Christopher South

Avijit Shee

Debashis Mukherjee

Angela K Wilson

Trond Saue
email: trond.saue@irsamc.ups-tlse.fr

January

4-Component relativistic calculations of L3 ionization and excitations for the isoelectronic

Introduction

Uranyl compounds are of great interest due to their unique coordination chemistry as well as their potential impact on the environment. Uranium has been shown to readily form a myriad of coordination complexes in three separate oxidation states (IV, V, and VI), though tends to prefer the IV and VI oxidation states, resulting in its rich chemistry [START_REF] Selbin | Chemistry of uranium (V)[END_REF][START_REF] Kraus | Chemistry of Aqueous Uranium(V) Solutions. I. Preparation and Properties. Analogy between Uranium(V), Neptunium(V) and Plutonium(V)[END_REF][START_REF] Kraus | Chemistry of Aqueous Uranium(V) Solutions. II. Reaction of Uranium Pentachloride with Water. Thermodynamic Stability of UO2+. Potential of U(IV)/(V), U(IV)/(VI) and U(V)/(VI) Couples[END_REF][START_REF] Nelson | Chemistry of Aqueous Uranium (V) Solutions. III. The Uranium (IV)-(V)-(VI) Equilibrium in Perchlorate and Chloride Solutions[END_REF]. Because uranium (along with thorium, actinium, and protactinium) is one of the few actinide elements that is stable and safe enough to be characterized in a laboratory, this allows for the physical effects of the f orbitals on the electronic structure, geometry and thermochemistry to be determined experimentally, giving much insight on the electronic structure of uranyl coordination complexes. [5][START_REF] Bailey | Uranyl-citrate speciation in acidic aqueous solutions -an XAS study between 25 and 200[END_REF][START_REF] Denecke | Actinide speciation using X-ray absorption fine structure spectroscopy[END_REF][START_REF] Belai | Pentavalent Uranium Oxide via Reduction of [UO 2 ] 2+ Under Hydrothermal Reaction Conditions[END_REF][START_REF] Fillaux | Investigating the electronic structure and bonding in uranyl compounds by combining NEXAFS spectroscopy and quantum chemistry[END_REF][START_REF] Vitova | High energy resolution x-ray absorption spectroscopy study of uranium in varying valence states[END_REF][START_REF] Barkleit | Coordination of uranium(VI) with functional groups of bacterial lipopolysaccharide studied by EXAFS and FT-IR spectroscopy[END_REF][START_REF] Atta-Fynn | Structure and Hydrolysis of the U(IV), U(V), and U(VI) Aqua Ions from Ab Initio Molecular Simulations[END_REF][START_REF] Poineau | X-ray absorption fine structure spectroscopic study of uranium nitrides[END_REF][START_REF] Walshe | An EXAFS and HR-XANES study of the uranyl peroxides [UO2(η 2 -O 2 )(H 2 O) 2 ]•nH 2 O (n = 0, 2) and uranyl (oxy)hydroxide [(UO 2 ) 4 O(OH)[END_REF][START_REF] Nelin | Theoretical analysis of the U L 3 -edge NEXAFS in U oxides[END_REF] The electronic properties of uranyl coordination complexes, as well as the nature of the compounds themselves, have been determined through electronic spectroscopy and x-ray diffraction methods. X-ray absorbance and photoelectron spectroscopy have been frequently used to provide knowledge about the electronic structure of the deep core, which results in an element specific electronic spectrum [START_REF] Denecke | Actinide speciation using X-ray absorption fine structure spectroscopy[END_REF]. X-ray absorbance near edge spectroscopy (XANES) has been used to probe low lying unoccupied core ionized states of the uranyl complexes, returning energetics of the energy levels of the low lying virtual orbitals, as well as information on the oxidation state of the molecule [START_REF] Denecke | Actinide speciation using X-ray absorption fine structure spectroscopy[END_REF][START_REF] Fillaux | Investigating the electronic structure and bonding in uranyl compounds by combining NEXAFS spectroscopy and quantum chemistry[END_REF][START_REF] Walshe | An EXAFS and HR-XANES study of the uranyl peroxides [UO2(η 2 -O 2 )(H 2 O) 2 ]•nH 2 O (n = 0, 2) and uranyl (oxy)hydroxide [(UO 2 ) 4 O(OH)[END_REF][START_REF] Nelin | Theoretical analysis of the U L 3 -edge NEXAFS in U oxides[END_REF]. Walshe and coworkers used both high resolution XANES spectroscopy as well as extended X-ray absorbance fine structure (EXAFS) spectroscopy to characterize uranyl peroxides and uranyl oxohydroxides in their mineral forms and provided the first experimental crystal structure of metastudtite, as well as subtle differences in the core spectra of the mineral forms studied [START_REF] Walshe | An EXAFS and HR-XANES study of the uranyl peroxides [UO2(η 2 -O 2 )(H 2 O) 2 ]•nH 2 O (n = 0, 2) and uranyl (oxy)hydroxide [(UO 2 ) 4 O(OH)[END_REF]. Given the highly local nature of the atomic core orbitals, the orbitals will be heavily affected by changes in the local electronic environment, such as relaxation effects which may be caused by changes in the formal oxidation state.

While uranium is not strongly radioactive (an alpha emitter with a half-life of over 44 billion years), it still presents a significant long term health risk, especially in a solvated form such as uranyl [START_REF] Lawrence | Uranium toxicity and chelation therapy[END_REF]. Computational modeling of uranium complexes circumvents this risk and has been used to model the thermochemistry and electronic structure of uranium coordination complexes [START_REF] Atta-Fynn | Structure and Hydrolysis of the U(IV), U(V), and U(VI) Aqua Ions from Ab Initio Molecular Simulations[END_REF][START_REF] Kovács | Quantum Chemical Calculations and Experimental Investigations of Molecular Actinide Oxides[END_REF][START_REF] Jackson | Density Functional Theory Study of the Complexation of the Uranyl Dication with Anionic Phosphate Ligands with and without Water Molecules[END_REF][START_REF] Vallet | Probing the Nature of Chemical Bonding in Uranyl(VI) Complexes with Quantum Chemical Methods[END_REF][START_REF] Tecmer | Charge-Transfer Excitations in Uranyl Tetrachloride ([UO 2 Cl 4 ] 2-): How Reliable are Electronic Spectra from Relativistic Time-Dependent Density Functional Theory?[END_REF][START_REF] Rios | Roles of Acetone and Diacetone Alcohol in Coordination and Dissociation Reactions of Uranyl Complexes[END_REF][START_REF] Averkiev | How accurate are electronic structure methods for actinoid chemistry?[END_REF][START_REF] Wei | Geometries, electronic structures, and excited states of UN2, NUO + , and UO 2+ 2 : a combined CCSD(T), RAS/CASPT2 and TDDFT study[END_REF][START_REF] Ruipérez | An ab Initio Theoretical Study of the Electronic Structure of UO + 2 and[END_REF][START_REF] Réal | Benchmarking Electronic Structure Calculations on the Bare UO 2+ 2 Ion: How Different are Single and Multireference Electron Correlation Methods?[END_REF][START_REF] Jackson | Prediction of Vibrational Frequencies of UO 2+ 2 at the CCSD(T) Level[END_REF][START_REF] Sonnenberg | Theoretical Investigations of Uranyl-Ligand Bonding: Four-and Five-Coordinate Uranyl Cyanide, Isocyanide, Carbonyl, and Hydroxide Complexes[END_REF][START_REF] Pierloot | Electronic structure and spectrum of UO 2+ 2 and UO 2 Cl 2-4[END_REF][29][START_REF] Craw | Ab Initio Quantum Chemical Calculations on Uranyl UO 2+ 2 , Plutonyl PuO 2+ 2 , and Their Nitrates and Sulfates[END_REF][START_REF] Heinemann | NUO + , a New Species Isoelectronic to the Uranyl Dication UO 2+ 2[END_REF][START_REF] Pyykkö | Quasirelativistic Pseudopotential Study of Species Isoelectronic to Uranyl and the Equatorial Coordination of Uranyl[END_REF].

Analogous to experiment, computational chemistry can be used to calculate electronic transitions, allowing for the creation of electronic spectra from calculated excitation energies. Multireference computational methods have been used to directly solve for electronic states and transitions between states in the molecule [START_REF] Wei | Geometries, electronic structures, and excited states of UN2, NUO + , and UO 2+ 2 : a combined CCSD(T), RAS/CASPT2 and TDDFT study[END_REF][START_REF] Ruipérez | An ab Initio Theoretical Study of the Electronic Structure of UO + 2 and[END_REF][START_REF] Réal | Benchmarking Electronic Structure Calculations on the Bare UO 2+ 2 Ion: How Different are Single and Multireference Electron Correlation Methods?[END_REF][START_REF] Pierloot | Electronic structure and spectrum of UO 2+ 2 and UO 2 Cl 2-4[END_REF]. Reál et al. compared preexisting CASPT2 electronic transitions of uranyl to intermediate Hamiltonian Fock space coupled cluster (IHF-SCC) excitations and found that the two methods perform similarly for low level excitations, but the methods deviate from each other for higher excitation energies [START_REF] Réal | Benchmarking Electronic Structure Calculations on the Bare UO 2+ 2 Ion: How Different are Single and Multireference Electron Correlation Methods?[END_REF]. However, multireference methods scale unfavorably with the size of the system, and as a result, can only be utilized for the smallest of molecular systems. A commonly used alternative to multi-reference methods for the simulation of molecular spectra is time-dependent DFT (TDDFT), which have been used to great effect to calculate the electronic transitions in uranyl complexes [START_REF] Wei | Geometries, electronic structures, and excited states of UN2, NUO + , and UO 2+ 2 : a combined CCSD(T), RAS/CASPT2 and TDDFT study[END_REF][START_REF] Pierloot | Electronic spectrum of UO 2+ 2 and [UO 2 Cl 4 ] 2calculated with time-dependent density functional theory[END_REF][START_REF] Tecmer | Electronic spectroscopy of UO 2+ 2 , NUO + and NUN: an evaluation of time-dependent density functional theory for actinides[END_REF][START_REF] Tecmer | Reliable modeling of the electronic spectra of realistic uranium complexes[END_REF]. Tecmer et al. performed TDDFT calculations to simulate the UV-Vis spectrum of uranyl as well as the isoelectronic analogues OUN + and UN 2 . They found that among the functionals used, CAMB3LYP, M06, and PBE0 gave the lowest mean errors relative to IHFSCC, performing similarly to CASPT2 overall [START_REF] Tecmer | Electronic spectroscopy of UO 2+ 2 , NUO + and NUN: an evaluation of time-dependent density functional theory for actinides[END_REF]. The complex polarization propagator (CPP) method [START_REF] Norman | Near-resonant absorption in the time-dependent self-consistent field and multiconfigurational self-consistent field approximations[END_REF][START_REF] Norman | Nonlinear response theory with relaxation: the first hyperpolarizability[END_REF][START_REF] Autschbach | Time-Dependent Density Functional Calculations of Optical Rotatory Dispersion Including Resonance Wavelengths as a Potentially Usedful tool for Determining Absolute Configurations of Chiral Molecules[END_REF][START_REF] Devarajan | Two-component relativistic density functional method for computing nonsingular complex linear response of molecules based on the zeroth order regular approximation[END_REF][START_REF] Kasper Kristensen | Quasienergy formulation of damped response theory[END_REF][START_REF] Coriani | Coupled-cluster response theory for near-edge x-ray-absorption fine structure of atoms and molecules[END_REF] is similar to time-dependent methods, but explicitly accounts for the linewidth of the peaks via an imaginary damping factor which, when graphed, generates the spectrum. At the Hartree-Fock level, 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] Carravetta | Computational x-ray spectroscopy[END_REF][START_REF] Hans Ågren | Static exchange and cluster modeling of core electron shakeup spectra of surface adsorbates: Co/cu(100)[END_REF] models core excitations as an electron being acted upon by the core ionized molecule, the latter of which is optimized separately in order to give a complete account of the orbital relaxation in the molecule.

Accounting for relativistic effects is of great importance in calculations on actinide elements, especially in the calculation of properties of deep core orbitals as spin-orbit is incredibly strong for these orbitals. The most direct means of accounting for relativity is the full four-component Dirac-Coulomb (DC) Hamiltonian. The DC Hamiltonian gives a full account of the one-electron relativistic effects in a molecule, whereas the fully relativistic two-electron interaction is truncated, but includes the instantaneous Coulomb interaction as well as spin-own orbit interaction. [START_REF] Saue | Relativistic Hamiltonians for Chemistry: A Primer[END_REF] It has been used to calculate properties of uranyl and its complexes [START_REF] Tecmer | Charge-Transfer Excitations in Uranyl Tetrachloride ([UO 2 Cl 4 ] 2-): How Reliable are Electronic Spectra from Relativistic Time-Dependent Density Functional Theory?[END_REF][START_REF] Ruipérez | An ab Initio Theoretical Study of the Electronic Structure of UO + 2 and[END_REF][START_REF] Réal | Benchmarking Electronic Structure Calculations on the Bare UO 2+ 2 Ion: How Different are Single and Multireference Electron Correlation Methods?[END_REF]. Relativistic effective core potentials (RECPs) are another means of accounting for relativistic effects for heavy atoms, and have been used frequently to recover these effects and to simplify the costly calculations involved [START_REF] Atta-Fynn | Structure and Hydrolysis of the U(IV), U(V), and U(VI) Aqua Ions from Ab Initio Molecular Simulations[END_REF][START_REF] Jackson | Density Functional Theory Study of the Complexation of the Uranyl Dication with Anionic Phosphate Ligands with and without Water Molecules[END_REF][START_REF] Wei | Geometries, electronic structures, and excited states of UN2, NUO + , and UO 2+ 2 : a combined CCSD(T), RAS/CASPT2 and TDDFT study[END_REF][START_REF] Réal | Benchmarking Electronic Structure Calculations on the Bare UO 2+ 2 Ion: How Different are Single and Multireference Electron Correlation Methods?[END_REF][START_REF] Jackson | Prediction of Vibrational Frequencies of UO 2+ 2 at the CCSD(T) Level[END_REF][START_REF] Pyykkö | Quasirelativistic Pseudopotential Study of Species Isoelectronic to Uranyl and the Equatorial Coordination of Uranyl[END_REF][START_REF] Tecmer | Reliable modeling of the electronic spectra of realistic uranium complexes[END_REF][START_REF] Dolg | Accurate Relativistic Small-Core Pseudopotentials for Actinides. Energy Adjustment for Uranium and First Applications to Uranium Hydride[END_REF][START_REF] Evarestov | Electronic structure of crystalline uranium nitrides UN, U 2 N 3 and UN 2 : LCAO calculations with the basis set optimization[END_REF][START_REF] Gagliardi | Uranium triatomic compounds XU Y (X,Y=C,N,O): a combined multiconfigurational second-order perturbation and density functional study[END_REF][START_REF] Küchle | Energy-adjusted pseudopotentials for the actinides. Parameter sets and test calculations for thorium and thorium monoxide[END_REF]. Dolg and Cao developed a relativistic small core pseudopotential for uranium constructed from Dirac-Coulomb-Breit (DCB) orbitals, compared it to all-electron DKH calculations with perturbative spin-orbit (+BP), and noted that the vertical excitation energies agreed well with the values obtained at the all-electron level, at far reduced computational cost [START_REF] Dolg | Accurate Relativistic Small-Core Pseudopotentials for Actinides. Energy Adjustment for Uranium and First Applications to Uranium Hydride[END_REF]. However, since the chemical core is simplified and treated by a simple function, part of the core level description is lost and no core level properties (such as coreionization energies) can be calculated. The large and small component of the wave function can also be decoupled yielding two (or one) component Hamiltonians for the purpose of reducing computational cost relative to the full four-component calculation [START_REF] Averkiev | How accurate are electronic structure methods for actinoid chemistry?[END_REF][START_REF] Wei | Geometries, electronic structures, and excited states of UN2, NUO + , and UO 2+ 2 : a combined CCSD(T), RAS/CASPT2 and TDDFT study[END_REF][START_REF] Réal | Benchmarking Electronic Structure Calculations on the Bare UO 2+ 2 Ion: How Different are Single and Multireference Electron Correlation Methods?[END_REF][START_REF] Pierloot | Electronic structure and spectrum of UO 2+ 2 and UO 2 Cl 2-4[END_REF]29,[START_REF] Pierloot | Electronic spectrum of UO 2+ 2 and [UO 2 Cl 4 ] 2calculated with time-dependent density functional theory[END_REF]. The eXact 2-Component relativistic Hamiltonian (X2C) [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] has been used by Reál et al. to calculate relativistic effects for uranyl compounds and has resulted in similar effectiveness to Dirac-Coulomb in the calculation of vertical excitation energies [START_REF] Réal | Benchmarking Electronic Structure Calculations on the Bare UO 2+ 2 Ion: How Different are Single and Multireference Electron Correlation Methods?[END_REF]. Klooster et al. [START_REF] Klooster | Calculation of X-ray photoelectron spectra with the use of the normalized elimination of the small component method[END_REF] have reported calculations of X-ray photoelectron spectra, including U 5+ using the analogous normalized elimination of the small component (NESC) Hamiltonian. Calculations on uranium compounds including scalar relativistic effects through the Darwin and mass-velocity terms have also been reported [29,[START_REF] Heinemann | NUO + , a New Species Isoelectronic to the Uranyl Dication UO 2+ 2[END_REF].

Two isoelectronic homologues of uranyl have been synthesized, OUN + and UN 2 , and have been characterized using experimental and theoretical methods [START_REF] Wei | Geometries, electronic structures, and excited states of UN2, NUO + , and UO 2+ 2 : a combined CCSD(T), RAS/CASPT2 and TDDFT study[END_REF][START_REF] Heinemann | NUO + , a New Species Isoelectronic to the Uranyl Dication UO 2+ 2[END_REF][START_REF] Pyykkö | Quasirelativistic Pseudopotential Study of Species Isoelectronic to Uranyl and the Equatorial Coordination of Uranyl[END_REF][START_REF] Tecmer | Electronic spectroscopy of UO 2+ 2 , NUO + and NUN: an evaluation of time-dependent density functional theory for actinides[END_REF][START_REF] Evarestov | Electronic structure of crystalline uranium nitrides UN, U 2 N 3 and UN 2 : LCAO calculations with the basis set optimization[END_REF][START_REF] Michael | Formation of Bare UO 2+ 2 and NUO + by Fragmentation of Gas-Phase Uranyl-Acetonitrile Complexes[END_REF]. Both of these molecules possess uranium in the +6 formal oxidation state and, analogous to uranyl, have also been shown to have linear structures [START_REF] Kovács | Quantum Chemical Calculations and Experimental Investigations of Molecular Actinide Oxides[END_REF][START_REF] Wei | Geometries, electronic structures, and excited states of UN2, NUO + , and UO 2+ 2 : a combined CCSD(T), RAS/CASPT2 and TDDFT study[END_REF][START_REF] Jackson | Prediction of Vibrational Frequencies of UO 2+ 2 at the CCSD(T) Level[END_REF]29,[START_REF] Heinemann | NUO + , a New Species Isoelectronic to the Uranyl Dication UO 2+ 2[END_REF][START_REF] Pyykkö | Quasirelativistic Pseudopotential Study of Species Isoelectronic to Uranyl and the Equatorial Coordination of Uranyl[END_REF][START_REF] Tecmer | Electronic spectroscopy of UO 2+ 2 , NUO + and NUN: an evaluation of time-dependent density functional theory for actinides[END_REF][START_REF] Evarestov | Electronic structure of crystalline uranium nitrides UN, U 2 N 3 and UN 2 : LCAO calculations with the basis set optimization[END_REF][START_REF] Gagliardi | Uranium triatomic compounds XU Y (X,Y=C,N,O): a combined multiconfigurational second-order perturbation and density functional study[END_REF][START_REF] Zhe-Yan | A CCSD(T) Study on Structures and Harmonic Frequencies of the Isoelectronic Uranium Triatomic Species OUO 2+ , NUN and NUO +[END_REF][START_REF] Hunt | Matrix infrared spectra of NUN formed by the insertion of uranium atoms into molecular nitrogen[END_REF]. The geometries and valence electronic spectra for these molecules have been determined computationally. However, less is known about the core spectra of these molecules from a theoretical standpoint (especially for OUN + and UN 2 ).

Therefore, to classify and characterize these new molecules, we have in the present work simulated the uranium L 3 edge XANES spectrum for UO 2+ 2 , OUN + and UN 2 using restricted excitation window (REW) TDDFT, the CPP method, as well as STEX. We have also investigated the position of the uranium L 3 ionization threshold using both wave function and density functional methods. The paper is outlined as follows: In section 2 we present the theory behind the methods employed in this work. In section 3 we provide computational details and then, in section 4, present and discuss our results. In section 5 we provide conclusions and perspectives.

Theory

Core ionization

Koopmans' theorem [START_REF] Koopmans | Über die Zuordnung von Wellenfunktionen und Eigenwerten zu den Einzelnen Elektronen Eines Atoms[END_REF] provides a reasonable estimate of valence ionization energies, although it is based on the difference of Hartree-Fock energies between the ionized and the parent state, using the orbitals of the parent state. There are accordingly two major sources of error, that is, i) lack of orbital relaxation of the core-ionized state and ii) lack of electron correlation:

IP i (M ) = -ε i (M ) + ∆ relax + ∆ corr .
The relaxation contribution ∆ relax (-∆ relax is denoted the contraction error by Koopmans [START_REF] Koopmans | Über die Zuordnung von Wellenfunktionen und Eigenwerten zu den Einzelnen Elektronen Eines Atoms[END_REF]) will be negative since orbital relaxation lowers the energy of the core-ionized state. The correlation contribution ∆ corr is, on the other hand, expected to be positive since there is one more electron to correlate in the parent state (see for instance [START_REF] Mulliken | Quelques Aspects de la théorie des orbitales moléculaires[END_REF]). In practice, for valence ionizations, the two contributions are found to be of the same order of magnitude, there by providing fortunate error cancellation, as illustrated for the HOMO (1b 1 ) ionization of the water molecule in table 1. For core ionizations, on the other hand, the Koopmans estimate is known to be poor [START_REF] Bagus | Self-Consistent-Field Wave Functions for Hole States of Some Ne-Like and Ar-Like Ions[END_REF] since the correlation contribution is typically an order of magnitude smaller than the relaxation contribution, as seen in table 1 for the 1a 1 (oxygen 1s) ionization energy.

In the present work the correlation contribution is calculated by ∆MP2. The core-ionized state is first obtained by a Kramers restricted average-of-configuration Hartree-Fock (HF) calculation [START_REF] Thyssen | Development and Applications of Methods for Correlated Relativistic Calculations of Molecular Properties[END_REF] starting from the orbitals of the parent state. Convergence is straightforward and obtained by first 12.61 [START_REF] Siegbahn | ESCA applied to free molecules[END_REF] reordering the orbitals such that the target core orbital is in the position of the open shell and then kept there by overlap selection. This method has been used since the molecular ∆SCF calculations of Bagus and Schaefer [START_REF] Bagus | Direct Near-Hartree-Fock Calculations on the 1s Hole States of NO +[END_REF][START_REF] Almlöf | MOLECULE-ALCHEMY program package[END_REF], but has more recently been rediscovered under the name Maximum Overlap Method by Gill and co-workers [START_REF] Andrew | Self-Consistent Field Calculations of Excited States Using the Maximum Overlap Method (MOM)[END_REF]. For the ∆MP2 calculation we have employed the RELCCSD module of DIRAC [START_REF] Dirac | a relativistic ab initio electronic structure program, Release DIRAC14[END_REF] which uses an Kramers unrestricted formalism [START_REF] Visscher | Formulation and implementation of a relativistic unrestricted coupled-cluster method including noniterative connected triples[END_REF] and thereby allows simple open-shell calculations. However, since the incoming molecular orbitals are optimized under Kramers restriction, the reconstructed Fock matrix for the core-ionized state is not diagonal and will have have a non-zero occupied-virtual (ov ) block. We have therefore extended the MP2 algorithm to handle this case. We start from the electronic Hamiltonian normal-ordered with respect to the Fermi vacuum defined by the current (Kramers restricted) orbital set

H N = pq f p q a † p a q + 1 4 pqrs V pr qs a † p a † r a s a q ; V pr qs = pr || qs .
Following Lauderdale et al. [START_REF] Walter | Many-body perturbation theory with a restricted open-shell Hartree-Fock reference[END_REF], we then define the zeroth-order Hamiltonian to be the diagonal blocks (oo and vv ) of the Fock matrix. Setting up perturbation theory in a coupled cluster (CC) framework we subsequently derive the non-canonical MP2 energy

E ncM P 2 = ai f i a t a(1) i + 1 4 ij ab V ij ab t ab(1) ij . ( 1
)
Here and in the following we employ indices i, j, k, l for occupied orbitals, a, b, c, d for virtual orbitals and p, q, r, s for general orbitals. The equations of the first-order CC amplitudes are

b f a b t b(1) i - j t a(1) j f j i = -f a i c f b c t ac(1) ij + f a c t bc(1) ij - k f k j t ab(1) ik + f k i t ab(1) jk = -V ab ij .
Starting from these, one may show that the non-canonical MP2 energy is invariant under separate rotation of occupied and virtual orbitals. The equation for the first-order T 1 -amplitudes may be recognized as the Sylvester equation and can therefore be solved in a direct fashion, but we have for convenience chosen to use the existing iterative scheme in RELCCSD for the solution of the amplitude equations.

Core excitation

In this work we are exploring three different methods for the calculation of core excitation spectra: restricted excitation window time-dependent density functional theory (REW-TDDFT), complex polarization propagator (CPP) and the static exchange approximation (STEX). In this section we give a brief presentation and comparison of these methods.

A common starting point for the three methods is the frequency-dependent linear response function which in the exact state formalism [START_REF] Olsen | Linear and nonlinear response functions for an exact state and for an MCSCF state[END_REF][START_REF] Olsen | Time-dependent response theory with applications to selfconsistent field and multiconfigurational self-consistent field wave functions[END_REF][START_REF] Bast | Relativistic adiabatic time-dependent density functional theory using hybrid functionals and noncollinear spin magnetization[END_REF] A; B ω = -

1 m>0 A * m B m ω m -ω + B * m A m ω m + ω (2)
involves an explicit sum over the excited states |m of the zeroth-order Hamiltonian. In the above expression ω m = E m -E 0 are excitation energies with respect to the unperturbed ground state |0

and P m = m ĤP 0 , (P = A, B) the corresponding transition moments with respect to property operator ĤP . In the present work we restrict ourselves to the electric dipole approximation, so that both property operators are limited to components of the electric dipole operator, although the short wave length of X-ray radiation may require the inclusion of terms beyond this approximation [START_REF] Bernadotte | Origin-independent calculation of quadrupole intensities in X-ray spectroscopy[END_REF][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]. It is clear from the above expression that a scan of the linear response function through a frequency window will display poles corresponding to excitations in the range allowed by the property operators ĤA and ĤB and whose transition moments can be extracted from the residues. The singularities are unphysical, though, in that they correspond to infinitely long lifetimes of the excited states. This feature may be amended by the introduction of inverse lifetimes γ m through the substitution ω m → ω m -iγ m in the above expression [START_REF] Kraus | Chemistry of Aqueous Uranium(V) Solutions. I. Preparation and Properties. Analogy between Uranium(V), Neptunium(V) and Plutonium(V)[END_REF]. The response function then becomes generally complex, with the real part corresponding to refractive properties such as polarizabilities and the imaginary part associated with absorption processes. Within the framework of Hartree-Fock and Kohn-Sham methods we may employ an exponential parametrization of orbitals (and thereby density and energy)

ϕ i (κ) = a ϕ a exp [-κ] ai ; κ pq = -κ * qp
which allows for unconstrained optimization and straightforward identification of redundancies [START_REF] Levy | Multi-configurational Self-Consistent Wavefunctions of Formaldehyde[END_REF][START_REF] Joergensen | Time-dependent hartree-fock calculations in the pariserparr-pople model. applications to aniline, azulene and pyridine[END_REF][START_REF] Dalgaard | Optimization of orbitals for multiconfigurational reference states[END_REF][START_REF] Saue | Four-component relativistic Kohn-Sham theory[END_REF]. In the present case we restrict ourselves to closed-shell references in which only rotations between occupied and virtual orbitals, with amplitudes κ ai , are non-redundant; all other amplitudes may therefore be set to zero. At the SCF level of theory the frequency-dependent linear response function may be formulated as

A; B ω = E [1] † A X B (ω) ,
where the vector X B (ω) contains the first-order orbital rotation amplitudes

X B (ω) = K(ω) K * (-ω) ; K ai (ω) = κ (1) ai (ω).
It is a solution of the linear response equation

E [2] 0 -ωS [2] X B (ω) = -E [1] B ,
where appears the property gradient

E [1] B = g g * ; g ai = -ϕ a ĤB ϕ i
as well as the the generalized metric S [2] and the electronic Hessian E

[2] 0 , with structures

S [2] = I 0 0 -I ; E [2] 0 = A B B * A * ; A ai,bj = ∂ 2 E0 ∂κ * ai ∂κ bj κ=0 B ai,bj = ∂ 2 E0 ∂κ * ai ∂κ * bj κ=0 . ( 3
)
Further discussion of the SCF linear response formalism can be for instance be found in references [START_REF] Olsen | Linear and nonlinear response functions for an exact state and for an MCSCF state[END_REF][START_REF] Christiansen | Response Functions from Fourier Component Variational Perturbation Theory Applied to a Time-Averaged Quasienergy[END_REF][START_REF] Saue | Post Dirac-Hartree-Fock Methods -Properties[END_REF]. Since the SCF linear response function do not contain any sum over states, its complex extension can not be obtained by the introduction of state-specific inverse lifetimes γ m . It is therefore common practice to employ a single damping parameter γ which can be interpreted as an imaginary extension of the perturbing frequency ω → ω +iγ [START_REF] Norman | Near-resonant absorption in the time-dependent self-consistent field and multiconfigurational self-consistent field approximations[END_REF][START_REF] Norman | Nonlinear response theory with relaxation: the first hyperpolarizability[END_REF][START_REF] Autschbach | Time-Dependent Density Functional Calculations of Optical Rotatory Dispersion Including Resonance Wavelengths as a Potentially Usedful tool for Determining Absolute Configurations of Chiral Molecules[END_REF][START_REF] Devarajan | Two-component relativistic density functional method for computing nonsingular complex linear response of molecules based on the zeroth order regular approximation[END_REF][START_REF] Kasper Kristensen | Quasienergy formulation of damped response theory[END_REF][START_REF] Coriani | Coupled-cluster response theory for near-edge x-ray-absorption fine structure of atoms and molecules[END_REF]. A relativistic implementation of complex response, including spin-orbit interaction, has been reported by Devarajan et al. [START_REF] Devarajan | Two-component relativistic density functional method for computing nonsingular complex linear response of molecules based on the zeroth order regular approximation[END_REF]. However, it is based on the zeroth order regular approximation (ZORA), which may not be very accurate for core excitations, as pointed out by the authors themselves. In the present contribution we employ the complex response implementation of the dirac code [START_REF] Villaume | Linear complex polarization propagator in a four-component Kohn-Sham framework[END_REF], which can be used with the more accurate 4-component Dirac-Coulomb and eXact 2-Component (X2C) Hamiltonians.

Working within the electric dipole approximation the isotropic oscillator strength f iso , including a Lorentzian linewidth defined by the damping parameter γ, is obtained directly by a scan of the imaginary part of the isotropic electric dipole polarizability α iso through the desired frequency window f iso (ω) = 2mω πe 2 Im α iso (ω + iγ) . Alternatively, excitation energies can be found by TDDFT (or TDHF), that is, by solving the generalized eigenvalue problem

E [2] 0 -ω m S [2] X m = 0.
An inconvenience with this approach is that excitation energies are typically found by a "bottomup" approach, which becomes highly impractical for core excitations. A solution is to restrict the occupied orbitals entering the orbitals rotation amplitudes {κ ai } to the desired core orbitals. This is referred to a the "restricted excitation window" [START_REF] Stener | Time dependent density functional theory of core electrons excitations[END_REF] or "restricted channel" [START_REF] Ekström | Time-dependent molecular properties in the optical and x-ray regions[END_REF] approach. In the present work we employ the relativistic adiabatic TDDFT implementation reported by Bast et al. [START_REF] Bast | Relativistic adiabatic time-dependent density functional theory using hybrid functionals and noncollinear spin magnetization[END_REF], where restriction are possible both on occupied and virtual orbitals, such that the extension to REW-TDDFT is straightforward.

Transition moments are found by contracting the eigenvectors X m with the corresponding property gradient

P m = X † m (ω)E [1]
P . The isotropic oscillator strength associated with excitation m is then obtained as

f iso m = 2mω m 3 e 2 α=x,y,z |µ m,α | 2 . (4)
Cumulated isotropic oscillator strengths, including a Lorentzian broadening ∆ L , are then obtained as

f iso (ω) = m f iso m ∆ L (ω; ω m , γ) ; ∆ L (ω; ω m , γ) = 1 γπ γ 2 (ω -ω m ) 2 + γ 2 .
The resulting simulated spectra obtained by complex response and REW-TDDFT are expected to be identical to the extent that no other occupied orbitals are involved in the excitation processes with the selected frequency window (channel coupling) and to the extent that the REW-TDDFT calculation includes a sufficient number of excitation to cover the frequency window. In passing we note that upon a change of energy units the Lorentz-broadened oscillators strengths are scaled down by the same factor as the energy is scaled up, in order to conserve the integrated oscillator strength.

A third method investigated in the present work is the static exchange approximation (STEX). The name originated in the context of early theoretical investigations of the scattering of electrons by hydrogen atoms [START_REF] Burke | The Low-Energy Scattering of Electrons and Positrons by Hydrogen Atoms[END_REF][START_REF] Mott | The Theory of Atomic Collisions[END_REF]. The two-electron wave function of the system was expanded in products of hydrogen atom orbitals and orbitals of the projectile electron [START_REF] Massey | The Collision of Slow Electrons with Atoms. I. General Theory and Elastic Collisions[END_REF]. The static exchange approximation was obtained by restricting the atomic orbital in this expansion to the ground state 1s orbital of the target hydrogen atom, thus neglecting polarization of the atomic charge density during collision, yet retaining exchange effects, shown by Morse and Allis in 1933 to have some importance upon scattering with slow electrons [START_REF] Morse | The Effect of Exchange on the Scattering of Slow Electrons from Atoms[END_REF]. This is apparently the first STEX calculation. A further development was the observation by Hunt and Goddard [START_REF] Hunt | Excited States of H 2 O using improved virtual orbitals[END_REF] that the optimal virtual orbital ϕ a in the otherwise frozen N -electron singly-excited determinant Φ a i is obtained by diagonalization of the orbital-specific Fock operator

F (N -i) = ĥ + N j =i Ĵj -Kj
where Ĵj and Kj are the usual Coulomb and exchange operators, respectively. The diagonalization is carried out in the space of virtual orbitals, thus keeping the occupied orbitals frozen. The assumption that the other occupied orbitals are hardly modified upon excitation is an instance of the STEX approximation, as pointed out by Langhoff [START_REF] Rescigno | K-shell photoionization in molecular nitrogen[END_REF][START_REF] Langhoff | Applications of the Stieltjes-Tchebycheff procedure for atomic and molecular photoionization calculations in Hilbert space[END_REF][START_REF] Sheehy | Correlation of molecular valence-and K-shell photoionization resonances with bond lengths[END_REF]. The so-called improved virtual orbitals (IVO) generated in this manner contrasts with the canonical virtual HF orbitals generated from the usual Fock operator

F N = F (N -i) + Ĵi -Ki
and which are more appropriate for the (N +1)-electron system. Not surprisingly then, the orbitalspecific Fock operator F (N -i) is also the conventional Fock operator for the (N -1)-electron system obtained by removal of the occupied orbital ϕ i from the system. Based on the above observations, Ågren and co-workers [START_REF] Hans Ågren | Direct, atomic orbital, static exchange calculations of photoabsorption spectra of large molecules and clusters[END_REF][START_REF] Carravetta | Computational x-ray spectroscopy[END_REF] proposed to extend the IVO approach to core excitations and notably to build the STEX operator F (N -i) using the occupied orbitals of the corresponding core ionized system, thus capturing orbital relaxation essentially missing in TD-DFT/HF. Transition moments are calculated between the parent ground state and the core excited states, the latter built from the core ionized orbitals. Since two non-orthogonal orbital sets are used, special techniques, such as a cofactor expansion [START_REF] Löwdin | Quantum Theory of Many-Particle Systems. I. Physical Interpretations by Means of Density Matrices, Natural Spin-Orbitals, and Convergence Problems in the Method of Configurational Interaction[END_REF], must be used. In the present work we are using the 4-component relativistic STEX implementation of Ekström et al. [START_REF] Ekström | Relativistic four-component staticexchange approximation for core-excitation processes in molecules[END_REF].

The core excitation energies obtained by a STEX calculation can be reproduced by a REW-TDHF calculation using the orbitals of the core ionized system and invoking the Tamm-Dancoff approximation (TDA), that is, setting B = 0 in the electronic Hessian [START_REF] Kraus | Chemistry of Aqueous Uranium(V) Solutions. II. Reaction of Uranium Pentachloride with Water. Thermodynamic Stability of UO2+. Potential of U(IV)/(V), U(IV)/(VI) and U(V)/(VI) Couples[END_REF]. If excitations are restricted to a single (core) orbital ϕ i the elements of the remaining A block can be expressed as

A ai,bi = Φa i ĤN Φb i -δ ab Φ0 ĤN Φ0 = F N ab -δ ab F N ii -ãĩ bĩ = F N -i ab -δ ab F N -i ii ,
In the above expression we employ the tilde symbol to indicate quantities calculated in the orbitals of the core ionized system. Upon diagonalization of the A block, we obtain the eigenvalues of the orbital-specific Fock operator F (N -i) , shifted by F N -i ii , which can be recognized as the negative of the core ionization energy, calculated in the frozen orbitals of the core ionized system, contrary to Koopmans' theorem, who uses the frozen orbitals of the parent system. It is corrected by rather using the ionization energy obtained by ∆SCF.

Projection analysis

In order to elucidate the electronic structure of the title species as well as to assign the simulated core excitation spectra we have performed projection analysis [START_REF] Dubillard | Bonding analysis using localized relativistic orbitals: Water, the ultrarelativistic case and the heavy homologues H 2 X, (X = Te, Po, eka-Po)[END_REF]. This method is akin to Mulliken population analysis, but the strong basis-set dependence of the latter method is avoided by expanding molecular orbitals in pre-calculated orbitals of the atoms constituting the molecule

|ψ i = Aj ψ A j c A ji + ψ pol i , (5)
where indices A and j refer to atoms and atomic orbitals (AOs), respectively. Charges and populations of atoms in the molecule are subsequently calculated in analogous manner to Mulliken population analysis, but starting from molecular orbitals given as linear combinations of true and well-defined atomic orbitals, rather than in terms of atom-centered basis functions. The atoms are calculated in their proper basis and by default in their ground state configuration, either by average-of-configuration at the HF level or by using fractional occupation at the DFT level. In order to make the projection analysis chemically meaningful, the expansion in Eq. ( 5) is normally limited to AOs occupied in the atomic ground state configuration. However, for the assignment of the calculated core excitation spectra these orbitals, in the case of uranium, were supplemented by selected improved virtual orbitals, as discussed in section 4.3. In either case, the selected set of AOs is not guaranteed to fully span a given molecular orbital. The orthogonal complement ψ pol i , which we denote the polarization contribution, can be eliminated using the Intrinsic Atomic Orbital scheme of Knizia [START_REF] Knizia | Intrinsic Atomic Orbitals: An Unbiased Bridge between Quantum Theory and Chemical Concepts[END_REF].

Computational details

Reference geometries were optimized at the scalar-relativistic CCSD(T) level using the MOLPRO 09 [START_REF] Werner | MOLPRO, version 2009.1, a package of ab initio programs[END_REF] package and numerical gradients. For uranium we employed a relativistic small core potential (ECP60MDF) with a (14s13p10d8f6g)/[6s6p5d4f3g] quadruple zeta level valence ANO basis set developed by Dolg and Cao [START_REF] Dolg | Accurate Relativistic Small-Core Pseudopotentials for Actinides. Energy Adjustment for Uranium and First Applications to Uranium Hydride[END_REF], and the (11s6p3d2f)/[5s4p3d2f] aug-cc-pVTZ basis sets [START_REF] Thom | Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen[END_REF][START_REF] Kendall | Electron-affinities of the 1st-row atoms revisited -systematic basis-sets and wave-functions[END_REF] for oxygen and nitrogen.

All other calculations were carried out with the dirac code [START_REF] Dirac | a relativistic ab initio electronic structure program, Release DIRAC13[END_REF] and are, unless otherwise stated, based on the 4-component Dirac-Coulomb Hamiltonian using the simple Coulombic correction [START_REF] Visscher | Approximate molecular relativistic Dirac-Coulomb calculations using a simple Coulombic correction[END_REF] to avoid the explicit calculation of two-electron integrals involving the small components only. For uranium we employed the dyall.v3z basis set [START_REF] Kenneth | Relativistic double-zeta, triple-zeta, and quadruple-zeta basis sets for the actinides Ac-Lr[END_REF] (large component 33s29p20d13f4g2h) and for oxygen and nitrogen the cc-pVTZ basis set [START_REF] Thom | Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen[END_REF] (large components 10s5p2d1f). For MP2 calculations we switched to the slightly larger dyall.ae3z basis (large component 33s29p20d13f7g3h) for uranium. All basis sets were uncontracted and the small components generated by restricted kinetic balance. A finite nucleus model in the form of a Gaussian charge distribution was employed [START_REF] Visscher | Dirac-Fock atomic electronic structure calculations using different nuclear charge distributions[END_REF].

Uranium 2p ionization energies were calculated by ∆SCF calculations [START_REF] Bagus | Self-Consistent-Field Wave Functions for Hole States of Some Ne-Like and Ar-Like Ions[END_REF], both at the Kramersrestricted Hartree-Fock and Kohn-Sham level, the latter using the BLYP [START_REF] Becke | Density-functional exchange-energy approximation with correct asymptotic behavior[END_REF][START_REF] Lee | Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density[END_REF][START_REF] Miehlich | Results obtained with the correlation energy density functionals of becke and Lee, Yang and Parr[END_REF], 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], PBE [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF], PBE0 [START_REF] Adamo | Toward reliable density functional methods without adjustable parameters: The PBE0 model[END_REF], and CAM-B3LYP [START_REF] Yanai | A new hybrid exchange-correlation functional using the Coulomb-attenuating method (CAM-B3LYP)[END_REF] functionals. Convergence of the core excited states was straightforward using initial reordering of orbitals followed by selection of orbitals based on overlap with starting orbitals during the SCF cycles.

The uranium L 3 edge XANES spectrum of the selected molecules was simulated by REW-TDDFT, CPP and STEX calculations, the former two using the CAM-B3LYP [START_REF] Yanai | A new hybrid exchange-correlation functional using the Coulomb-attenuating method (CAM-B3LYP)[END_REF] functional. Transition moments have been calculated within the electric dipole approximation, more specifically in the length gauge, that is, as integrals over the electric dipole operator. The nature of the excitations was determined from the excitation amplitudes combined with Mulliken and projection analysis [START_REF] Dubillard | Bonding analysis using localized relativistic orbitals: Water, the ultrarelativistic case and the heavy homologues H 2 X, (X = Te, Po, eka-Po)[END_REF] of the involved molecular orbitals. In the REW-TDDFT and STEX calculations finite linewidths of the individual peaks were introduced by Lorentzian functions of half-width at half-maximum (HWHM) γ = 0.0367 E h (∼ 1eV). The same value of γ was taken as damping parameter in the CPP calculations.

Results and discussions 4.1 Molecular and electronic structures

Prior to the calculation of core ionization and excitation energies we optimized the geometries of the title species and investigated their electronic structures by projection analysis [START_REF] Dubillard | Bonding analysis using localized relativistic orbitals: Water, the ultrarelativistic case and the heavy homologues H 2 X, (X = Te, Po, eka-Po)[END_REF]. In Table 2 we report our calculated bond lengths for the isoelectronic series together with selected literature values. Wei et al. [START_REF] Wei | Geometries, electronic structures, and excited states of UN2, NUO + , and UO 2+ 2 : a combined CCSD(T), RAS/CASPT2 and TDDFT study[END_REF] reported bond lengths for NUO + and UN 2 calculated at the CCSD(T) level using the aug-cc-pVTZ basis set [START_REF] Thom | Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen[END_REF] for the ligands. For uranium the authors employed the rela-tivistic small core potential ECP60MWB with the accompanying (12s11p10d8f)/[8s7p6d4f] valence basis, although it was developed for SCF calculations. [START_REF] Küchle | Energy-adjusted pseudopotentials for the actinides. Parameter sets and test calculations for thorium and thorium monoxide[END_REF] Jackson et al. [START_REF] Jackson | Prediction of Vibrational Frequencies of UO 2+ 2 at the CCSD(T) Level[END_REF] reported bond distances for UO 2+ 2 with the same computational setup, except that they added two g functions to the valence basis. Switching to the larger segmented valence basis set (14s13p10d8f6g)/[10s9p5d4f3g] developed by Cao and co-workers [START_REF] Cao | Valence basis sets for relativistic energyconsistent small-core actinide pseudopotentials[END_REF][START_REF] Cao | Segmented contraction scheme for small-core lanthanide pseudopotential basis sets[END_REF] and further augmentation by h and i functions was found to have only a small effect on calculated bond lengths. More recently, Tu et al. reported bond lengths for the entire isoelectronic series with basically the same computational setup. [START_REF] Zhe-Yan | A CCSD(T) Study on Structures and Harmonic Frequencies of the Isoelectronic Uranium Triatomic Species OUO 2+ , NUN and NUO +[END_REF] They reproduce the uranyl bond length reported by Jackson et al. [START_REF] Jackson | Prediction of Vibrational Frequencies of UO 2+ 2 at the CCSD(T) Level[END_REF], and get slightly shorter bond distances than reported by Wei et al. [START_REF] Wei | Geometries, electronic structures, and excited states of UN2, NUO + , and UO 2+ 2 : a combined CCSD(T), RAS/CASPT2 and TDDFT study[END_REF] for the other species. We have optimized bond lengths for the isoelectronic species using the more recent ECP60MDF core potential with the accompanying valence basis [START_REF] Dolg | Accurate Relativistic Small-Core Pseudopotentials for Actinides. Energy Adjustment for Uranium and First Applications to Uranium Hydride[END_REF] and interestingly get somewhat longer bond lengths, closer to those reported by Gagliardi and Roos [START_REF] Gagliardi | Uranium triatomic compounds XU Y (X,Y=C,N,O): a combined multiconfigurational second-order perturbation and density functional study[END_REF] at the CASPT2 level with ANO basis sets. Particularly noteworthy is that the U-N bond in the nitridooxouranium cation is shorter than the U-O bond, although experiment suggest that that the former is weaker than the latter (bond dissociation energies BDE[OU + -N] = 4.44±1.27 eV vs. BDE[NU + -O] = 7.66±1.70 eV). [START_REF] Heinemann | NUO + , a New Species Isoelectronic to the Uranyl Dication UO 2+ 2[END_REF] Bond CCSD(T)[pw] CCSD(T) CCSD(T) [START_REF] Zhe-Yan | A CCSD(T) Study on Structures and Harmonic Frequencies of the Isoelectronic Uranium Triatomic Species OUO 2+ , NUN and NUO +[END_REF] CASPT2 [START_REF] Gagliardi | Uranium triatomic compounds XU Y (X,Y=C,N,O): a combined multiconfigurational second-order perturbation and density functional study[END_REF] We have also investigated the electronic structure of the title compounds by projection analysis [START_REF] Dubillard | Bonding analysis using localized relativistic orbitals: Water, the ultrarelativistic case and the heavy homologues H 2 X, (X = Te, Po, eka-Po)[END_REF] at the HF level using the pre-calculated atomic orbitals occupied in the electronic ground state of the constituent atoms. Polarization contributions have been eliminated by polarizing the atomic orbitals in the molecule according to the Intrinsic Atomic Orbital scheme of Knizia [START_REF] Knizia | Intrinsic Atomic Orbitals: An Unbiased Bridge between Quantum Theory and Chemical Concepts[END_REF], yet conserving overlap between atomic orbitals on different centers. The charge and electronic configuration of uranium in the three molecules are given in Table 3. Concerning the electronic configurations, one in particular notes the 6p-hole [START_REF] Pyykko | Relativistically parameterized extended Hueckel calculations. 8. Double-.zeta. parameters for the actinoids thorium, protactinium, uranium, neptunium, plutonium, and americium and an application on uranyl[END_REF], primarily arising from overlap between the 6p 3/2 orbital with the ligands, and which is basically identical for the three species. The calculated atomic charges, which do not suffer from the strong basis set dependence of Mulliken charges, are far from the formal oxidation state +VI of uranium in these molecules, in agreement with previous theoretical and experimental studies. [START_REF] Denning | Electronic structure and bonding in actinyl ions[END_REF] The uranium charge furthermore reduces according to the total molecular charge, as expected. The ligand charge is -0.42e and in -0.73e in UO 2+ 2 and UN 2 , respectively. In NUO + the charge on oxygen and nitrogen is -0.63e and -0.49e, respectively. The calculated dipole moment of NUO + is -1.43 D, when the uranium atom is placed at the origin with the nitrogen atom along the positive axis. Interestingly, for certain initial start guesses the HF SCF procedure converges to a solution 0.2 E h higher in energy and with atomic charges very slightly modified (Q U =+2.16e, Q O =-0.70e, Q N = -0.45e), but enough to switch the sign of the calculated dipole moment (+1.55 D).

Molecule

Q U Atomic configuration UO 2+ 2 +2.84 5f 2.26 6p 5.67 6d 1.20 7s 0.04 NUO + +2.12 5f 2.52 6p 5.67 6d 1.60 7s 0.10 UN 2 +1.45 5f 2.68 6p 5.66 6d 2.01 7s 0.23

Table 3: Charge and electronic configuration of uranium in the title compounds obtained by projection analysis at the HF level.

Canonical orbitals are quite suitable for the description of electron detachment and excitation processes, such as XPS and XAS, respectively. However, in order to "see" chemical bonds one needs to rotate the occupied molecular orbitals to form localized ones [START_REF] Hund | Zur Frage der chemischen Bindung[END_REF][START_REF] Mulliken | Spectroscopy, molecular orbitals and chemical bonding[END_REF], although there is no unique localization criterion. In Table 4 we present a projection analysis of bonding orbitals obtained by Pipek-Mezey localization [START_REF] Pipek | A fast intrinsic localization procedure applicable for ab initio and semiempirical linear combination of atomic orbital wave functions[END_REF]. The bonding orbitals are identificed as localized molecular orbitals with significant contributions from both the uranium center and a (single) ligand. Approximate orbital eigenvalues have been calculated as expectation values of the converged Fock operator. For each ligand we find three such bonding orbitals, of which two are almost degenerate and with ω = 1/2 and 3/2, respectively. Based on our analysis we conclude that each ligand is bound to the central uranium atom by triple (σ,π) bonds, where the π bond has been split by spin-orbit interaction into π 1/2 and π 3/2 , and where the metal center contributes df hybrid atomic orbitals.

ω ε U 6p 3/2 U 5f 5/2 U 5f 7/2 U 6d 3/2 U 6d 5/2 U 7s 1/2 X

Uranium 2p binding energy

In table 5 we report relaxation ∆ relax and correlation ∆ corr contributions to the uranium 2p ionization energies of the title species, for uranium using the dyall.ae3z basis set which include correlation functions for all occupied orbitals. The calculations are based on the default Hamiltonian of the dirac package, that is, the 4-component relativistic Dirac-Coulomb Hamiltonian with a simple Coulomb correction [START_REF] Visscher | Approximate molecular relativistic Dirac-Coulomb calculations using a simple Coulombic correction[END_REF], thus avoiding the calculations of two-electron integrals (SS|SS) containing small component basis functions only. At the HF level the uranium 2p orbitals are split by 3838 eV due to spin-orbit interaction, and the 2p 3/2 orbital further split by a mere 0.3 eV due to the molecular field. As illustrated by figure 1, the ∆HF uranium 2p binding energies are a linear function, all with slope 13.4 eV/e, of the atomic charges reported in table 3, thus demonstrating the chemical shift of X-ray photoelectron spectroscopy [START_REF] Sokolowski | Chemical Shift Effect in Inner Electronic Levels of Cu Due to Oxidation[END_REF][START_REF] Hagström | Electron spectroscopic determination of the chemical valence state[END_REF] as well as the usefulness of the atomic charges obtained from projection analysis. We have defined ∆ corr as the difference between the ionization energy (IP) obtained at the ∆MP2 and ∆HF level, that is,

Method UO 2+ 2 OUN + UN2 2p -1 1/2,1/2 2p -1 3/2,1/2 2p -1 3/2,3/2 2p -1 1/2,1/2 2p -1 3/2,1/2 2p -1 3/2,3/2 2p -1 1/2,1/2 2p -1 3/2,1/2 2p -1 3/2,
∆ corr = IP (∆MP2 ) -IP (∆HF ) ,
but it should be emphasized that due to the non-canonical nature of the orbitals of the core ionized states, there will be non-zero T 1 contributions to the MP2 energy, that are more properly associated with relaxation (see Eq.( 1)). However, these contributions are negligible in the calculations reported in Table 5, as will be discussed and demonstrated below. Table 5 shows that the correlation contribution ∆ corr is an order of magnitude smaller than the relaxation contribution ∆ relax and that the Koopmans estimate gives errors on the order of 75 eV. What is striking, though, is that the correlation contribution has the same sign as the relaxation contribution, that is, it is negative. If we consider the canonical MP2 energy expression

E M P 2 = i<j e ij ; e ij = a<b | ij || ab | 2 ε i + ε j -ε a -ε b ,
with N o occupied and N v virtual orbitals for the parent state, then the core-ionized state has (N o -1) fewer pair energies e ij , all of them negative in the parent state. In addition, the remaining pair energies has N v new contributions containing the now virtual core orbital, such that denominators may be zero or even positive. Indelicato and co-workers [START_REF] Indelicato | Relativistic effects, correlation, and QED corrections on K α transitions in medium to very heavy atoms[END_REF][START_REF] Mooney | Precision measurements of K and L transitions in xenon: Experiment and theory for the K , L , and M levels[END_REF][START_REF] Indelicato | Relativistic and many-body effects in K, L, and M shell ionization energy for elements with 10 ≤ Z ≤ 100 and the determination of the 1s Lamb shift for heavy elements[END_REF], in the framework of many-body perturbation theory, makes a distinction between contributions to the ionization energy for which |ε i + ε j | > |ε h |, where ε h is the energy of the virtual core orbital, say b, and contributions for which |ε i + ε j | < |ε h | and refer to them core-core and Auger effects, respectively. Core-core contributions only occur if there are core orbitals lower in energy than the ionized one. The denominator is generally negative, but may change sign if the second virtual orbital, say a is bound, which are precisely the orbitals associated with pre-edge structure in X-ray absorption spectroscopy. The denominator of Auger effect contributions, on the other hand, starts off positive with increasing energy of the second virtual orbital, but eventually become negative for sufficiently high-lying virtuals. Such contributions were observed by Nooijen and Bartlett [START_REF] Nooijen | Description of core-excitation spectra by the openshell electron-attachment equation-of-motion coupled cluster method[END_REF] to lead to convergence problems in coupled-cluster calculations of core-ionized states and were therefore ignored.

In the present non-canonical MP2 calculations we do not make a distinction between Auger and core-core contributions, but simply monitor contributions to the pair correlation energy of positive sign. For the 1a 1 ionization energy of water the total contribution is quite small (0.13 eV), whereas we obtain 3.43 eV for the uranium 2p The above discussion confirms the expectation that the correlation contribution should be positive. However, a negative contribution, arising from a strong coupling of correlation and relaxation, can not be excluded. To investigate this, we carried out non-canonical MP2 calculations on core-ionized uranyl as well as water using the molecular orbitals of the parent state. The results are shown in table 6. Using the orbitals of the parent state (denoted "frozen" in the table) the relaxation contribution ∆ relax is by definition zero, to within numerical noise. The correlation contribution ∆ corr , on the other hand, becomes negative for both water and uranyl. However, when decomposing ∆ corr further into T 1 and T 2 contributions, according the the non-canonical MP2 expression of Eq. ( 1), one observes that the T 1 -contribution, which can be associated with relaxation, is completely dominating. For water the T 2 -contribution ∆ corr (T 2 ) remains positive and is basically the same as when using relaxed orbitals. For uranyl, on the other, hand the T 2contribution using frozen orbitals is positive, leading us to conclude that the negative correlation contribution obtained with the relaxed orbitals is indeed due to a strong coupling of correlation and relaxation. Table 7: Ionization energies (in eV) obtained using the dyall.v3z basis set for uranium.

Method UO 2+ 2 OUN + UN2 2p -1 1/2,1/2 2p -1 3/2,1/2 2p -1 3/2,3/2 2p -1 1/2,1/2 2p -1 3/2,1/2 2p -1 3/2,3/2 2p -1 1/2,1/2 2p -1 3/2,1/2 2p -1 3/2,
We now turn to the effect of extensions to the default Hamiltonian of the dirac package. In table 7 we report the effect of explicit inclusion of the (SS|SS) class of integrals (∆∆DOSSSS). Although the effect is sizable, causing a reduction of binding energies on the order of 18 and 6 eV for uranium 2p 1/2 and 2p 3/2 orbitals, respectively, it is constant for all three isoelectronic species, and thus does not contribute to the chemical shift. Even more important is the effect of the inclusion of the Gaunt two-electron interaction, but again the chemical shift is not affected, and so we have ignored these both (SS|SS) integrals and the Gaunt term in the subsequent calculations. In passing we note that the Gaunt term reduces the spin-orbit splitting of the uranium 2p manifold by about 40 eV, which makes sense, since the Gaunt term contains the spin-other-orbit interaction [START_REF] Saue | Relativistic Hamiltonians for Chemistry: A Primer[END_REF].

We have also investigated the performance of a selection of DFT functionals for the calculation of uranium 2p binding energies of the title species. These are reported in table 7 relative to the HF binding energies. To the extent that the difference between HF and DFT binding energies can be interpreted as pure correlation contributions, we note that these are significantly larger in magnitude than the correlation contributions extracted from the ∆MP2 calculations. It should be noted that the core ionized species have been calculated under Kramers restriction such that spin polarization, which is expected to reduce ionization energies, is missing. The GGA functionals PBE and BLYP reduce both uranium 2p 1/2 and 2p 3/2 binding energies, whereas the global hybrid functionals PBE0 and B3LYP, as well as the long-range corrected hybrid CAMB3LYP, decrease the 2p 3/2 binding energies and increase the 2p 1/2 ones. No DFT functional have a performance similar to MP2.

No experimental uranium L 2 or L 3 binding energies are available for the title species. The 2p 1/2 and 2p 3/2 binding energies of metallic uranium is 20948 and 17166 eV, respectively, relative to the Fermi level [START_REF] Bearden | Reevaluation of X-Ray Atomic Energy Levels[END_REF]. If we add the (SS|SS) and Gaunt contributions as corrections, then the MP2 ionization energies for UN 2 , in which uranium has the smallest charge, agree to within 10 eV with the cited experimental numbers. If we instead linearly extrapolate the MP2 ionization energies to zero uranium nuclear charge at the HF level and add the cited corrections, we underestimate the experimental numbers by about 30 eV. In this section we present and analyze simulated uranium L 3 edge XANES spectra of the title species. A good discussion of the electronic structure of actinyls has been given by Denning [START_REF] Robert | Electronic Structure and Bonding in Actinyl Ions and their Analogs[END_REF]. Here focus will be on the bound virtual orbitals. We start by considering the calculated uranium L 3 edge XANES spectrum for uranyl obtained by the STEX method. In the left panel of figure 2 we compare the spectra obtained with three different basis sets. These are local Gaussian basis sets which are not appropriate for the description of continuum states [START_REF] Cacelli | The calculation of photoionisation cross sections of simple polyatomic molecules by L 2 methods[END_REF], as can be seen from the lack of any convergence of the spectra with respect to basis sets beyond the L 3 edge. Such artifacts have been observed previously, and it was suggested by Ekström and Norman that the meaningful energy range of a simulated spectrum in a local basis can be ascertained by exponent scaling [START_REF] Ekström | X-ray absorption spectra from the resonant-convergent first-order polarization propagator approach[END_REF]. In the present case, a zoom into the pre-edge region of the spectrum, as shown in the right panel of figure 2 suggests that the spectrum is not fully converged for quasi-bound states in the vicinity of the ionization threshold. Convergence in this region would probably require a rather extensive set of diffuse functions. In the following we shall therefore focus on the first three peaks of the spectrum. We shall also compare the performance of the three different methods discussed in section 2.2. We start by demonstrating numerically that the STEX excitation energies can be obtained by using a REW-TDHF calculation within the Tamm-Dancoff approximation and using the orbitals of the core ionized state. In figure 3 the uranium L 3 edge XANES spectrum for uranyl calculated by the two methods is displayed. As discussed in section 2.2 both the STEX and the TDHF/TDA excitation energies has been corrected as

Uranium L 3 edge XANES spectra

ω m → ω m + E 0 -Ẽ0
where E 0 and Ẽ0 is the energy of the parent state calculated in the parent and core ionized orbitals, respectively. The same correction is applied to the oscillator strengths, Eq.( 4). The excitation energies are indeed seen to match perfectly, whereas the REW-TDHF/TDA Lorentz-broadened oscillator strengths are systematically smaller than the STEX ones.

A direct comparison of STEX on the one hand and CPP and REW-TDDFT on the other hand is complicated by the fact that the latter spectra are significantly shifted with respect to the experimental L 3 edge due to the combined effect of missing orbital relaxation and self-interaction errors [START_REF] Tu | Core ionization potentials from self-interaction corrected Kohn-Sham orbital energies[END_REF]. In order to align the spectra we have therefore considered the position of the L 3 ionization threshold within a linear response regime. We may consider the process of ionization as Molecule Estimated IP(U2p an extreme case of a charge-transfer excitation in which the separation between donor and acceptor tends towards infinity. Following the arguments of Dreuw et al. [START_REF] Dreuw | Long-range charge-transfer excited states in time-dependent density functional theory require non-local exchange[END_REF], it then becomes clear that the excitation energy reduces, for any DFT functional, to the energy difference between the acceptor and donor orbital. Since this excitation energy is also equal to the difference between the ionization potential of the donor and the electron affinity of the acceptor we are led to the conclusion that within TDDFT (and TDHF) the ionization threshold is de facto given by Koopmans' theorem, that is, as the negative of the donor orbital. We have tested this conclusion numerically by forcing an excitation from the selected core orbital to a remote tight ghost function placed 100 a 0 away from the uranium atom along the molecular axis and then subtracting the ghost orbital eigenvalue from the resulting excitation energy. The results are given in table 8 and clearly confirms the validity of our conclusion. On the other hand, it should be emphasize that the meaning of orbital eigenvalues is different in Hartree-Fock and exact Kohn-Sham theory. [START_REF] Chong | Interpretation of the Kohn-Sham orbital energies as approximate vertical ionization potentials[END_REF][START_REF] Gritsenko | Physical interpretation and evaluation of the Kohn-Sham and Dyson components of the ε-I relations between the Kohn-Sham orbital energies and the ionization potentials[END_REF][START_REF] Gritsenko | The analog of Koopmans' theorem for virtual Kohn-Sham orbital energies[END_REF] As a consequence we have shifted the excitation energies obtained with CPP and REW-TDDFT according to

ω m → ω m + IP i (∆HF ) + ε KS i (6)
From figures 4, 5 and 6 it is seen that these shifts, on the order of 150 eV, clearly bring the CPP and REW-TDDFT spectra into the same energy region as the STEX one. The CPP and REW-TDDFT spectra agree perfectly, as they should (cf. section 2.2); the slight deviation observed towards higher photon energies in figure 5 is simply due to an insufficient number of excitations calculated at the REW-TDDFT level. For uranyl the CPP and REW-TDDFT spectra are seen to match the STEX one very well, but the agreement deteriorates as the molecular charge is reduced along the isoelectronic series. For OUN + we note in particular new pre-edge features with respect to STEX. It should be pointed out, however, that the uranium 2p 3/2 natural width is 7.43 eV [START_REF] Krause | Natural widths of atomic K and L levels, Kα X-ray lines and several KLL Auger lines[END_REF], although the experimental energy resolution can be reduced down to about 4 eV using partial fluorescence yield techniques [START_REF] Vitova | High energy resolution x-ray absorption spectroscopy study of uranium in varying valence states[END_REF][START_REF] Hämäläinen | Elimination of the innershell lifetime broadening in x-ray-absorption spectroscopy[END_REF]. This still means that the above-mentioned pre-edge structures of the theoretical spectrum can not be resolved by present-day experiment.

Amongst the three methods STEX offers perhaps the most straightforward assignment of spectra. This is because XANES spectroscopy, in an orbital picture, probes bound virtual orbitals which, as discussed in section 2.2, has been optimized by calculating the core ionized state. As an illustration we may note that in the dyall.v3z basis an HF calculation on the uranyl ground state gives 41 bound virtual orbitals (Kramers pairs). whereas for OUN + this number is reduced to 15 and for UN 2 there are none. In contrast, the corresponding numbers for the uranium 2p 3/2ionized state are 46, 35 and 15, respectively. We have been able to carry out a detailed assignment of the STEX uranium L 3 XANES spectra using projection analysis [START_REF] Dubillard | Bonding analysis using localized relativistic orbitals: Water, the ultrarelativistic case and the heavy homologues H 2 X, (X = Te, Po, eka-Po)[END_REF]. For the ligands we have used the ground state orbitals. For uranium the ground state occupied orbitals were supplemented by the bound improved virtual orbitals generated by freezing the ground state orbitals and then recalculating the virtual orbitals for the 2p 3/2 -ionized state. The first peak of the uranyl STEX spectrum (cf. figure 4) is thereby found to be dominated by excitations to uranium 6d orbitals, but also a virtual orbitals of uranium 7s character. These virtual orbitals have essentially no ligand character. The second peak is assigned as excitations to virtual orbitals dominated by uranium 6d, but about 25 % ligand character. The third and final peak before the ionization threshold is dominated by excitations to uranium 7d orbitals. Moving to OUN + the molecular charge is reduced by one unit and the number of peaks before the ionization threshold to two; both are dominated by excitations to uranium 6d orbitals, but the second peak also has some ligand character (23 % nitrogen and 6% oxygen). Finally, for neutral UN 2 there is a single peak before the ionization threshold dominated by excitations to uranium 6d and with no ligand character. The same assignment basically carries over to the CPP and REW-TDDFT spectra, but with less precision since the virtual orbitals are less optimal. The extra features of the second peak of the OUN + spectrum (cf. figure 5) appears to be due to a larger splitting of the ω = 3/2 and ω = 5/2 components of the uranium 6d 5/2 orbitals.

Conclusions and perspectives

In the present work we have studied the processes of ionization and excitation out of the uranium 2p 3/2 orbital in the isoelectronic species UO 2+ 2 , OUN + and UN 2 at the 4-component relativistic level. Molecular geometries were reoptimized at the CCSD(T) level using small-core scalar relativistic pseudopotentials and correlation-consistent basis sets, and the electronic structure studied by projection analysis in localized orbitals. Using the extracted uranium atomic charges we find a perfectly linear chemical shift of uranium 2p 3/2 ionization energies obtained by ∆HF. We confirm the failure of Koopmans' theorem for core ionization due to the dominance of relaxation contributions over correlation ones. More unexpected is that the correlation contribution ∆ corr is negative for all three species, meaning that the parent state has less correlation energy than the core ionized state. Our analysis suggests that this is due to a strong coupling of relaxation and correlation. Uranium 2p 3/2 ionization energies calculated by ∆SCF using different DFT functionals do not agree very well with our ∆MP2 values, but this situation might improve by the introduction of spin polarization in a Kramers unrestricted formalism.

To describe core excitations we have investigated three methods and shown how they are related. In particular, we show how STEX excitation energies, but not intensities, can be reproduced by TDHF calculations within the Tamm-Dancoff approximation. We also show that for the same Lorentz broadening REW-TDDFT and CPP give identical spectra. The CPP method has a certain ease of application in that the spectrum is directly simulated by scanning the desired frequency region, without any worry about the appropriate number of excitations to include. On the other hand, CPP lacks some flexibility in that spectra are only simulated for one specific damping parameter and would therefore have to be recalculated if another value was chosen. Although Koopmans' theorem fails for core excitations, it is the correct approximation of ionization potentials in the linear response regime, and this observation has allowed us to introduce shifts (cf. Eq (6)), on the order of 150 eV, to align REW-TDDFT and CPP uranium L 3 XANES spectra with the STEX ones. Since orbital relaxation dominates over electron correlation for core excitations, the ionization threshold of STEX spectra are in the vicinity of experimental ones. The interpretation of STEX spectra is furthermore more straightforward in that the virtual orbitals of the core ionized state are optimal. We accordingly obtain a detailed assignment of our calculated STEX spectra using projection analysis, notably with improved virtual orbitals of the uranium atom. On the other hand, it has been claimed (see for instance [START_REF] Zhang | Core and valence excitations in resonant X-ray spectroscopy using restricted excitation window timedependent density functional theory[END_REF]) that REW-TDDFT (and thus CPP) gives better relative peak positions and intensities than STEX compared to experiment due to the inclusion of electron correlation. In the present work no direct comparison with experiment was made. In future work we plan to address this issue in detail. It should also be pointed out that the molecules in the present study are closed shell in their parent state, which is rather the exception in the domain of f-elements. As pointed out by Roemelt et al. [START_REF] Roemelt | A combined DFT and restricted open-shell configuration interaction method including spin-orbit coupling: Application to transition metal L-edge X-ray absorption spectroscopy[END_REF], TDDFT (and therefore also CPP and STEX) has simply not enough parameters to handle the general open-shell case. A challenge for the future is therefore to develop cost-effective methods for the simulation of X-ray spectra of actinide species.

Figure 1 :

 1 Figure 1: Shifted ∆HF uranium 2p ionization energies of the title species (in eV) as a function of the uranium charge (in a.u.) from projection analysis.

Figure 2 :

 2 Figure 2: (left) UO 2+ 2 uranium L 3 edge XANES spectra simulated by STEX using different basis sets and a Lorentzian broadening of ∼ 1eV. The vertical line indicates the ionization threshold obtained at the ∆HF level (the value changes less than 0.1 eV with the indicated basis sets). To the right a zoom of the pre-edge region.

Figure 3 :

 3 Figure 3: Comparison of UO 2+ 2 uranium L 3 edge XANES spectra obtained by STEX and REW-TDHF, in both cases adding a Lorentzian broadening of ∼ 1eV. The REW-TDHF calculations has been carried out within the Tamm-Dancoff approximation and using the orbitals of the core ionized state. The vertical line indicates the ionization threshold obtained at the ∆HF level. The REW-TDHF and STEX excitation energies are both corrected by the difference between the ground state energy calculated in ground state and core ionized orbitals.

Figure 4 :

 4 Figure 4: UO 2+ 2 uranium L 3 edge XANES spectra simulated by STEX, CPP(CAM-B3LYP) and REW-TDDFT(CAM-B3LYP), including a Lorentzian broadening of ∼ 1 eV. The two latter spectra have been shifted by 152.63 eV according to (6).

Figure 5 :

 5 Figure 5: OUN + uranium L 3 edge XANES spectra simulated by STEX, CPP(CAM-B3LYP) and REW-TDDFT(CAM-B3LYP), including a Lorentzian broadening of ∼ 1 eV. The two latter spectra have been shifted by 151.73 eV according to (6).

Figure 6 :

 6 Figure 6: UN 2 uranium L 3 edge XANES spectra simulated by STEX, CPP(CAM-B3LYP) and REW-TDDFT(CAM-B3LYP), including a Lorentzian broadening of ∼ 1 eV. The two latter spectra have been shifted by 150.98 eV according to (6).

Table 1 :

 1 Valence and core ionization energies (in eV) of gaseous water obtained using the dyall.ae3z basis set.

			539.7[61]
	Koopmans	13.84	560.1
	∆HF	11.05	539.6
	∆MP2	12.75	540.5
	∆ relax	-2.79	-20.5
	∆ corr	1.70	0.9

Table 2 :

 2 Calculated bond lengths (in Å) for the title species.

	PBE[34]

  X2s 1/2 X2p 1/2 X2p 3/2

	UO 2+ 2	1/2 -1.487	0.09	0.19	0.18	0.08	0.08	0.00	O	0.10	0.34	0.88
		1/2 -1.078	0.00	0.10	0.12	0.12	0.09	0.00	O	0.00	1.09	0.43
		3/2 -1.075	0.00	0.07	0.16	0.05	0.16	0.00	O	0.00	0.00	1.52
	NUO + 1/2 -1.147	0.06	0.13	0.12	0.09	0.08	0.00	O	0.12	0.35	0.98
		1/2 -0.790	0.00	0.08	0.10	0.11	0.09	0.00	O	0.00	1.16	0.43
		3/2 -0.787	0.00	0.06	0.13	0.04	0.16	0.00	O	0.00	0.00	1.59
		1/2 -1.146	0.19	0.28	0.20	0.11	0.07	0.01	N	0.02	0.23	0.76
		1/2 -0.689	0.01	0.13	0.19	0.20	0.17	0.00	N	0.00	0.94	0.31
		3/2 -0.683	0.01	0.12	0.23	0.09	0.28	0.00	N	0.00	0.00	1.24
	UN2	1/2 -0.826	0.16	0.19	0.16	0.10	0.09	0.03	O	0.03	0.31	0.81
		1/2 -0.427	0.00	0.12	0.14	0.21	0.16	0.00	O	0.00	0.96	0.38
		3/2 -0.425	0.00	0.09	0.19	0.08	0.28	0.00	O	0.00	0.00	1.33

Table 4 :

 4 Projection analysis of Pipek-Mezey localized bonding orbitals in the title compounds at the HF level. ε refers to the expectation value (in E h ) of the converged Fock operator. X refers to the ligand.

Table 5 :

 5 Ionization energies (in eV) obtained with dyall.ae3z basis set for uranium.

Table 6 :

 6 3/2,3/2 ionization of uranyl. Relaxation and correlation contributions (in eV) to core ionization energies of water and uranyl. See text for further details.

	Molecule	IP	Orbital set ∆ relax	∆ corr ∆ corr (T 1 ) ∆ corr (T 2 )
	H 2 O	1a 1	relaxed	-20.51	0.92	-0.17	1.08
			frozen	0.00	-34.58	-35.50	0.92
	UO 2+ 2	2p 3/2,3/2	relaxed	-64.54	-5.17	-0.28	-4.89
			frozen	0.06 -103.91	-106.08	2.17

Table 8 :

 8 3/2 ) in eV -ε 2p 3/2 (eV) Estimated ionization potential of uranium 2p 3/2 orbital (in eV) based on excitation to a tight ghost function placed 100 a 0 away from uranium, compared to the negative orbital energy -ε 2p 3/2 (averaged over the m j components). All results have been obtained with the CAMB3LYP functional.

	UO 2+ 2 OUN +	17104.31 17095.03	17104.41 17095.13
	UN 2	17087.25	17087.34

Acknowledgements TS and DM acknowledges financial support from the Indo-French Centre for the Promotion of Advanced Research (IFCPAR project No. 4705-3), including the PhD-grant for AS . Computing time from CALMIP (Calcul en Midi-Pyrenées) is gratefully acknowledged. TS would like to acknowledge helpful discussions with Paul Bagus (Denton) , Vincenzo Carravetta (Pisa), Ulf Ekström (Oslo), Eva Lindroth (Stockholm), Patrick Norman (Linköping) and Tonya Vitova (Karlsruhe). AKW acknowledges support of the National Science Foundation under Grant No. CHE-1362479. Computing support, in part, was provided by the Computing and Information Technology Center at the University of North Texas. Support is provided by the U.S. Department of Energy (DOE) for the Center for Advanced Scientific Computing and Modeling (CASCaM) at the University of North Texas. AKW and CS acknowledge support from the Global Research Experiences, Exchange, and Training (GREET) program, a program of the American Chemical Society. We are happy to dedicate this paper to Evert Jan Baerends on the occasion of his 70th birthday.