . 3 :

 33 Figure 3.3: Electric component of right-(m odd) and left handed (m even) circularly polarized light, traced out over space at a fixed position and phase (t = 0 and δ = 0).

χ a→b 1 Figure 3 . 4 :

 134 Figure 3.4: graph that represents the two consecutive gauge transformations that are required to transform (φ a , A a ) to (φ b , A b ).

. 86 )

 86 and the magnetic multipole moment operator (Eqn.(3.59)). At first order, Eqn. (3.85) yields the length representation (Eqn.(3.24)) and is thus referred to as the generalized length representation.

( 3 .

 3 109) in terms of the anti-symmetrized two-electron integrals L pq,rs = (pq|rs) -(pr|qs).(3.110)

Figure 3 . 5 :

 35 Figure 3.5: Schematic representation of an ensemble of freely moving water molecules.

Figure 3 . 6 :

 36 Figure 3.6: fixed molecule with variable propagation direction of light, represented by the blue line and parametrized by the angles θ and φ.

  (3.22) does not depend on χ ∆f (ω) = -i 2m e ω e 2 e k;i (θ, φ) • ε ijk f | e ω cα j e ik(θ,φ)•r |i f | e ω cα k e ik(θ,φ)•r |i *

Figure 4 . 1 :

 41 Figure 4.1: flow chart the full light-matter interaction part of the BED implementation. Rectangular boxes indicate an operation or modules, whereas the arrows and the symbols besides them indicate the output or input. X n and ω are the solution vector and frequency obtained from Eqn. (1.114). The property gradient, E T , is defined according to Eqn. (1.103) and (3.16).

Figure

  Figure 4.1 depicts a flow chart of the BED implementation responsible for the calculation of the full oscillator strength. In the isotropic case, the code starts with a loop over the points on the

Figure 4 . 2 :

 42 Figure 4.2: code flow chart of the implementation of the truncated light-matter interaction. The boxes indicate an operation or module, whereas the arrows and the symbols besides them indicate the output or input. Here, the operator Ŷ [n] denotes the multipole operators in either representation (Eqn. (3.57) and (3.85)) stripped of its frequency prefactors. X n and ω are the solution vector and frequency obtained from Eqn. (1.114). The property gradient, E Ω , is defined according to Eqn. (1.103) with Ŷ [n] as the operator.

  r e c u r s i v e s u b r o u t i n e bedloop (m, k2 , i n d i c e s , iterm , jterm , FOSC,ATMPPF,OMEGA) #i n c l u d e " dgroup . h" #i n c l u d e " dcbxpp . h" r e a l * 8 : : FOSC(MAXEXC,NBSYM) , ATMPPF(MAXEXC,NPPAPT,NBSYM) ,OMEGA(MAXEXC,NBSYM) i n t e g e r : : i n d i c e s ( k2 ) , i n d ( 3 ) , k2 i f (m==k2 ) then do i =1 ,3 i n d i c e s (m) = i c a l l r o t f a c _ b e d ( i n d i c e s , k2 , iterm , jterm , FOSC,ATMPPF,OMEGA) enddo r e t u r n e l s e do i =1 ,3 i n d i c e s (m) = i c a l l bedloop (m+1,k2 , i n d i c e s , iterm , jterm , FOSC,ATMPPF,OMEGA) enddo e n d i f END SUBROUTINE BEDLOOP

Figure 4 . 4 :

 44 Figure 4.4: code snippet from the file src/prp/bed_osc.F90 in Dirac. The array FOSC contains the nth order oscillator strength for a given excitation and symmetry, ATMPFF contains the transition moments for a given excitation, operator and symmetry, while OMEGA denotes the frequencies. When the if statement is not met, a loop over 1,2 and 3 is initiated that fills the array INDICES at location m. In this loop the subroutine calls itself, while the index m is shifted by one, thus repeating the process for the next entry of INDICES. As a result, this subroutine will fill INDICES with all possible combinations of 1,2 and 3, until it has reached the final index of INDICES. At this point, the subroutine will call ROTFAC_BED, which creates the rotational tensor for that specific index combination and contracts it with the relevant transition moments. The recursive loop repeats this for all possible combinations in INDICES, thus accumulating the isotropic truncated oscillator strength iteratively.

Figure 4 . 5 :

 45 Figure 4.5: Non-dipolar effects on electronic absorption of radium: (a) the valence and (b) K-edge spectra for Ra within and beyond the ED approximation (ED and BED, respectively) at the 4c-TD-PBE0/dyall.ae3z level of theory, using Coulomb gauge (velocity representation) for the former and the full interaction operator in Eqn.(3.134) and an 86-point (L max = 12) Lebedev grid for the latter. The labels indicate the character of the receiving orbital. Note the differences in scales on the axes in the valence and X-ray region. Oscillator strengths are summed over contributions from transitions within each degenerate (same ∆J components) and near-degenerate (different ∆J components) set, and the sticks have been convoluted with a Lorentzian lifetime broadening of 1000 cm -1 . The experimental 1s ionization energy is 103922 ± 7.2 eV.[START_REF] Bearden | Reevaluation of X-ray atomic energy levels[END_REF]

Figure 4 . 6 :

 46 Figure 4.6: Convergence behavior of the oscillator strengths for ns 1/2 → 7p 1/2 transitions of radium at various orders (colored lines) in the wave vector within the Coulomb gauge (velocity representation) : (a)-(g) correspond to n = 7, 6, . . . , 1. Except for the valence transitions, the oscillator strengths seem to diverge after ω = c, which is indicated by a vertical dashed line. Excitation energies (ω f i ) are in a.u.

  [START_REF] Pyykko | Relativistic effects in structural chemistry[END_REF] in terms of the dimensionless parameter Q (Eqn. (4.7)) The right-hand expression has the form of an alternating series, which converges according to the Leibniz criterion provided that lim m→∞ a m = 0 and the absolute value of the coefficients, |a m |, decreases monotonically. The first condition follows readily from Eqn. (4.23), whereas the second condition holds, if we have a m+1 am < 1; ∀m. This fraction evaluates to

  Figure 4.7) further confirm this.

Figure 4 . 7 :

 47 Figure 4.7: Radial distribution of the electric-velocity dipole moment (R [1] Q ) for the 1s 1/2 (top panel) and 7s 1/2 → 7p 1/2 (bottom panel) transitions. Expectation values of the radial position for the relevant s orbitals are indicated as vertical sticks where the labels represent the index of the s orbital, i.e. n i s 1/2 (n i = 1, . . . , 7). The percentages in the upper right corner of each box are the relative errors of the transition moments, i.e. | T bas -Tnum Tnum |×100%.

7

 7

  and 4.8, or the lack thereof, can be understood by considering the radial part of the atomic orbitals involved in the core and valence transition (Figures

Figure 4 .

 4 Figure 4.8: Radial distribution of the full interaction (R T ) for the 1s 1/2 (top panel) and 7s 1/2 → 7p 1/2 (bottom panel) transitions. To assess the validity of the electric-dipole approximation, the numerical reference curve from the full interaction (GRASP BED) is compared with the reference curve from the electric-velocity dipole (GRASP ED). Expectation values of the radial position are shown for the relevant s orbitals. The labels represent the index of the s orbital, i.e. n i s 1/2 (n i = 1, . . . , 7). The percentages in the upper right corner of each box are the relative errors of the transition moments, i.e. | T bas -Tnum

8 :

 8 Figure 4.8: Radial distribution of the full interaction (R T ) for the 1s 1/2 (top panel) and 7s 1/2 → 7p 1/2 (bottom panel) transitions. To assess the validity of the electric-dipole approximation, the numerical reference curve from the full interaction (GRASP BED) is compared with the reference curve from the electric-velocity dipole (GRASP ED). Expectation values of the radial position are shown for the relevant s orbitals. The labels represent the index of the s orbital, i.e. n i s 1/2 (n i = 1, . . . , 7). The percentages in the upper right corner of each box are the relative errors of the transition moments, i.e. | T bas -Tnum Tnum |×100%.

Figure 4 . 9 :

 49 Figure 4.9: Radial functions of the large and small component of the 7s 1/2 ( r = 4.365a 0 ) and 7p 1/2orbital (top row) and their deviations from the numerical reference (bottom row). These orbitals were taken from the radium atom with [Rn]7s1 1/2 7p 1 1/2 configuration. The basis set orbitals were calculated at the 4c-HF level using Dirac, while the numerical reference was calculated with GRASP. Note that the scaling is different for each individual box and that the error curves are plotted in a different range than the radial functions.

Figure 4 . 10 :

 410 Figure 4.10: Radial functions of the large and small component of the 1s 1/2 ( r = 0.01454 a 0 ) and 7p 1/2 orbital (top row) and their deviations from the numerical reference (bottom row). These orbitals were taken from the radium atom with [Ra]1s -11/2 7p 1 1/2 configuration. The basis set orbitals were calculated at the 4c-HF level using Dirac, while the numerical reference was calculated with GRASP. Note that the scaling is different for each individual box and that the error curves are plotted in a different range than the radial functions.

Figure 4 . 11 :

 411 Figure 4.11: Valence transition (7s 1/2 → 7p 1/2 ): radial distributions of electric multipole moments in the generalized length representation, Q[2n+1] , n ∈ [0, 6]. In each box, the upper panel contains the radial distribution, while the lower panel contains the deviation with the numerical reference. Note that each subfigure has different scales. The percentages in the upper right corner of each box are the relative errors of the transition moments, i.e. | T bas -Tnum Tnum |×100%.

Figure 4 .

 4 Figure 4.13 contains the radial distributions of the magnetic multipole moments m[2n] for the valence transition. For the lowest two orders, these distributions have distinct shapes, whereas their shapes start to be reminiscent of the electric multipole distributions in Figures 4.11and 4.12 for higher orders. Likewise, the peaks of the radial distributions move away from the origin and become higher upon increasing the order. The overall errors indicate that the dyall.ae3z basis set is preferable for 2 < n < 5, whereas the dyall.ae4z basis set should be used at higher orders. However, for n = 6 and n = 7, the dyall.ae4z basis does not reduce the overall error below 1%. The magnetic multipole moments for the core transition (Figure4.14) follow the same trend, although the higher order multipole moments tend to converge with more difficulties towards the numerical reference. Already at n = 4, the dyall.ae4z basis set is not enough to reduce the error below 1%. In general, the basis set convergence of the magnetic multipole moments seem to follow the same trends as the electric-length multipole moments, although the convergence of the former is much more demanding. Furthermore, note that the dyall.ae4z curve has two peaks for n = 5, whereas the reference curve

  Figure 4.13 contains the radial distributions of the magnetic multipole moments m[2n] for the valence transition. For the lowest two orders, these distributions have distinct shapes, whereas their shapes start to be reminiscent of the electric multipole distributions in Figures 4.11and 4.12 for higher orders. Likewise, the peaks of the radial distributions move away from the origin and become higher upon increasing the order. The overall errors indicate that the dyall.ae3z basis set is preferable for 2 < n < 5, whereas the dyall.ae4z basis set should be used at higher orders. However, for n = 6 and n = 7, the dyall.ae4z basis does not reduce the overall error below 1%. The magnetic multipole moments for the core transition (Figure4.14) follow the same trend, although the higher order multipole moments tend to converge with more difficulties towards the numerical reference. Already at n = 4, the dyall.ae4z basis set is not enough to reduce the error below 1%. In general, the basis set convergence of the magnetic multipole moments seem to follow the same trends as the electric-length multipole moments, although the convergence of the former is much more demanding. Furthermore, note that the dyall.ae4z curve has two peaks for n = 5, whereas the reference curve

Figure 4 . 12 :

 412 Figure 4.12: Core transition (1s 1/2 → 7p 1/2 ): radial distributions of electric multipole moments in the generalized length representation, Q[2n+1] , n ∈ [0, 6]. In each box, the upper panel contains the radial distribution, while the lower panel contains the deviation with the numerical reference. Note that each box has a different scale. The percentages in the upper right corner of each box are the relative errors of the transition moments, i.e.| T bas -Tnum Tnum |×100%.

  and

Figure 4 . 13 :

 413 Figure 4.13: Valence transition (7s 1/2 → 7p 1/2 ): radial distributions of magnetic multipole moments m[2n] , n ∈ [1, 7]. In each box, the upper panel contains the radial distribution, while the lower panel contains the deviation with the numerical reference. Note that each box has a different scale. The percentages in the upper right corner of each box are the relative errors of the transition moments, i.e. | T bas -Tnum Tnum |×100%.

Figure 4 . 14 :

 414 Figure 4.14: Core transition (1s 1/2 → 7p 1/2 ): radial distributions of magnetic multipole moments m[2n] , n ∈ [1, 7]. In each box, the upper panel contains the radial distribution, while the lower panel contains the deviation with the numerical reference. Note that each box has different scales. The percentages in the upper right corner of each box are the relative errors of the transition moments, i.e. | T bas -Tnum Tnum |×100%.

Figure 4 . 15 :

 415 Figure 4.15: Valence transition (7s 1/2 → 7p 1/2 ): radial distributions of electric multipole moments in the generalized velocity representation Q[2n+1] , n ∈ [0, 6]. In each box, the upper panel contains the radial distribution, while the lower panel contains the deviation with the numerical reference. Note that each box has a different scale. The percentages in the upper right corner of each box are the relative errors of the transition moments, i.e. | T bas -Tnum Tnum |×100%.

Figure 4 . 16 :

 416 Figure 4.16: Core transition (1s 1/2 → 7p 1/2 ): radial distributions of electric multipole moments in the generalized velocity representation Q[2n+1] , n ∈ [0, 6]. In each box, the upper panel contains the radial distribution, while the lower panel contains the deviation with the numerical reference. Note that each box has different scales. The percentages in the upper right corner of each box are the relative errors of the transition moments, i.e. | T bas -Tnum Tnum |×100%.

Figure 4 . 17 :

 417 Figure 4.17: Isotropic linear (top) and differential (bottom) absorption spectra of (CH 3 S) 2 : (a) valence, (b) L 2,3 -edge, (c) L 1 -edge and (d) K-edge spectra using the full interaction operator in Eqs. (3.16) (green shadings) or the lowest non-vanishing generalized velocity representation (orange lines). Left axes correspond to (differential) absorption cross-sections, whereas (differential) oscillator strengths (sticks) are shown on the right axes. Black sticks indicate the location of all computed transitions whereas black lines are experimental spectra.[START_REF] Tokue | Photoexcitation of dimethyl sulfide and dimethyl disulfide in the vacuum ultraviolet region: Rydberg states and photofragment emissions[END_REF][START_REF] Hearn | The near U.V. absorption spectra of dimethyl sulfide, diethyl sulfide and dimethyl disulfide at T=300 K. Atmospheric Environment[END_REF][START_REF] Schnorr | Tracing the 267 nm-Induced Radical Formation in Dimethyl Disulfide Using Time-Resolved X-ray Absorption Spectroscopy[END_REF][START_REF] Ochmann | UV-Photochemistry of the Disulfide Bond: Evolution of Early Photoproducts from Picosecond X-ray Absorption Spectroscopy at the Sulfur K-Edge[END_REF][START_REF] Hitchcock | Sulfur and Chlorine K-shell spectra of gases[END_REF] The stick spectra were convolved with a Gaussian lineshape with FWHM of 0.4 eV. The theoretical absorption spectra have been uniformly shifted to align with the experimental counterparts (shift values indicated in the top panels). The same shifts were applied to the ECD spectra.

3 S) 2

 32 for the full semi-classical light-matter interaction operator and in first-order within the Coulomb gauge, computed at the 4c-TD-PBE0 level of theory and the uncontracted aug-pcx-3/aug-pc3 basis set. For the oriented case, k(π/2, π/2) for the valence and L 2,3 -edge transitions and k(π/2, π/4) for the L 1 -and K-edge transitions. For each spectral region, the accumulated (differential) oscillator strength (ac) across the two pairs of transitions (A/B symmetry) is given in the third line. Numbers in parentheses are exponents of 10.

Figure 4 . 18 :

 418 Figure 4.18: Comparison of full and truncated differential oscillator strength ∆f (θ, φ) across the spectral regions (valence, L 3 -, L 1 -and K-edges). Transitions of (a-d) A and (e-h) B symmetry in H 2 S 2 . The black arrow points along the direction of the wave vector for the anisotropic ECD intensity given in Table 4.8. The truncated ECD is represented by the smooth surface, whereas the full ECD is shown as individual points generated with a 5810-point Lebedev grid (L max = 131). Blue: negative; red: positive ECD signal. Note that different scaling factors (upper right corner) have been applied. The corresponding isotropic differential oscillator strengths are indicated in each subfigure.

Figure 4 . 19 :

 419 Figure 4.19: Convergence of the truncated anisotropic ECD distributions, i.e., ∆f [2n+1] (θ, φ), for the K-edge transitions in H 2 S 2 . The superscript notation ∆f (→m) indicates accumulated contributions up through mth order. The black arrows point along the direction of the wave vector for the anisotropic ECD intensity given in Table D.3.

Figure 4 . 20 :

 420 Figure 4.20: Isotropic linear (top) and differential (bottom) absorption spectra of H 2 X 2 , X=S, Se and Te: (a-c) L 1 -edge and (d-f ) K-edge spectra using the full interaction operator (green shadings) or the lowest non-vanishing generalized velocity representation (orange lines). Left axes correspond to (differential) absorption cross-sections, whereas (differential) oscillator strengths are shown on the right axes. Black sticks indicate the location of all computed transitions. The stick spectra were convolved with a Gaussian lineshape with FWHM of 0.4 eV. No shifts have been applied to the theoretical spectra.

e c 2 dφ

 2

( 2 .

 2 [START_REF] Donovan Hedegård | Theoretical 57 Fe Mössbauer spectroscopy: isomer shifts of [Fe]-hydrogenase intermediates[END_REF]. To start off, I will take the complex conjugate of the minimally-coupled time-dependent Dirac equation[89, Section 1.6][90, Chapter 5][110, Section 2.8.1]cα * • p * + m e c 2 βeφ + ecα * • A ψ * (r, t) = -i ∂ t ψ * (r, t).(B.11) Taking the complex conjugate of the Dirac matrices only changes its y-component α * y = -α y , (B.12) whereas all components of the momentum operator change sign due to its imaginary prefactor p * =p. (B.13) q = a + b j; a, b ∈ C, (B.22)

√ 2 ĵ-, ĵz , 1 √ 2

 212 ĵ+ }, the j-dependent integral can be found from the relation jm|j z |j m = m δ jj δ mm , (C.18) thus implying (j|| ĵ(1) ||j ) = j(j + 1)(2j + 1)δ jj . (C.19)

3 Y

 3 11 }, (C.22) which I will insert in Eqn. (C.15)

  p e ik•r = 4π e ω ∞ =0 m=-i j (kr)Y m * (e k )cα p Y m (θ, φ), (C.30)

Figure D. 1 :

 1 Figure D.1: Core transition (1s 1/2 → 7p 1/2 ): radial distributions of magnetic multipole moments m[2n] , n ∈ [1, 7]. These distributions are calculated with the default linear dependence thresholds: 10 -6 and 10 -8 for the large and small component space. In each box, the upper panel contains the radial distribution, while the lower panel contains error relative to the numerical reference. Note that each box has different scales. The percentages in the upper right corner of each box are the relative errors of the transition moments, i.e. | T bas -Tnum Tnum |×100%.

Figure D. 3 :

 3 Figure D.3: Scheme I. Radial distributions of electric transition multipole moments in the generalized length representation for the 1s 1/2 → 7p 1/2 transition. The basis sets for these calculations were generated by augmenting the dyall.ae2z basis set with diffuse functions in an even-tempered fashion. The percentages in the upper right corner of each box are the relative errors of the transition moments, i.e. | T bas -Tnum Tnum |×100%.

Figure D. 4 :

 4 Figure D.4: Scheme I. Radial distributions of magnetic transition multipole moments for the 1s 1/2 → 7p 1/2 transition. The basis sets were generated by augmenting the dyall.ae2z basis set with diffuse functions in an even-tempered fashion. The percentages in the upper right corner of each box are the relative errors of the transition moments, i.e. | T bas -Tnum Tnum |×100%.

Figure D. 5 :

 5 Figure D.5: Scheme II. Radial distributions of electric transition multipole moments in the generalized length representation for the 1s 1/2 → 7p 1/2 transition. The basis sets for these calculations were generated by augmenting the dyall.ae2z basis set with the diffuse functions of either the dyall.ae3z or dyall.ae4z basis sets. The percentages in the upper right corner of each box are the relative errors of the transition moments, i.e. | T bas -Tnum Tnum |×100%.

Figure D. 6 :Figure D. 7 :

 67 Figure D.6: Scheme II. Radial distributions of magnetic transition multipole moments for the 1s 1/2 → 7p 1/2 transition. The basis sets for these calculations were generated by augmenting the dyall.ae2z basis set with the diffuse functions of either the dyall.ae3z or dyall.ae4z basis sets. The percentages in the upper right corner of each box are the relative errors of the transition moments, i.e. | T bas -Tnum Tnum |×100%.

[ 2 ]

 2 xy = f | Q[2]xy |i . For excitations of B symmetry the corresponding matrices are R

  . . 82 3.6.1 Exact-State-Formalism: a Gauge Theoretical Approach . . . . . . . . 82 3.6.2 Conversion at the SCF Level . . . . . . . . . . . . . . . . . . . . . . . 84 3.7 Rotational Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.7.1 Full Oscillator Strength . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.7.2 Truncated Oscillator Strength . . . . . . . . . . . . . . . . . . . . . . 91 3.7.3 Angular Dependence of Truncated the Differential Oscillator Strength 94 4 Applications 95 4.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.4 Systems with Symmetry Equivalent Centers . . . . . . . . . . . . . . . . . . 126 4.4.1 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.4.2 Full vs. Truncated Light-Matter Interaction . . . . . . . . . . . . . .

4.1.1 Full Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.1.2 Truncated Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.2 Convergence of the Multipole Expansion . . . . . . . . . . . . . . . . . . . . 101 4.2.1 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.2.2 Full Light-Matter Interaction . . . . . . . . . . . . . . . . . . . . . . 104 4.2.3 Truncated Light-Matter Interaction . . . . . . . . . . . . . . . . . . . 105 4.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.3 Basis Set Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.3.1 Radial Transition Moment Distributions . . . . . . . . . . . . . . . . 111 4.3.2 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.3.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.3.4

  [START_REF] Saue | Post Dirac-Hartree-Fock Methods -Properties[END_REF] From a pragmatic point of view, it is completely valid to use this relativistic coupling in this setting. Starting from the Schrödinger Hamiltonian, minimal substitution gives the following result

				Ĥp =	σ • ( p + eA) 2m e	2	-eφ,	(2.67)
	which gives the Pauli Hamiltonian in its more familiar form after applying the Dirac identity
	Ĥp =	p2 2m e	+	e 2m e	( p • A + A • p) +	e 2 A 2 2m e	+	e 2m e	(σ • B) -eφ.	(2.68)

  dθ sin θe r;p e r;q e k;j 1 e k;j 2 • • • e k;j 2n .

																• • • e k;j 2n	θ,φ	, where
	the χ-dependent part assumes the same form as Eqn. (3.135)
		p q χ e k;j 1 e k;j 2 • • • e k;j 2n	θ,φ	=	1 2	δ pq e k;j 1 e k;j 2 • • • e k;j 2n	θ,φ	-	1 2	e r;p e r;q e k;j 1 e k;j 2 • • • e k;j 2n	θ,φ
																(3.139)
	=	1 8π	δ pq	0	2π	dφ	0	π	dθ sin θe k;j 1 e k;j 2 • • • e k;j 2n -	1 8π	0	2π	dφ	0	π
	Therefore, we need to find expressions for integrals of the form
							E uvw =	1 8π	0	2π	dφ	0	π	dθ sin θe u k;x e v k;y e w k;z

Table 4 .

 4

	1: Anisotropic oscillator strengths for radium ns 1/2 → 7p 1/2 excitations at various orders in Coulomb gauge (velocity representation) as well as with the full semi-classical light-matter interaction operator at the 4c-TD-HF/dyall.ae3z level of theory.	n ∆E (eV) ω/c (a.u.) 0 2 4 6 8 10 12 f full	7 1.82142 4.88(-04) 6.664(-01) 9.370(-07) -2.335(-12) 6.710(-19) 2.819(-24) -5.422(-30) 5.928(-36) ac 6.664(-01) 6.664(-01) 6.664(-01) 6.664(-01) 6.664(-01) 6.664(-01) 6.664(-01) 6.664(-01)	6 41.04937 1.10(-02) 9.850(-05) 8.799(-08) 1.253(-11) 4.800(-15) -9.481(-19) 1.982(-22) -4.937(-26) ac 9.850(-01) 9.859(-05) 9.859(-05) 9.859(-05) 9.859(-05) 9.859(-05) 9.859(-05) 9.841(-05)	5 269.05081 7.22(-02) 3.947(-04) -3.662(-07) -1.568(-10) 6.514(-14) 4.330(-15) -7.432(-17) 9.082(-19) ac 3.947(-04) 3.943(-04) 3.943(-04) 3.943(-04) 3.943(-04) 3.943(-04) 3.943(-04) 3.943(-04)	4 1240.4258 3.33(-01) 1.754(-04) -3.161(-07) -1.242(-08) 1.028(-08) -5.164(-09) 1.683(-09) 3.974(-10) ac 1.754(-04) 1.751(-04) 1.751(-04) 1.751(-04) 1.751(-04) 1.751(-04) 1.751(-04) 1.751(-04)	3 4888.5865 1.31(+00) 6.018(-05) 2.170(-08) -3.342(-06) 4.760(-05) -3.680(-04) 1.849(-03) -6.737(-03) ac 6.018(-05) 6.020(-05) 5.686(-05) 1.045(-04) -2.635(-04) 1.585(-03) -5.152(-03) 6.011(-05)	2 19398.588 5.20(+00) 1.784(-05) 5.379(-07) -1.750(-04) 3.918(-02) -4.758(+00) 3.758(+02) -2.155(+00) ac 1.784(-05) 1.838(-05) -1.567(-04) 3.903(-02) -4.719(+00) 3.711(+02) 3.689(+02) 1.807(-05)	1 104647.71 2.81(+01) 3.143(-06) 5.310(-08) 6.767(-03) -4.469(+01) 1.594(+05) -3.688(+08) 6.190(+11) ac 3.143(-06) 3.196(-06) 6.770(-03) -4.468(+01) 1.593(+05) -3.686(+08) 6.186(+11) 3.579(-06)

Numbers in parentheses are exponents of 10. For each excitation, the second line contains the accumulated oscillator strength (ac) to given order. The wave k and polarization vectors were oriented along the y-and z-axes, respectively.

  2. Alternatively, a smaller basis could

	R M (r) R bas R num R M (r) R bas R num	0.0 0.4 0.8 1.2 0.00 0.09 0.18 0.27 ×10 3 0.11% 0.49% 0.77% n=1 r (a 0 ) 0.0 1.7 0.8 0.8 ×10 6 0.6 0.0 0.6 1.2 1.8 ×10 2 32.75% 25.34% 11.74% n=5 0 10 20 30 r (a 0 ) 8.1 4.1 0.0 4.1 ×10 1	0 2 4 6 ×10 4 0 7 n=2 2.43% 1.00% 0.93% 14 21 r (a 0 ) 0.0 4.2 2.1 2.1 ×10 5 0.0 0.8 1.6 2.4 ×10 2 0 7 n=3 8.73% 3.89% 0.12% 14 21 r (a 0 ) 0.0 3.7 1.8 1.8 1.0 1.5 0.5 0.0 ×10 3 0.5 1 0 1 2 3 ×10 4 41.27% 44.58% 22.53% n=6 0 10 20 30 r (a 0 ) 1.4 0.7 0.0 0.7 ×10 4 0 2 4 6 ×10 6 40.24% 66.43% 32.62% n=7 0 r (a 0 ) 12 24 36 2.8 1.4 0.0 1.4 ×10 6	0	n=4 19.91% 11.57% 3.88% 9 ×10 1 18 27 2.6 0.0 2.6 5.1 r (a 0 ) numerical dyall.ae2z dyall.ae3z dyall.ae4z

Table 4 .

 4 3 contains results obtained after lifting restrictions on the virtual orbital space. I have

	R (r) R bas R num	0.4 0.0 0.4 0.8 1.2 0.00 0.04 0.08 0.12 ×10 1 0.09% 0.00% 0.00% n=0 r (a 0 ) 0.0 1.6 0.8 0.8 ×10 2 0.5 0.0 0.5 1.0 1.5 0.00 0.09 0.18 0.27 ×10 3 0.25% 0.25% 0.84% n=1 r (a 0 ) 1.6 ×10 6 0.8 0.0 0.8 1.6	2 3 ×10 4 1 0 1 0 6 n=2 1.37% 0.35% 2.11% 12 18

Table 4 .

 4 10 3 f full 6: Comparison of isotropically averaged oscillator strengths for Cl 1s →Ti 3d transitions of TiCl 4 for the full BED operator and at various orders within multipolar gauge (lr: length representation) and Coulomb gauge (vr: velocity representation) gauges as computed at the 4c-TD-PBE0 level of theory with the dyall.ae3z basis set. Numbers in parentheses are exponents of 10. F.S. stands for final state. Contributions from degenerate states have been summed. A 86-point (L max= 12)

	1 T 1	2773.351719	lr vr	0.000 0.000	17.976 17.976	-7.344 -7.372	8.560 8.561	2.879 2.866	4.191 4.183	3.979 3.970	3.993
	1 E	2773.351723	lr vr	0.000 0.000	7.199 7.156	-1.251 -1.229	3.569 3.548	1.945 1.943	2.308 2.298	2.249 2.242	2.248
	1 T 2	2773.351725	lr vr	7.825 7.781	-17.413 -17.380	16.369 16.353	-4.360 -4.358	2.948 2.942	1.272 1.271	1.543 1.540	1.510
		Sum	lr vr	7.825 7.781	7.763 7.752	7.775 7.752	7.769 7.752	7.772 7.752	7.772 7.752	7.771 7.752	7.752

Table 4 .

 4 9: Weights of contributions from solid harmonics (Eqn.(3.153)) to the differential cross section ∆f (θ, φ) of selected transitions in H 2 S 2 . The weights have been scaled by the absolute value of the s-contribution. Numbers in parentheses are exponents of 10.

	Irrep Excitation	∆f	[1] iso	s	d z 2	d x 2 -y 2	d xy
	A	Valence	5.176(-05) 1.000 -0.447 0.300 -0.056
		L 3 -edge	-9.118(-05) -1.000 0.447 0.414 -0.031
		L 2 -edge	-2.824(-04) -1.000 0.447 0.408 -0.024
		L 1 -edge	7.316(-05) 1.000 -0.447 -0.862 8.817
		K-edge	1.657(-04) 1.000 -0.447 -0.772 9.108
	B	Valence	-7.624(-05) -1.000 -0.328 -0.090 0.314
		L 3 -edge	9.186(-05) 1.000 -0.132 -0.592 -0.005
		L 2 -edge	2.883(-04) 1.000 -0.128 -0.591 -0.008
		L 1 -edge	-7.312(-05) -1.000 2.503 -0.347 -8.262
		K-edge	-1.657(-04) -1.000 0.377 0.813 -9.125

Table 4 .

 4 10: Contributions to the isotropic linear and differential oscillator strength (∆f

	[2n+1] iso	and

  2 S 2 Linear and differential absorption spectra for H 2 S 2 are depicted in Figure D.7. Corresponding oscillator strengths are given in Table D.3.

	R M (r) R bas R num R M (r) R bas R num	4 0 4 8 0.00 0.09 0.18 0.27 ×10 4 2.4% 2.4% 2.7% 2.3% 2.3% 2.3% n=1 r (a 0 ) 0 1 2 ×10 6 0.8 0.0 0.8 1.6 2.4 ×10 2 32.9% 14.4% 17.3% 18.3% 18.2% 18.2% n=5 0 14 28 42 r (a 0 ) 0 4 8 ×10 1	2 1 8 4	2 0 2 4 2 0 2 4	×10 4 0 9 n=2 3.3% 3.3% 4.1% 18 27 1.8% 1.8% 1.8% r (a 0 ) 0.0 5.0 2.5 2.5 ×10 5 0.0 0.8 1.6 0.8 ×10 4 82.2% 22.0% 23.3% 17.1% 19.0% 19.6% n=6 0 16 32 48 r (a 0 ) 2 1 0 1 2 ×10 4 4 0 4 8	×10 2 0 11 22 33 3.6% 3.2% 4.7% 3.4% 3.2% 3.2% n=3 r (a 0 ) 0 2 ×10 3 ×10 6 40.2% 9.0% 7.8% 3.5% 10.8% 13.0% n=7 0 r (a 0 ) 19 38 57 0 3 6 ×10 6	4 2 6 3	0.6 0.6 1.2 0.0	0	10.0% 6.1% 8.6% n=4 13 26 39 11.6% 11.2% 11.1% ×10 1 2.5 0.0 2.5 5.0 r (a 0 ) numerical dyall.ae2z + 1 s dyall.ae2z + 2 s dyall.ae2z + 3 s dyall.ae2z + 1 p dyall.ae2z + 2 p dyall.ae2z + 3 p

Acknowledgments

Appendix A Complements to Chapter 1

A.1 Solving the Generalized Eigenvalue Problem

In this appendix, a possible technique will be discussed to solve the generalized eigenvalue problem from Eqn. (1.114). The derivations provided in this section will mainly follow ref. [START_REF] Saue | Linear response at the 4-component relativistic level: Application to the frequency-dependent dipole polarizabilities of the coinage metal dimers[END_REF]. As discussed in Section 1.3.3, the dimensionality of the problem forms a major obstacle to solve this problem using a numerical scheme. To avoid this issue, the solution vectors will be expressed in a set of trial vectors

where h determines the hermiticity of the trial vectors. Accordingly, the electronic Hessian and generalized metric can be projected onto this basis

thus yielding the following reduced problem Ẽ [START_REF] Martin Van Horn | Probing chirality across the electromagnetic spectrum with the full semi-classical light-matter interaction[END_REF] 0 Xn = ωn0 S [START_REF] Martin Van Horn | Probing chirality across the electromagnetic spectrum with the full semi-classical light-matter interaction[END_REF] Xn , (A. [START_REF] Martin Van Horn | Transition moments beyond the electric-dipole approximation: Visualization and basis set requirements[END_REF] which can be solved readily using conventional diagonalization packages.

For the full problem, the residual equals zero R = Ẽ [START_REF] Martin Van Horn | Probing chirality across the electromagnetic spectrum with the full semi-classical light-matter interaction[END_REF] 0 -ωn0 S [START_REF] Martin Van Horn | Probing chirality across the electromagnetic spectrum with the full semi-classical light-matter interaction[END_REF] Xn , (A. [START_REF] Yuval | 21 lessons for the 21st century[END_REF] thus forming a suitable diagnostic to assess whether the reduced problem is sufficiently large to yield accurate results. If this is not the case, the residual vector can be used to extend the space of trial vectors. Starting from an initial set of trial vectors, this procedure is repeated iteratively until the residuals fall below a user-defined threshold. As a main advantage, this procedure avoids explicit storage of the full electronic Hessian. To construct the reduced electronic Hessian from Eqn. (A.3), it suffices to compute the vector quantity Σ j = E [2] B j , which are to be contracted with the trial vectors. Furthermore, since the upper and lower half of Σ j are related by complex conjugation, I will only compute the upper half in the following

Following a similar procedure as in Section 3.6.2, it can be shown that the inner commutators evaluate to

where appears the one-index transformed Hamiltonian Ĥ0 = hpq a † p a q + 1 2 (pq|rs)a † p a † r a s a q , (A.7)

in terms of the one-index transformed integrals hpq = W pt h tqh pt W tq (A.8)

(pq|rs) = W pt (tq|rs) -(pt|rs)W tq + W rt (pq|ts) -(pq|rt)W ts . (A.9)

Here, the matrix W is defined in terms of the solution vectors W ai = b ai ; W ia = -b ia ; W ij = W ab = 0. Alternatively, these vectors can be expressed in terms of the Fock matrix σ ai = F ab W bi -W aj F ji + L ai,jb W bj -W jb L ai,bj , (A. [START_REF] Moseley | XCIII. The high-frequency spectra of the elements[END_REF] and the anti-symmetrized two-electron integrals (see Eqn. (3.109)).

Solving the reduced problem iteratively thus hinges on the calculation of the sigma vectors, which in turn are obtained from the one-index transformed quantities. Applying this scheme allows us to calculate the lowest few eigenvalues of a matrix which otherwise would be too large to explicitly store. However, being a bottom-up approach, this method is cumbersome to calculate core transitions, requiring us to calculate all lower excitations first. A slight modification to this method that allows the calculation of core excitations is provided in Section C. [START_REF] Yuval | 21 lessons for the 21st century[END_REF].

Appendix B

Complements to Chapter 2

B.1 The Dirac Equation and Spin

In this appendix, I will demonstrate that the Dirac wave function can be expressed as two Pauli spinors (Eqn. (2.84)). To start of, I will express the Dirac equation in this basis

where the potential is given by V = -eφ. The Dirac equation is thus constructed from two coupled differential equations, where the second equation has the following solution for the small component

Note that for positive-energy solutions the small component is a factor of 1 c smaller than the large component, hence its name. In the non-relativistic limit, it is clear that the small component vanishes, suggesting that we should focus on the large component. Using Eqn. (B.2), we can obtain a single equation for the large component

Further applying the identity

yields the equation

which gives the Pauli Hamiltonian upon taking the non-relativistic limit. Therefore, it seems only reasonable to identify the first and last two components with a spin degree of freedom. However, it should be stressed that in the Dirac equation, spin appears naturally, contrary to the ad hoc treatment of spin in the Pauli equation. In addition, the Dirac equation incorporates non-perturbative spin-orbit coupling in its formalism.

Spin-orbit coupling arises from the relative motion of electrons and nuclei. In the molecular frame, the electrons encircle the (stationary) nuclei, whereas the opposite is true in the rest frame of the electrons. From the viewpoint of the electrons, the moving nuclei induce a magnetic field that interacts with the spin magnetic moment, hence locally steering the direction of electron spin. In Fig. B.1, this effect is represented schematically. Due to this effect, spin-and spatial degrees of freedom become non-separable, which renders the spin density, i.e. the density that integrates up to the spin, position dependent.

Figure B.1:

Right: from the perspective of the (core) electron, the nucleus is not stationary, but rather moving (see Fig. 2.1 for a nuclear centric model). Left: this relative motion generates magnetic induction, which couples with the spin magnetic moment of the electron, resulting in an effect more commonly referred to as spin-orbit coupling.

As pointed out in the work of Thomas, the classical analogue of this effect is magnetic induction. [START_REF] Thomas | The kinematics of an electron with an axis . The London, Edinburgh[END_REF] In his work, Thomas considered a charged ball that rotates around its own axis, while simultaneously orbiting around a central potential. Furthermore, he defined a rest frame of the ball by consecutive Lorentz transformations that follow the instantaneous motion of the ball. From the transformations in Eqs. (2.28), it follows that that the magnetic induction is given by

where v is the instantaneous velocity of the charged ball. In this expression, it is assumed that the electron moves slow, such that the Lorentz factor is unity. The factor of one half results from

Inserting the second relation into Eqn. (B.11), yields the result

The Dirac equation with opposite external charges can be retrieved from this expression by introducing the operator

which has the following commutation relation with the Dirac matrices

Application of these relations yields the desired result

B.3 Quaternion Scheme

The purpose of this appendix is to lay out the fundamental concepts of the quaternion scheme. This derivation closely follows refs. 145, 146, so the reader is referred to these sources for more detail.

In the following, it will be assumed that we have a time-reversal symmetric system, such that the eigenfunctions come in Kramers partners and the matrix representation of time-reversal symmetric operators assume the form in Eqn. (2.132). In most quantum-chemical applications, it is relevant to compute the eigenvectors and -values of such matrices

As expected from Kramers theorem, the eigenvectors come in degenerate pairs

that are related by time-reversal. Typically, degeneracies that are related to a symmetry suggest a block-diagonalization of the matrix representations. In our example, this is certainly possible, albeit at the expense of complicating the algebra in these matrices from complex to quaternion. Before proceeding to demonstrate this transformation explicitly, I will first briefly discuss quaternions. Quaternions can be thought of as generalizations of complex numbers

constructed from three different types of imaginary units, ȋ2 = j2 = k2 = ȋjk = -1. However, the increase in dimensionality compared to complex algebra comes at the cost of losing commutativity

and the quaternion wave function

As alluded to before, the quaternion scheme shows its true virtue in combination with point-group symmetry.

However, in a four-component formalism, point-group symmetry deserves special consideration, compared to the more straightforward symmetry considerations of scalar wave functions. After all, the four-component wave function transforms as a fermion function, which have the rather unusual feature of developing a phase of minus one after a rotation of 2π R(2π)ψ = -ψ.

(B.33)

Here, the rotation could be around any axis, so the axis of rotation is left out in the rotation operator, R. Interestingly, this result implies that a fermion ought to be rotated around an angle of 4π before returning to its initial state. Therefore, in our point-group we should also take this symmetry operation into account. Suppose that we have a point-group, consisting of the symmetry operations { Ĝ}, that preserve the structure of our Hamiltonian, ĥD , Ĝ = 0. For every member of this group, we can create a new member by combining it with the operation from Eqn. (B.33), Ĝ = R(2π) Ĝ, which results in a group of double dimensions. The amount of irreducible representations (irreps) in this group, however, is not doubled and can generally be divided into bosonic irreps and fermionic irreps, the former is symmetric and the latter is anti-symmetric with respect to R(2π).

It can be shown, however, that not all of these irreps are unique, because in some cases they are related by time-reversal. Therefore, we may want to include time-reversal into our double groups to obtain a more complete description of the symmetry structure. Including time-reversal turns out to be rather problematic for the group structure, because there is not anymore a one-to-one correspondence between the product of representation matrices and symmetry operations if anti-Hermitian operators are included in the group. [START_REF] Saue | Quaternion symmetry in relativistic molecular calculations: The Dirac-Hartree-Fock method[END_REF] It is still possible to construct these matrices, now referred to as a corepresentation, and decompose them into irreducible corepresentations (irrcoreps). However, the symmetry reductions that can be gained from the fermion irr(co)reps is rather limited. More reductions can be gained from symmetry considerations of the individual components.

Although the spinors as a whole generally transform according to the fermion irreps, the realand imaginary part of each components transform as boson irreps [START_REF] Wigner | 26 -time inversion[END_REF]

In the following I will assume that our pointgroup is either D 2h or any of its subgroup, implying that at most we have eight irreps: the totally symmetric irrep, Γ 0 the Cartesian coordinates, (Γ x , Γ y , Γ z ), the rotations (Γ Rx , Γ Ry , Γ Rz ) and the function Γ xyz . From the structure of the Dirac equation, in and even the solution vectors and the property gradient

Appendix C

Complements to Chapter 3

C.1 Chirality and ECD Beyond First Order

As alluded to in the introduction of this chapter, there seems to be an intricate relation between ECD and the isotropy of the sample. Contrary to what is mentioned in most textbooks, ECD can yield non-zero signal for achiral molecules under anisotropic conditions. To illustrate this, consider the first-order truncated anisotropic differential oscillator strength ∆f [1] = 2m e ω c 2 ω c e k;j e k;i

which is given in terms of the electric-dipole, magnetic-dipole and electric-quadrupole moment. Expressed in the irreps of D 2h and subgroups, the electric-dipole moment transforms as the Cartesian coordinates (Γ x , Γ y , Γ z ), the magnetic-dipole as the rotations (Γ Rx , Γ Ry , Γ Rz ) and the electricquadrupole as the rotations or the totally symmetric irep, Γ 0 . To obtain a non-vanishing ECD signal under anisotropic conditions, the transitions must simultaneously have non-vanishing contributions from the electric-dipole and the electric-quadrupole or from the electric-dipole and the magnetic-dipole. Therefore, for a non-vanishing anisotropic ECD, at least one of the Cartesian coordinates must either be totally symmetric Γ r i = Γ 0 , or transform as a rotation Γ r i = Γ R j , where the component of the rotation and coordinate do not necessarily have to be equal. Exactly which of these conditions holds true depends on the direction of the wave vector. In general, these conditions can also be met for non-chiral point groups, confirming that non-chiral molecules can have a nonzero ECD signal under anisotropic conditions. Upon taking the isotropic average, the first-order truncated ECD is given by Eqn. (3.150). Similar to previous example, the minimal condition for non-vanishing ECD is Γ r i = Γ R i , for at least one Cartesian coordinate, although in this case, the Cartesian coordinate and the rotation do have to be the same component. This small detail explains why isotropic ECD can only be measured for chiral molecules, since this condition is only met for chiral point groups. However, this derivation leans on the symmetry properties of the lowest-order multipole moments, whereas the full interaction and higher-order terms in the multipole expansion do not generally obey the same selection rules. Therefore, in the following, I will demonstrate that the isotropic ECD is zero for achiral molecules regardless of the order of the multipole expansion.

To start off, suppose that we are considering the isotropic ECD of some achiral molecule, A, consisting of n atoms, whose nuclear coordinates are stored in the matrix

In the previous Section, the effects of isotropy where taken into account by fixing the molecule in space and considering all possible propagation-and polarization directions of the incoming light. In this proof, it turns out to be useful to follow the alternative approach: I will fix the incoming light and rotate the coordinates, using the Euler angles θ, φ, χ

where the matrix O(θ, φ, χ) is constructed from three orthogonal rotations [START_REF] Holmgaard List | Rotationally averaged linear absorption spectra beyond the electric-dipole approximation[END_REF] O

In this expression, the matrix O(α, e n ) rotates the coordinates with an angle of α around the axis spanned by e n . Starting from R A , we can thus generate all possible orientations of A by varying the angles φ, θ and χ over their respective domains, [0, 2π], [0, π] and [0, 2π]. Let us further denote {R A } as the set of all possible orientation of A.

Because {R A } contains all possible orientations, we can choose any starting geometry from which we generate this set. Furthermore, because A is achiral, we can thus apply any orthogonal transformations to R A , without changing this set

This implies that this set is also invariant under transformations that do not preserve the handedness of the coordinate system, i.e. det(U ) = -1.

In particular, the parity inversion operator is of great relevance in physics, as it leaves electromagnetic interactions unchanged. Therefore, if we apply this operator to both the nuclear coordinates and the circularly polarized light, the oscillator strength should not change. For these purposes, I will write the anisotropic oscillator strength of circularly polarized light as a function of the wave vector and a parametric function of the nuclear geometry: f L/R (k; R A ). Due to the generality of this proof, this oscillator strength can either be derived from the full interaction operator, or a truncated interaction. The action of parity inversion on the oscillator strength can thus be expressed as

Let us further denote the set of oscillator strengths from all possible orientations of A as {f L/R (k; R A )}.

The isotropic differential oscillator strength can be obtained as the average of all members in

is constructed from all possible orientations of A, it is isotropic with respect to k. We can thus define k at will, without changing {f L/R (k; R A )}, with the only constraint being that k is consistent throughout this set. I will use this feature, together with Eqn. (C.6), to demonstrate that the isotropic differential oscillator strength is always zero for achiral molecules, regardless of the order in the multipole expansion. For these purposes, I will demonstrate that the sets {f L (k; R A )} and {f R (k; R A )} are the same, yielding the same value upon isotropic averaging, which implies that the isotropic differential oscillator strength is zero. Let us start off with the former set and apply parity inversion onto each of its members

where in the second equality I have inserted Eqn. (C.6). If we further exploit the fact that the set {R A } is invariant under orthogonal transformations and that the set {f L/R (k; R A )} does not depend on the choice of k, we arrive at the result

which thus concludes the proof that isotropic ECD vanishes for achiral molecules.

The situation is different, however, when our molecule is chiral. In this case, we have an enantiomeric pair, that I will label as L and R. Accordingly, the set of all their possible orientations is given by the sets {R L } and {R R }. A defining feature of chiral molecules, is that their point group only contain proper rotations. Any transformation with det( Û ) = -1 thus converts between the members of the enantiomeric pair. Unlike previous example, the parity operator thus has the following action on the sets of all orientations

(C.9)

Similar to previous example, we define the collections of oscillator strengths as {f L/R (ω; R L )} and {f L/R (ω; R R )}. From this, it follows that it is not possible to recreate the proof given in previous paragraph

Therefore, based on these symmetry considerations, there is no a priori reason to assume that the isotropic differential oscillator strength vanishes for chiral molecules. I recall that this derivation is entirely based on inversion symmetry, with no reference to selection rules arising from specific operators. Therefore, it is equally valid for the full-and truncated interaction.

C.2 Selection Rules

In this appendix, I will explore the selection rules of multipole moments and the full interaction operator. This section heavily leans on the findings from Appendix B from ref. [START_REF] Grant | The Dirac Equation[END_REF], which, in turn, will be applied to the various interaction operators introduced Chapter 3. Conventional electricdipole selection rules and those corresponding to higher-order multipole moments are a consequence of spherical symmetry. In the presence of this symmetry, one-electron states can be expressed as eigenfunctions of the total angular momentum (see Section 2. It should be noted that on the diagonal and each sub-diagonal, the index combination v = nm assumes a constant, distinct and positive value, suggesting that we can use it as a summation index. Summing over v implies that we are summing over the diagonals,

where we have the additional choice to run the second index over the columns or rows. In Figure C.3 a diagram of this type of summations is presented.

In some cases, it may also be useful to constrain our summation over the upper left triangle of the matrix

where the left-hand-side represents column-wise summation and the right-hand-side row-wise summation. Considering that along the any sub anti-diagonal and the anti-diagonal the index y = n+m is constant, distinct and positive, which makes it a suitable summation index. Therefore, it is also possible to carry out the summation over the sub anti-diagonals and the anti-diagonal Another useful technique to manipulate summations involves redefining of summation indices. Similar to previous examples, the main idea is that a summation can be carried out in any order, thus implying that we may shift the summation index, reverse it or even both. For example, suppose we are dealing with a simple row-wise summation and we want to shift the index of the column by a certain amount. This operation can be realized by applying the substitution

which implies that we have to insert m = ml into our summation Throughout the main text, it is advocated to use a tight linear dependence threshold, i.e. less functions deleted, for the calculations of radial distributions associated with truncated interaction. In this section, I will demonstrate what happens to the magnetic multipoles if the default linear dependence threshold in the Dirac code is used (10 -6 for the large component space and 10 -8 for the small component space). Figure D.1 shows the radial distributions of the magnetic multipole moments for the core transition. In these calculations, the default linear dependence threshold is applied, the effects of which are clearly visible: for 1 < n < 5, the radial distributions of the dyall.ae3z and dyall.ae4z basis sets oscillate considerably around the numerical reference. At higher orders, the oscillations appear to less pronounced because the r-dependent prefactor dominates the characteristics of the curve. Although the oscillations are diminished in these curves, the overall deviations vastly increase. Interestingly, the basis set convergence seems to be inverted: larger basis sets introduce larger deviations from the numerical reference.

To understand the origin of these oscillations, the involved orbitals need to be analyzed (Figure D.2). The 7p 1/2 orbital converges smoothly towards the numerical reference, whereas the error curve of the 1s 1/2 orbital exhibit similar oscillations as the magnetic multipoles. Especially the basis set convergence of the small component of the 1s 1/2 orbital is problematic: instead of reducing the error, the dyall.ae3z and dyall.ae4z deteriorate the description of the small component function. It seems that most of the oscillations in Figure D.1 stem from the small component of the 1s 1/2 orbital. Therefore, in the following, I inspect the Q 1,-1 function in terms of its basis set expansion.

Ideally, the large and small component are expanded in a two-component basis (Eqn. (2.99))

where the radial part is a Gaussian function

and the angular part a spherical spinor, depending on the angular quantum numbers κ and m. However, in its current form, this basis cannot be used for the small component function, because II Replacing the diffuse functions of the dyall.ae2z basis set with the diffuse functions of the larger basis sets. This scheme is inspired by recent work of Jensen and coworkers [START_REF] Ambroise | Probing basis set requirements for calculating core ionization and core excitation spectra using correlated wave function methods[END_REF] in which they augmented the cc-pVnZ basis set with the core polarization functions of the cc-pCV(n+1)Z basis set.

In Table D.1, the exponents and the shell of the augmentation functions from Scheme I are depicted. Because the dyall.aeXz basis sets are constructed from uncontracted functions, there is no need to give contraction coefficients in Table D.1. To generate the augmentation functions, the ratio was taken of the two most diffuse exponents. The exponents of the augmentation functions are obtained by multiplying an integer multiple of this ratio with the most diffuse exponent. Table D.2 summarizes the composition of the basis sets from Scheme II. These basis sets are constructed by replacing the N most diffuse functions from the dyall.ae2z basis set with the M most diffuse functions of the dyall.ae3z or dyall.ae4z basis set. Accordingly, these basis sets are named ae2z+dae3z and ae2z+dae4z, respectively. In the construction of these bases, M is always chosen larger than N , hence creating a larger basis set than the original. However, there remains a certain degree of arbitrariness in the construction of the basis sets in Scheme II, because there is not a general method to decide how many functions to remove and introduce. Jensen and coworkers did not face this problem, because the cc-pcVnZ basis sets are constructed by introducing core-polarization functions to the cc-pVnZ basis set. Therefore, the augmentation of the cc-pVnZ basis set with the core polarization functions of the cc-pCV(n+1)Z basis set is unambiguous. To assess the performance of these composite basis sets, I computed the radial distributions of the length representation electric and magnetic transition multipole moments for the 1s 1/2 → 7p 1/2 transition. The results of Scheme I are depicted in Figures D.3 and D.4, respectively. In both figures, the augmented basis sets have similar performance for the first two orders. For n = 2 and n = 3, the augmented basis sets seem to induce small oscillations with respect to the numerical reference. Numbers in parentheses are exponents of 10.

Transition ∆E (eV) Appendix E

Résumé en Français

Introduction Au fil des siècles, l'évolution de la chimie se caractérise par l'émergence de sous-disciplines distinctes dont les frontières se sont progressivement estompées au fil du temps. Un exemple notable est l'avènement de la chimie organique au XIXe siècle. Au début, ce domaine était principalement préoccupé par la chimie de la matière vivante, tandis que son cousin plus développé et plus ancien, la chimie inorganique, était plutôt centré sur les sels inorganiques et les acides minéraux. Pour cette raison, une division dichotomique de la chimie prévalait, selon laquelle la chimie organique repose sur des principes fondamentalement différents de la chimie inorganique. Cependant, une découverte clé qui a remis en question ce paradigme est la synthèse de l'urée à partir de matière inorganique par Wöhler en 1828. Associée aux avancées ultérieures de la théorie moléculaire, l'unification des deux domaines était inévitable. Les chimistes au coeur de ces développements peuvent être comparés à des détectives, déduisant méticuleusement les caractéristiques structurelles à l'aide de ressources minimalistes, telles que l'analyse de la décomposition et la réactivité chimique. Dans la chimie moderne, la résolution de la structure moléculaire est devenue une tâche routinière grâce à l'avènement de la spectroscopie. La spectroscopie mesure généralement les interactions lumière-matière en termes d'absorption, d'émission ou de diffusion en fonction de la longueur d'onde. Cependant, l'utilité de la spectroscopie va au-delà de la simple caractérisation structurale, car elle sert également de technique pour évaluer les propriétés moléculaires, dont la nature dépend généralement de la longueur d'onde de la lumière. Par exemple, dans la plage d'énergie inférieure du spectre, c'est-à-dire les micro-ondes et l'infrarouge (IR), le rayonnement induit des rotations et des vibrations moléculaires, tandis que le régime UV-Vis est associé à des excitations électroniques. En réduisant davantage la longueur d'onde, on obtient des rayons X, dont l'application en spectroscopie moléculaire a été pionnière grâce à de Broglie, Siegbahn et Stenström. Les rayons X ont une énergie suffisante pour révéler la structure des couches d électrons internes, au coeur des atomes et qui ne sont pas autrement affectés par le rayonnement UV-Vis. De plus, au-dessus d'un certain seuil, les rayons X peuvent même ioniser le système, ce qui se manifeste par des sauts verticaux soudains dans le spectre d'absorption. Les seuils constituent un outil particulièrement utile dans l'analyse structurale, car ils sont centrés sur les énergies d'ionisation des électrons internes, étant ainsi spécifiques aux éléments et bien séparés.

Cependant, les caractéristiques structurales les plus intéressantes ne sont pas facilement déduites Pour simplifier le calcul de l'état fondamental, des approximations peuvent être appliquées soit au niveau de la méthode de la structure électronique, soit au niveau de l Hamiltonien. Par exemple, les orbitales de valence généralement impliquées en spectroscopie UV-Vis sont les moins sensibles aux effets relativistes, suggérant qu'un Hamiltonien non relativiste, tel que l'Hamiltonien de Schrödinger ou l'Hamiltonien de Pauli, est généralement suffisant. Une fois que l Hamiltonien à utiliser a été décidé, il faut choisir une méthode de structure électronique pour approximer la complexité de la fonction d'onde à plusieurs corps. Il existe une pléthore de méthodes de structure électronique parmi lesquelles choisir, chacune ayant un rapport coût-précision distinct. Par exemple, l'état fondamental peut être obtenu de manière plus fiable à l'aide de méthodes de fonction d'onde telles que le coupled cluster (CC) ou l'interaction de configuration (CI), bien que cela implique des coûts de calcul élevés. Les coûts de calcul sont moindres pour la méthode de Hartree-Fock (HF), bien que sa précision soit plutôt faible. Un rapport coût-précision amélioré peut être obtenu en utilisant la théorie de la fonctionnelle de la densité (DFT), qui se situe généralement entre un calcul HF et peut même approcher la précision des méthodes de fonction d'onde dans certains cas. Dans cette thèse, le HF et la DFT seront les méthodes de choix pour décrire la fonction d'onde de l'état fondamental. Par la suite, le terme "méthodes auto-cohérentes" sera utilisé pour englober les deux approches.

En utilisant la fonction d'onde de l'état fondamental comme point de départ, diverses méthodes sont disponibles pour traiter les états excités, comme cela sera plus clair dans ce chapitre. Pour les états exacts et les fonctions d'onde CI, cette procédure est plutôt simple, car les états excités peuvent être obtenus directement par diagonalisation du Hamiltonien. Suivre cette approche au niveau de la SCF est plus fastidieux, car HF et DFT sont essentiellement des théories de l'état fondamental. Cependant, le calcul explicite des états excités peut être évité en appliquant la théorie de la réponse linéaire. Dans ce chapitre, il sera démontré comment les pôles et les résidus des fonctions de réponse linéaire peuvent être reliés aux énergies d'excitation et aux moments de transition, qui à leur tour peuvent être utilisés pour obtenir les intensités d'absorption. Un inconvénient majeur de cette méthode est qu'elle ne tient pas compte des effets des trous du coeur, ce qui ne devrait pas avoir beaucoup d'importance dans le régime UV-Vis.

Il reste donc à trouver une description adaptée de l'interaction lumière-matière. Dans ce travail, une description semi-classique sera poursuivie, bien qu'il existe des exemples dans la littérature où la nature quantique de la lumière est prise en compte. Une caractéristique particulière du régime UV-Vis est la grande taille de la longueur d'onde par rapport au système moléculaire, ce qui permet l'application de l'approximation du dipôle électrique.

Pour mieux comprendre les origines de cette approximation, considérons l'exemple suivant. Supposons que nous voulons modéliser l'absorption de lumière rouge polarisée linéairement (λ = 700 nm) par une molécule de benzène. Les champs électromagnétiques décrivant ce type de lumière sont montrés ci-dessous.

où apparaît le vecteur d'onde k avec magnitude

le vecteur de polarisation et la phase δ. La lumière peut être polarisée sous d'autres formes que celle montrée ci-dessus, mais cette discussion sera reportée au Chapitre 3.

Pour les solutions diluées, l'absorption de la lumière est donnée par la loi de Lambert-Beer.

Dans cette expression, I et I 0 sont les intensités sortantes et incidentes, tandis que l'exposant donne le nombre effectif de molécules absorbantes, exprimé en termes de densité de molécules absorbantes, N , de la longueur parcourue par la lumière, , et de la section efficace d'absorption, σ(ω). Cette dernière peut être considérée comme une surface couvrant toutes les molécules qui ont absorbé la lumière incidente, formant ainsi une mesure de l'intensité d'absorption. Les quantités exprimées dans la loi de Lambert-Beer sont de nature macroscopique et sont, en principe, uniquement accessibles par l'experience, tandis que les simulations de chimie quantique fournissent généralement des quantités microscopiques. Dans le contexte de l'absorption de la lumière, les deux échelles peuvent être reliées par la relation suivante.

où w i→f est le taux de transition et ω f i est la différence d'énergie entre l'état initial et final. Les deux côtés de cette équation donnent une mesure de l'énergie absorbée par le système. Le taux de transition peut être calculé à l'aide de méthodes de mécanique quantique, ce qui permet d'éviter de calculer directement la section efficace d'absorption. Avant de procéder au calcul de cette quantité, revenons d'abord en arrière et examinons le système en question (Figure 1.1).

Ce qui ressort, c'est la grande différence d'échelle entre la longueur d'onde et la molécule de benzène, cette dernière ne pouvant être perçue qu'à l'intérieur d'un cadre zoomé. Dans ce cadre, la plupart de la courbure de l'onde plane est perdue, ce qui implique que la molécule ressent effectivement un champ uniforme, dont le composant magnétique peut être négligé en raison de son préfacteur de 1/c. Formulé de manière plus mathématique, l'interaction lumière-matière est développée en séries de vecteurs d'onde et tronquée à l'ordre zéro, justifiée par la limite kr 1. À cet ordre, notre interaction est donnée par l'opérateur de dipôle électrique, correspondant à un champ électrique uniforme. Ces considérations forment la base de l'approximation du dipôle électrique, largement utilisée en spectroscopie UV-Vis. Pour décrire l'ECD, cependant, l'approximation du dipôle électrique ne suffit pas, car cette interaction dépend également de l'opérateur de dipôle magnétique, qui apparaît à un ordre supérieur dans notre expansion.

Tournons maintenant notre attention vers le Hamiltonien qui décrit ce système. Sans perte de généralité, notre Hamiltonien peut être séparé en une partie statique (donnée par un Hamiltonien non relativiste) et un opérateur d'interaction dépendant du temps.

le second étant exprimé de manière pratique sous forme d'une transformée de Fourier. De plus, l'hermiticité de l'opérateur d'interaction implique la relation suivante dans le domaine de fréquence.

Je vais reporter la construction explicite de cet opérateur au chapitre 3, car il suffit pour le moment de supposer que cet opérateur est périodique. En conséquence, la transformée de Fourier de l'opérateur d'interaction est discrétisée en composantes de fréquence qui sont des multiples entiers d'une fréquence fondamentale.

Par conséquent, l'opérateur d'interaction peut être exprimé comme suit.

où les indices y et α représentent les composantes de Fourier et cartésiennes. Pour les composantes cartésiennes, la convention de sommation d'Einstein est appliquée, tandis que la somme sur les composantes de fréquence est écrite explicitement. Comme on peut le déduire de l'hermiticité de l'opérateur global, la somme sur les composantes de fréquence contient des indices positifs et négatifs, correspondant à des paires égales mais de signe de fréquences opposées.

Ici, les composantes de fréquence zéro décrivent les contributions statiques, c'est-à-dire ω 0 = 0, à notre perturbation. De plus, dans cette expression, est un nombre infinitésimal garantissant que la perturbation est activée doucement, c'est-à-dire un commutateur adiabatique.

Dans l'approximation du dipôle électrique, les composantes de fréquence sont données par l'opérateur du moment dipolaire électrique

où -e est la charge de l'électron. Dans notre cas particulier de lumière monochromatique, une seule paire de fréquences contribue, ce qui nous permet d'exprimer notre opérateur d'interaction sous une forme plus simple

α cos(ωt)e t . (E.11)

Cependant, dans ce chapitre, la représentation de l'équation (E.8) est préférée pour préserver la généralité du formalisme.

En tenant compte de la parité, on peut faire valoir que le dichroïsme circulaire des échantillons isotropes, par exemple les gaz et les solutions, n'est différent de zéro que pour les molécules chirales. En fait, pour les énantiomères, le dichroïsme circulaire est égal mais du signe opposé, ce qui en fait l'une des rares techniques pour distinguer les énantiomères. Des effets similaires peuvent être obtenus pour les systèmes non chiraux en imposant des conditions anisotropes, comme dans les cristaux, ou en appliquant un champ magnétique externe supplémentaire. Cependant, cette dernière suggestion crée un type d'interaction différent, également appelé dichroïsme circulaire magnétique des rayons X.

S'appuyant sur les résultats du chapitre précédent, toutes les dérivations seront effectuées dans un cadre relativiste à quatre composantes. Ces dérivations reposent principalement sur des articles dont je suis co-auteur. Alternativement, il est également possible de dériver un formalisme similaire dans un cadre non relativiste, bien qu'il soit plus compliqué en raison du grand nombre de termes nécessaires pour décrire l'interaction lumière-matière. Par conséquent, le formalisme non relativiste ne sera pas poursuivi ici, sauf indication contraire. En outre, sauf indication contraire, j'utiliserai la notation de la théorie de l'état exact tout au long de mes dérivations, car la généralisation à la théorie SCF découle directement de la procédure décrite au chapitre 1. calculs numériques. De plus, j'ai constaté qu'avec le seuil de dépendance linéaire par défaut (10 -6 et 10 -8 pour les espaces des grandes et petites composantes, respectivement), plusieurs fonctions de base étaient supprimées. Surtout, la petite composante du 1s 1/2 était susceptible d'être supprimée. Cela nuit à la qualité des résultats, donc pour toutes les calculs, j'ai réglé le seuil de dépendance linéaire à 10 -9 et 10 -10 pour les espaces des grandes et petites composantes, respectivement.

Pour les ordres les plus bas, la base dyall.ae2z était suffisante pour converger les moments multipolaires électriques en représentation de longueur pour les deux transitions. À des ordres plus élevés, la base dyall.ae4z était nécessaire. Il a été couramment observé que la transition de coeur a tendance à être plus difficile à converger que la transition de valence. La convergence de la base des moments multipolaires magnétiques suit également ces tendances, bien qu'elle semble être encore plus difficile à converger pour la transition de coeur. Par conséquent, pour la représentation de la longueur généralisée, la base dyall.ae4z suffit pour décrire correctement les multipôles d'ordre supérieur, bien que le seuil de dépendance linéaire doive être ajusté.

De tous les types de multipôles, il semble que les multipôles électriques en représentation de vitesse soient les plus problématiques à converger. Même si les multipôles de valence convergent vers la référence numérique, les multipôles de coeur présentent des écarts par rapport à la référence sous forme d'oscillations et de pics supplémentaires. En raison de ces artefacts, la convergence de la base semble inversée : les bases plus grandes s'éloignent davantage de la référence. Cela suggère que la représentation de la vitesse généralisée doit être utilisée avec prudence pour n > 3. Pour la représentation de la vitesse généralisée, je recommande d'utiliser la base dyall.ae3z, qui ne semble pas souffrir autant des artefacts que la base dyall.ae4z. De plus, je recommande de ne pas aller au-delà de n = 3, car après cet ordre, la fiabilité de ce schéma devient douteuse.

L'opérateur d'interaction complet ne souffre pas de ces problèmes et est déjà convergé avec la base dyall.ae2z. Étant donné que l'interaction tronquée souffre également d'une convergence lente par rapport à l'expansion multipolaire, je recommande d'appliquer l'interaction complète lors du calcul des transitions aux rayons X.

Ces conclusions sont potentiellement moins évidentes pour le niveau relativiste à 2 composantes intermédiaire. L'approche actuelle exigerait formellement la génération de distributions radiales en utilisant des opérateurs d'interaction transformés correctement, et ceux-ci ne sont par exemple pas disponibles pour le Hamiltonien à 2 composantes eXact (X2C), pour lequel en général seule une représentation matricielle, et non une représentation en espace réel, est disponible. Cependant, l'élimination des petites composantes suggère à nouveau que la convergence de la base devrait s'améliorer.

E.4.4 Systèmes avec des Centres de Symétrie Équivalents

Pour les calculs sur l'atome de radium, on a observé que les transitions permises par le dipôle ne gagnaient que des corrections modestes lorsqu'on incluait des effets non dipolaires, l'explication étant que la compacité des orbitales du coeur annule ces effets. Cependant, si le système en question contient des centres de symétrie équivalents, les transitions du coeur qui en découlent sont délocalisées, ce qui, en principe, devrait renforcer les effets non dipolaires. À cette fin, je considérerai le seuil K du Cl dans TiCl 4 . De plus, ce système représente un cas où il n'y a pas de choix naturel d'origine du jauge, formant ainsi un système approprié pour tester l'indépendance de l'origine du jauge des deux représentations. Ce système a déjà été étudié dans le contexte des effets non dipolaires dans l'absorption des rayons X linéaires en utilisant des expansions multipolaires de faible ordre. En particulier, il a été utilisé pour démontrer l'apparition de forces oscillatoires négatives lors de la troncature de l'interaction lumière-matière dans la représentation de la vitesse généralisée dans un cadre non relativiste. Ci-dessous, je revisiterai ce cas et évaluerai la nécessité d'aller audelà de l'approximation électrique-dipolaire. J'étudie également numériquement l'indépendance de l'origine du jauge des trois schémas dans le cas de l'absorption des rayons X mous. La spectroscopie d'absorption de le seuil K du ligand suppose fournir des informations directes sur la covalence des liaisons métal-ligand en raison du mélange des orbitales p du ligand avec les orbitales d du métal. L'absorption de Cl K -edge de TiCl 4 a été étudiée à la fois expérimentalement et théoriquement dans le cadre et au-delà de l'approximation électrique-dipolaire en utilisant des expressions tronquées d'expansions multipolaires. Son spectre expérimental présente un pic pré-edge large qui nécessite un ajustement à deux pics (dans le toluène : à 2821,58 et 2822,32 eV avec un rapport d'intensité approximatif de 0,84). Dans la symétrie T d , les cinq orbitales 3d de Ti appartiennent aux représentations irréductibles e et t 2 , et les bandes pré-edge peuvent être attribuées à des excitations des orbitales 1s du Cl a 1 et t 2 vers les ensembles e et t 2 des orbitales 3d sur Ti, respectivement. Ici, je me concentre sur les huit transitions les plus basses (a 1 , t 2 → e) qui donnent naissance à trois ensembles dégénérés (E, T 1 et T 2 ), dont le dernier est permis en ED.

Les données rapportées dans cette section ont été obtenues avec une version de développement du code de structure électronique .

Les résultats calculés présentés dans cette section ont été obtenus par des calculs de théorie de la fonctionnelle de la densité dépendante du temps (TD-DFT), basés sur le Hamiltonien de Dirac-Coulomb et dans le cadre de la fenêtre d'excitation restreinte (REW) en utilisant la fonctionnelle d'échange-corrélation PBE0 et les bases dyall.ae3z. Les bases des petites composantes ont été générées en respectant la condition de l'équilibre cinétique restreint, et les intégrales (SS|SS) ont été remplacées par une correction SS interatomique. Un modèle gaussien a été utilisé pour la distribution des charges nucléaires. Une grille Lebedev à 86 points (L max = 12) a été utilisée pour la moyenne isotropique des forces oscillatoires basées sur l'opérateur complet de l'interaction lumière-matière. L'origine du jauge a été placée au centre de masse et la symétrie spatiale a été invoquée dans tous les cas, sauf pour les calculs de l'indépendance de l'origine du jauge.

Deux conclusions clés importantes découlent de ces calculs :

• Les transitions étudiées issues du seuil K du Cl dans TiCl 4 ont d'importantes corrections non dipolaires, principalement induites par la délocalisation des orbitales du coeur du Cl. Cependant, en raison de la proximité des transitions, la valeur cumulative de ces transitions est à peine affectée. Même dans le cas contraire, on peut remettre en question si la délocalisation de l'orbite du coeur est un effet physique réel. Pour l'évaluer correctement, il est nécessaire d'aller au-delà de la théorie SCF.

• L'invariance formelle de l'origine du jauge des forces oscillatoires en jauge multipolaire repose sur des expressions de commutateur qui ne tiennent pas nécessairement dans une base finie. Cela explique le manque notoire d'indépendance de l'origine du jauge d'ordre en ordre dans les calculs pratiques au-delà de l'approximation électrique-dipolaire basée sur n'importe quelle formulation tronquée du jauge multipolaire. Comme indiqué au chapitre 3, ces relations de commutateur, impliquant le Hamiltonien, correspondent à une transformation du jauge de la représentation de la longueur à la représentation de la vitesse. En d'autres termes, l'indépendance de l'origine du jauge en jauge multipolaire est démontrée en se transformant en une autre jauge pour laquelle l'indépendance de l'origine est maintenue. Jusqu'à présent, je n'ai pas été en mesure de démontrer l'invariance de l'origine du jauge tout en restant dans le jauge multipolaire.

E.4.5 Dichroïsme Circulaire Électronique

Jusqu'à présent, je n'ai pas encore exploité pleinement la polyvalence de la mise en oeuvre BED, car toutes les calculs jusqu'à présent portaient sur l'absorption de lumière polarisée linéairement. Dans cette section, je me concentrerai donc sur le ECD isotrope et anisotrope calculé en utilisant l'interaction lumière-matière semi-classique complète, ainsi que la représentation de la vitesse généralisée. Cela nous permet d'étudier la réponse ECD sur l'ensemble du spectre électromagnétique, de l'optique aux régimes des rayons X. Jusqu'à présent, les études théoriques précédentes sur le XNCD isotrope des molécules étaient basées sur une formulation non relativiste tout en ne considérant que l'interaction tronquée du premier ordre, qui est proportionnelle au produit scalaire du moment dipolaire électrique et magnétique).

Une estimation sommaire du XNCD du premier ordre peut être trouvée en supposant que la transition en question est effectivement de nature atomique, ce qui est justifié par le caractère de type atomique des orbitales du coeur. Dans cette hypothèse, les règles de sélection atomiques s'appliquent, ce qui signifie que les règles de transition des moments dipolaires électriques et magnétiques dictent si le XNCD s'annule. Cependant, les règles de sélection des moments dipolaires magnétiques non relativistes prescrivent que les transitions atomiques ne peuvent se produire qu'entre des états qui sont séparés par un couplage spin-orbite, suggérant ainsi que le XNCD est nul dans ce schéma simplifié. Dans des calculs plus réalistes, les orbitales du coeur sont polarisées hors de la symétrie atomique, ce qui rend le XNCD non nul, un effet pris en compte dans les études mentionnées cidessus. La mise en oeuvre actuelle tient compte de deux contributions possibles supplémentaires : (i) les effets des interactions lumière-matière au-delà du premier ordre et (ii) l'inclusion d'effets relativistes, notamment le couplage spin-orbite qui modifie les règles de sélection. Par conséquent, cela permet, pour la première fois, d'examiner de manière réaliste la réponse ECD des molécules sur l'ensemble des régimes de valence et des rayons X.

Cependant, contrairement aux exemples précédents, il n'est pas suffisant d'utiliser le même système cible atomique, qui a une ECD nulle en raison de sa grande symétrie. Par conséquent, en tant que systèmes de test, je considère les modèles chromophores les plus simples de disulfure, le dihydrogène disulfure H 2 S 2 et le diméthyl disulfure (CH 3 S) 2 . En raison des barrières de torsion faibles du disulfure (∼6-11 kcal/mol, les deux formes énantiomères (P -et M -hélice) ne peuvent pas être résolues expérimentalement. Cependant, le pont disulfure est un élément structural important dans les protéines, où il se produit de préférence dans des conformations non planes, chirales (de symétrie C 2 ) et affiche donc une chiralté axiale induite structuralement. Une perspective intéressante pour les systèmes complexes (par exemple, les protéines) est l'utilisation potentielle du XNCD comme une sonde locale de la chiralté. Cela pourrait éventuellement compléter les informations de conformation délocalisées encodées dans l'ECD de valence. En raison de sa facilité de calcul, H 2 S 2 a été largement utilisé pour évaluer les méthodes de structure électronique pour le calcul des propriétés chiroptiques. Pour la même raison, Goulon et al. l'ont également utilisé pour estimer les magnitudes relatives des réponses XNCD dans le cadre de l'interaction tronquée du premier ordre et du cadre non relativiste, en signalant des valeurs en dessous des limites de détection expérimentales. Ici, je revisite l'ECD du chromophore disulfure sur toute l'échelle de valence, les seuils L-et K, allant au-delà de ces approximations.

De manière critique, le traitement du premier ordre se détériore à des énergies de transition plus élevées (au-delà de ω=c∼3728 eV), comme cela peut être déduit des calculs supplémentaires impliquant les analogues plus lourds de H 2 S 2 . Dans ces cas, ce traitement peut même pas réussir à prédire le signe des intensités de force oscillatrice individuelles, bien que cela soit partiellement atténué si les pics sont suffisamment larges. À de telles énergies, passer à des ordres supérieurs n'est pas une solution pratique en raison de la convergence lente de l'interaction tronquée -l'interaction complète est incontournable.

E.5 Conclusion et Perspectives

Nous parvenons à la conclusion générale qu'au moins pour l'absorption de la lumière polarisée linéairement, l'approximation du dipôle électrique est une excellente approximation pour la plupart des systèmes. Même dans le régime des rayons X durs, l'approximation du dipôle électrique peut capturer les caractéristiques générales du spectre. Cependant, pour décrire avec précision les transitions autorisées par le dipôle et même inclure les transitions interdites par le dipôle, il est impératif d'aller au-delà de l'approximation du dipôle électrique. Une autre exception où l'approximation du dipôle électrique pourrait être insuffisante est donnée par des systèmes avec des centres symétriques équivalents, bien que l'élargissement spectral rende difficile d'observer ces effets. Les effets non dipolaires semblent être les plus pertinents pour le dichroïsme circulaire électronique, apparaissant déjà pour H2S2 et étant moins sensibles à l'élargissement spectral. Deux exemples de projets futurs sont l'extension au dichroïsme circulaire magnétique et la mise en oeuvre de la relaxation des trous de coeur.