Thomas J Penfold
email: tom.penfold@ncl.ac.uk

Etienne Gindensperger
email: egindensperger@unistra.fr

Chantal Daniel
email: c.daniel@unistra.fr

Christel M Marian
email: christel.marian@hhu.de

The Spin-Vibronic Mechanism for Intersystem Crossing

Intersystem crossing (ISC), formally forbidden within non-relativistic quantum theory, is the mechanism by which a molecule changes its spin state. It plays an important role in the excited state decay dynamics of many molecular systems and not just those containing heavy elements. In the simplest case ISC is driven by direct spin-orbit coupling between two states of different multiplicity. This coupling is usually assumed to remain unchanged by vibrational motion, i.e. the Condon approximation. It is also often presumed that spin-allowed radiationless transitions, i.e. internal conversion, and the nonadiabatic coupling that drives them, can be considered separately from ISC and spin-orbit coupling owing to the vastly different timescales upon which these 1 processes are assumed to occur. However, these assumptions are too restrictive. Indeed, the strong mixing brought about by the simultaneous presence of nonadiabatic and spin-orbit coupling means that often the spin, electronic and vibrational dynamics cannot be described independently. Instead of considering a simple ladder of states, as depicted in a Jablonski diagram, one must consider the more complicated spinvibronic levels. Despite the basic ideas being outlined in the 1960's, it is only with the advent of high-level theory and femtosecond spectroscopy that the importance of the spin-vibronic mechanism for ISC in both fundamental as well as applied research fields has been revealed with significant impact across chemistry, physics and biology.

In this review article, we present the theory and fundamental principles of the spinvibronic mechanism for ISC. This is followed by empirical rules to estimate the rate of ISC within this regime. The most recent developments in experimental techniques, theoretical methods and models for the spin-vibronic mechanism are discussed. These concepts are subsequently illustrated with examples, including the ISC mechanisms in transition metal complexes, small organic molecules and organic chromophores.

Introduction

The development of molecular spectroscopy in the mid-twentieth century made it possible to address many fundamental issues relating to the response of complex polyatomic molecules to light. However, this response often becomes quite complicated and involves many intertwined radiative and non-radiative decay processes, such as luminescence, internal conversion (IC), intersystem crossing (ISC), electron/energy transfer, which are often accompanied or driven by structural rearrangements including dissociation. Knowledge of these primary events, which often occur within the first picoseconds (ps) after excitation, is critical for understanding fundamental excited-state processes, but also controlling the functional prop-erties of molecular excited states that are exploited for a wide number of applications across photochemistry, photophysics and photobiology.

The complexity of excited-state dynamics means that early empirical concepts, 1,2 developed for interpreting the spectral features, kinetics, lifetime and luminescence efficiency, are a matter of debate. Indeed, the interpretation of spectroscopic observables founded on the simple qualitative analyses of Kasha 3 and El-Sayed 4 can lead to erroneous conclusions as far as the photophysics and luminescent properties of polyatomic molecules are concerned. In the case of El-Sayed's rules directly concerned with ISC, early examples of this breakdown were observed in the 1960's for the phosphorescence properties of small organic molecules, such as benzene. [START_REF] Albrecht | Vibronic Spin-Orbit Perturbations and the Assignment of the Lowest Triplet State of Benzene[END_REF][START_REF] Lim | Vibronic Interaction between (n, π*) and (π, π*) States and Spin Orbit Coupling in Nitrogen Heterocyclics[END_REF][START_REF] Lim | Vibronic Spin-Orbit Interactions in Heteroaromatic Molecules. II. Phosphorescence of Quinoxaline and Other Diazanaphthalenes[END_REF][START_REF] Henry | Spin-Orbit Coupling in Aromatic Hydrocarbons. Calculation of the Radiative Triplet Lifetimes of Naphthalene, Anthracene, and Phenanthrene[END_REF][START_REF] Henry | Spin-Orbit Coupling in Aromatic Hydrocarbons. Analysis of Nonradiative Transitions between Singlet and Triplet States in Benzene and Naphthalene[END_REF] Importantly, understanding the dynamics of high-spin states and the mechanisms by which they are generated is of great importance as they can be both detrimental, such as non-emissive decay pathways [START_REF] Rose | Unusually short excited state lifetimes of indenofluorene and fluorenofluorene derivatives result from a conical intersection[END_REF] or fluorescence quenching [START_REF] Kamkaew | BODIPY dyes in photodynamic therapy[END_REF] and exploited, e.g. triplet-triplet up conversion, [START_REF] Singh-Rachford | Photon upconversion based on sensitized triplet-triplet annihilation[END_REF] spin crossover, [START_REF] Shatruk | Symmetry-breaking structural phase transitions in spin crossover complexes[END_REF][14][15] photodynamic therapy, 16 energy transfer triggering, 17 building of luminescent assemblies, 18 photocatalysis 19 or triplet exciton harvesting. [START_REF] Goswami | π-Conjugated Organometallic Isoindigo Oligomer and Polymer Chromophores: Singlet and Triplet Excited State Dynamics and Application in Polymer Solar Cells[END_REF][START_REF] Chen | Triplet Energy Transport in Platinumacetylide Light Harvesting Arrays[END_REF] A step change in our understanding was achieved through the advent of femtosecond laser spectroscopy, [22][START_REF] Zewail | Femtochemistry: Atomic-Scale Dynamics of the Chemical Bond[END_REF][START_REF] Mukamel | Principles of Nonlinear Optical Spectroscopy[END_REF] which made it possible to probe, in real time, the ultrafast dynamics in photoexcited molecules and to decipher complicated excited state dynamics over a large range of time scales. In this context, one of the most striking examples is the ultrafast ISC kinetics in transition metal complexes , which are often characterised by a high number of various electronic excited states of different multiplicities in a limited domain of energy. [START_REF] Chergui | On the Interplay Between Charge, Spin and Structural Dynamics in Transition Metal Complexes[END_REF] This is exemplified by the ultrafast light induced spin-crossover dynamics of [Fe(bpy) 3 ] 2+ . [START_REF] Cannizzo | Light-induced spin crossover in Fe(II)-based complexes: The Full Photocycle Unraveled by Ultrafast Optical and X-ray Spectroscopies Coord[END_REF][START_REF] Zhang | Tracking Excited State Charge and Spin Dynamics in Iron Coordination Complexes[END_REF][START_REF] Auböck | Sub-50-fs Photoinduced Spin Crossover in [Fe (bpy) 3] 2+[END_REF][START_REF] Lemke | Coherent Structural Trapping Through Wavepacket Dispersion During Photoinduced Spin State Switching[END_REF] Photoexcitation of the 1 MLCT band of [Fe(bpy) 3 ] 2+ leads to population of a non-emissive quintet MC ( 5 T 2 ) state within 100 fs. [START_REF] Consani | Vibrational Coherences and Relaxation in the High-Spin State of Aqueous [Fe II (bpy) 3 ] 2+ Angew[END_REF][START_REF] Mccusker | Sub picosecond 1MLCT-5T2 Intersystem Crossing of Low-Spin Polypyridyl Ferrous Complexes[END_REF][START_REF] Bressler | Femtosecond XANES Study of the Light-Induced Spin Crossover Dynamics in an Iron(II)[END_REF] Such dynamics involve two spin transitions and therefore are remarkable given that the timescales involved are significantly faster than the timescale usually expected for a single ISC event. This is accompanied in other molecules by intriguing effects such as ISC rates varying with the vibrational period of important normal modes rather than heavy atom effects [START_REF] Cannizzo | Femtosecond Fluorescence and Intersystem Crossing in Rhenium (I) Carbonyl-Bipyridine Complexes[END_REF] or which are strongly solvent dependent. [START_REF] Van Der Veen | Vibrational Relaxation and Intersystem Crossing of Binuclear Metal Complexes in Solution[END_REF] For a long time it was believed that spin-forbidden non-radiative transitions in purely organic compounds were slow in comparison to their spin-allowed counterparts and often their presence in excited state processes was neglected. However, there is increasing evidence that even in compounds composed of only light elements, ISC can occur on the subpicosecond time scale and is thus competitive with IC. This is strongly illustrated by aromatic carbonyl compounds of the xanthone type and nitro aromatic compounds. The unusual photophysical and photochemical properties of nitroarenes stimulated a series of femtosecond-resolved experiments. [START_REF] Takezaki | Nonradiative Relaxation Processes and Electronically Excited States of Nitrobenzene Studied by Picosecond Time-Resolved Transient Grating Method[END_REF][START_REF] Takezaki | Relaxation of Nitrobenzene from the Excited Singlet State[END_REF][START_REF] Morales-Cueto | Singlet Excited-State Dynamics of Nitropolycyclic Aromatic Hydrocarbons: Direct Measurements by Femtosecond Fluorescence Up-Conversion[END_REF][START_REF] Zugazagoitia | Ultrafast Intersystem Crossing in 1-nitronaphthalene. An Experimental and Computational Study[END_REF][START_REF] Zugazagoitia | Relaxation in the Triplet Manifold of 1-Nsitronaphthalene Observed by Transient Absorption Spectroscopy[END_REF][START_REF] Reichardt | On the Origin of Ultrafast Nonradiative Transitions in Nitro-polycyclic Aromatic Hydrocarbons: Excited State Dynamics in 1-nitronaphthalene[END_REF][START_REF] Crespo-Hernández | Environmental Photochemistry of Nitro-PAHs: Direct Ibservation of Ultrafast Intersystem Crosing in 1-nitropyrene[END_REF][START_REF] Heinz | Excited State Processes of Nitrobenzaldehydes Probed by Ultrafast Fluorescence and Absorption Spectroscopy[END_REF][START_REF] Cheng | Fluorescence and Solvent-Dependent Phosphorescence Studies of o-nitrobenzaldehyde: A Combined Experimental and Theoretical Investigation[END_REF][START_REF] Vogt | Excited-State Dynamics in Nitro-Naphthalene Derivatives: Intersystem Crossing to the Triplet Manifold in Hundreds of Femtoseconds[END_REF][START_REF] Vogt | Conformational Control in the Population of the Triplet State and Photoreactivity of Nitronaphthalene Derivatives[END_REF][START_REF] Larsen | Transient IR Spectroscopic Observation of Singlet and Triplet States of 2-Nitrofluorene: Revisiting the Photophysics of Nitroaromatics[END_REF] Their photoexcitation decay kinetics are characterized by a biphasic decay of the lowest excited singlet state with time constants of ≤ 100 fs and < 10 ps and a very high quantum yield of triplet formation. The identity of the coordinate that tunes the initially excited singlet and the receiving triplet states into a potential curve crossing remains a matter of debate. Some authors favour the pyramidalisation, [START_REF] Takezaki | Nonradiative Relaxation Processes and Electronically Excited States of Nitrobenzene Studied by Picosecond Time-Resolved Transient Grating Method[END_REF][START_REF] Takezaki | Relaxation of Nitrobenzene from the Excited Singlet State[END_REF][START_REF] Orozco-Gonzalez | Theoretical Study of the Absorption and Nonradiative Deactivation of 1-nitronaphthalene in the Low Lying Singlet and Triplet Excited States Including Methanol and Ethanol Solvent Effects[END_REF][START_REF] Giussani | Understanding of the Photophysics and Photochemistry of 1-Nitronaphthalene under Solar Radiation: The First Theoretical Evidence of a Photodegradation Intramolecular Rearrangement Mechanism Involving the Triplet States[END_REF] others the torsional motion of the nitro group [START_REF] Morales-Cueto | Singlet Excited-State Dynamics of Nitropolycyclic Aromatic Hydrocarbons: Direct Measurements by Femtosecond Fluorescence Up-Conversion[END_REF][START_REF] Crespo-Hernández | Environmental Photochemistry of Nitro-PAHs: Direct Ibservation of Ultrafast Intersystem Crosing in 1-nitropyrene[END_REF][START_REF] Vogt | Excited-State Dynamics in Nitro-Naphthalene Derivatives: Intersystem Crossing to the Triplet Manifold in Hundreds of Femtoseconds[END_REF][START_REF] Vogt | Conformational Control in the Population of the Triplet State and Photoreactivity of Nitronaphthalene Derivatives[END_REF][START_REF] Orozco-Gonzalez | Theoretical Study of the Absorption and Nonradiative Deactivation of 1-nitronaphthalene in the Low Lying Singlet and Triplet Excited States Including Methanol and Ethanol Solvent Effects[END_REF][START_REF] Giussani | Understanding of the Photophysics and Photochemistry of 1-Nitronaphthalene under Solar Radiation: The First Theoretical Evidence of a Photodegradation Intramolecular Rearrangement Mechanism Involving the Triplet States[END_REF][START_REF] Quenneville | Quantum Chemistry Studies of Electronically Excited Nitrobenzene, TNA, and TNT[END_REF] while yet others believe the ONO bending to be the decisive coordinate. [START_REF] Mewes | On the Molecular Mechanism of Non-radiative Decay of Nitrobenzene and the Unforeseen Challenges this Simple 111 Molecule Holds for Electronic Structure Theory[END_REF] Importantly, it is clear, however, that a strong coupling between nuclear motion, electronic structure and the electron spin magnetic moment determines the photopysics of these compounds.

All of these time-resolved experiments studying ISC dynamics have benefited from a strong insight from theory and computation. Indeed, in the aforementioned case of aromatic carbonyl compounds, quantum chemical studies along side rate constants calculated using perturbation theory have been pivotal for the understanding of the underlying mechanistic picture. [START_REF] Rai-Constapel | Isolated and Solvated Thioxanthone: A Photophysical Study[END_REF][START_REF] Rai-Constapel | Photophysics of Xanthone: A Quantum Chemical Perusal[END_REF][START_REF] Rai-Constapel | Chimeric Behavior of Excited Thioxanthone in Protic Solvents: II. Theory[END_REF][START_REF] Rai-Constapel | Solvent Tunable Photophysics of Acridone: A Quantum Chemical Perspective[END_REF][START_REF] Mundt | Thioxanthone in Apolar Solvents: Ultrafast Internal Conversion Precedes Fast Intersystem Crossing[END_REF] Pioneering simulations combining quantum chemistry with molecular or quantum dynamics have been proposed for both organic molecules [START_REF] Morita | Theoretical Study on the Intersystem Crossing Mechanism of a Diradical in Norrish type II Reactions in Solution[END_REF] and transition metal complexes. [START_REF] Daniel | Spin Orbit Induced Radiationless Transitions in Organometallics: Quantum Simulation of the 1E 3A1 Intersystem Crossing Process in HCo(CO)4[END_REF][START_REF] Heitz | Spin Orbit Induced Radiationless Transitions in Organometallics: Quantum Simulation of the Intersystem Crossing Processes in the of HCo (CO)4[END_REF] Indeed, prior to any experimental evidence it was shown that fast ISC (<50 ps) in a first-row transition metal complex could be described as a succession of elementary transitions occurring at critical geometries or crossing points between the singlet and the triplet potential energy surfaces (PESs) and that the efficiency of the radiationless transitions was governed by the overlap of the propagated wave packet with the critical region of the coupled potential energy surfaces. [START_REF] Heitz | Spin Orbit Induced Radiationless Transitions in Organometallics: Quantum Simulation of the Intersystem Crossing Processes in the of HCo (CO)4[END_REF] More recently by combining time-resolved photoelectron spectroscopy and quantum dynamics (QD) it was demonstrated that ISC in hydrocarbons could compete with IC. [START_REF] Parker | Ultrafast Dynamics of the S1 Excited State of Benzene[END_REF][START_REF] Minns | Competing Iltrafast Intersystem Crossing and Internal Conversion in the "Channel 3" Region of Benzene[END_REF] A significant theoretical activity in this area has emerged within the past five years, of which a significant portion has focused upon studying ISC processes using non-adiabatic molecular dynamics (MD) including spin-orbit coupling (SOC). [START_REF] Tavernelli | Nonadiabatic Molecular Dynamics with Solvent Effects: A LR-TDDFT QM/MM Study of Ruthenium (II) tris(bipyridine) in Water[END_REF][START_REF] Mai | Non-adiabatic and Intersystem Crossing Dynamics in SO2. II. The Role of Triplet States in the Bound State Dynamics Studied by Surface-Hopping Simulations[END_REF][START_REF] Polyak | Modeling Processes of Non-Radiative Relaxation of Electronically Excited States of 113 Fluorescent Probe 4-dimethylaminochalcone and its Complexes With Water Using Non Adiabatic Molecular Dynamics[END_REF][START_REF] Richter | Ultrafast Intersystem Crossing Dynamics in Uracil Unravelled by Ab Initio Molecular Dynamics[END_REF][START_REF] Martínez-Fernández | Competing Ultrafast Intersystem Crossing and Internal Conversion: A Time Resolved Picture for the Deactivation of 6-Thioguanine[END_REF][START_REF] Cui | Generalized Trajectory Surface-Hopping Method for Internal Conversion and Intersystem Crossing[END_REF][START_REF] Freitag | Theoretical Spectroscopy and Photodynamics of a Ruthenium Nitrosyl Complex[END_REF][START_REF] Mai | A General Method to Describe Intersystem Crossing Dynamics in Trajectory Surface Hopping[END_REF][START_REF] Crespo-Hernandez | Electronic and Structural Elements that Regulate the Excited-State Dynamics in Purine Nucleobase Derivatives[END_REF][START_REF] Iuchi | Insight into the Light-Induced Spin Crossover of [Fe(bpy)3] 2+ in Aqueous Solution from Molecular Dynamics Simulation of d-d Excited States[END_REF][START_REF] Franco De Carvalho | Nonadiabatic Dynamics with Intersystem Crossings: A Time-Dependent Density Functional Theory Implementation[END_REF][START_REF] Mai | Intersystem Crossing Pathways in the Noncanonical Nucleobase 2-Thiouracil: A time-Dependent Picture[END_REF][START_REF] Cao | Internal Conversion and Intersystem Crossing in α, β-enones: A Combination of Electronic Structure Calculations and Dynamics Simulations[END_REF][START_REF] Pederzoli | A New Approach to Molecular Dynamics with Non-Adiabatic and Spin Orbit Effects with Applications to QM/MM Simulations of Thiophene and Selenophene[END_REF][START_REF] Mai | The DNA Nucleobase Thymine in Motion-Intersystem Crossing Simulated with Surface Hopping[END_REF][START_REF] Borin | Ab Initio Molecular Dynamics Relaxation and Intersystem Crossing Mechanisms of 5-azacytosine[END_REF] Examples of ISC dynamics based upon quantum wavepacket dynamics are more scarce. [START_REF] Ando | Theoretical Study on Ultrafast Intersystem Crossing of Chromium (III) Acetylacetonate[END_REF][START_REF] Papai | High-Efficiency Iron Photosensitizer Explained with Quantum Wavepacket Dynamics[END_REF][START_REF] Pápai | Effect of tert-Butyl Functionalization on the Photoexcited Decay of a Fe (II)-N-Heterocyclic Carbene Complex[END_REF][START_REF] Lévêque | Communication: Theoretical Prediction of the Importance of the 3B2 state in the Dynamics of Sulfur Dioxide[END_REF][START_REF] Eng | Spin-Vibronic Quantum Dynamics for Ultrafast Excited-State Processes[END_REF][START_REF] Fumanal | Ultrafast Excited-State Decays in [Re (CO) 3 (N, N)(L)] n+: Nonadiabatic Quantum Dynamics[END_REF][START_REF] Harabuchi | Exploring the Mechanism of Ultrafast Intersystem Crossing in Rhenium (I) Carbonyl Bipyridine Halide Complexes: Key Vibrational Modes and Spin-Vibronic Quantum Dynamics[END_REF][START_REF] Xie | Ab initio Determination of Potential Energy Surfaces for the First Two UV Absorption Bands of SO2[END_REF][START_REF] Mondal | Infrared Absorption Spectra of Jahn-Teller Systems: Application to the Transition-Metal Trifluorides MnF3 and NiF3[END_REF] On-the-fly dynamics simulations including SOC are very powerful for the interpretation of the experiments highlighting the importance of ISC in a variety of non-adiabatic photochemical and photobiological processes with the possibility of including atomistic environment effects.

But for such simulations, it remains much harder than full quantum approaches to decipher the early time photophysics and discover the subtle effects that drive the spin-vibronic mechanism in complex molecules.

In this article, we review the most recent developments in experimental techniques, theoretical methods and models for describing ISC beyond the direct spin-orbit interaction, namely the spin-vibronic mechanism. To understand this mechanism one must explicitly consider the coupling between the spin, vibrational and electronic subsystems and its role in a wide variety of different systems is being demonstrated to be increasingly important. The concepts outlined are illustrated with examples including the ISC mechanisms in transition metal complexes, small organic molecules and organic chromophores.

Background and Theory

It is often assumed that radiationless transitions, especially on the pico-to femtosecond timescale, are due to the breakdown of the Born-Oppenheimer (BO) approximation, and are thus governed by matrix elements acting between states of the same multiplicity, i.e. IC. However, as discussed throughout this review, this statement is too restrictive. For many systems, including those containing only light elements such as C, N and H, radiationless transitions also include singlet-triplet transitions, i.e. ISC. In this regime one must not only consider scenarios in which the states are coupled by SOC, but also the possibility for states to be mixed by nonadiabatic and spin-orbit coupling simultaneously. Indeed, within a basis of spin-orbit-coupled BO states, there is no difference between IC and ISC. It is the main ideas and theory behind this which is now discussed.

The Coupling Elements

Consider an electronic Hamiltonian under the zeroth-order BO approximation: Ĥ = Ĥ0 + ĤSO (1) where Ĥ0 contains the usual kinetic and potential energies for the electrons. ĤSO represents the spin-orbit operator coupling the electronic and spin degrees of freedom together and mixing states of different multiplicity. Strictly speaking, the multiplicity classification for eigenfunctions of Ĥ does not hold with the presence of ĤSO , as it would for Ĥ0 , i.e. upon inclusion of ĤSO a singlet can not longer be classified as a pure singlet state. However throughout the remaining we assume it would in order to retain simplicity when discussing the states.

In this form, the nuclear coordinate, Q, is only included in Ĥ0 and ĤSO as a parameter.

This describes the Condon approximation, i.e. assuming that the electronic and vibrational degrees of freedom can be strictly separated, making Q a parameter. However, if this breaks down, we must consider the vibrational dependence of these Hamiltonian operators, written to first order as:

Ĥ = Ĥ0 + ĤSO + α ∂ Ĥ0 ∂Q α + ∂ ĤSO ∂Q α Q α ( 2
)
where the coupling elements now also depend on the vibrational degree of freedom α. Tn denotes nonadiabatic coupling between states of the same multiplicity and ĤSO denotes spin-orbit coupling. In the case that ĤSO between the S 1 and T 1 states was zero, decay could occur via a more complicated, secondorder channel, S 1 →T 2 →T 1 , involving both spin-orbit and nonadiabatic coupling. This is an example of the spin-vibronic mechanism as described in this review.

Spin-Orbit Coupling

SOC is a relativistic effect that acts on both angular momentum and spin, leading to its defining characteristic of mixing orbital and spin degrees of freedom, thus allowing electronic states of different multiplicities to couple. It is naturally incorporated within the Dirac equation, but is more commonly included as a perturbative correction to the standard, non-relativistic electronic structure theories. SOC has a wide range of important effects in chemistry and physics, such as fine-structure and band splitting in molecules, semi-conductors, and metals, molecular magnetism, spin transport in spintronics and magnetoelectronics, spin quantum dots, and qubits dynamics. In the context of this present review, SOC is also very important for photophysics and photochemistry because it turns spin-forbidden processes, such as ISC and phosphorescence into allowed transitions. [START_REF] Baryshnikov | Theory and Calculation of the Phosphorescence Phenomenon[END_REF] In its most common, Breit-Pauli form, the SOC operators is expressed:

ĤBP SO = 1 2m 2 e c 2 I i Z I r 3 iI (r iI × pi ) • ŝi - 1 2m 2 e c 2 i j =i Z I r 3 ij (r ij × pi ) • (ŝ i + 2ŝ j ) (3)
where riI × pi is the angular momentum of electron i with respect to nucleus I, rij × pi is the angular momentum of electron i with respect to electron j and ŝi is the spin-operator of electron i. The first term is the one-electron term, involving spin-same-orbit coupling only. This describes the interaction between the magnetic spin moment of an electron with the magnetic moment induced by its orbiting in the nuclear electrostatic field. The second term is the two-electron term, describing spin-same-orbit and spin-other-orbit coupling. The former is analogous to the one-electron terms, but instead the electric field is generated by different electron. The two-electron spin-other-orbit terms arise from the interaction of the spin magnetic moment of one electron with the orbital magnetic moment of a second one, and vice versa. The two electron terms provide screening of the one-electron term and in compounds composed of light elements only, this screening is substantial and cannot be neglected. However, it can be incorporated in the one-electron terms to good approximation in a mean-field manner. [START_REF] Hess | A Mean-Field Spin-Orbit Method Applicable to Correlated Wavefunctions[END_REF][START_REF] Marian | A New Mean-Field and ECP-based Spin-Orbit Method. Applications to Pt and PtH[END_REF] Importantly, because the one-electron contributions increase much faster with nuclear charge [START_REF] Pyykkö | Relativistic Effects in Chemistry: More Common Than You Thought[END_REF] than the two-electron terms for many cases involving heavier elements, the one-electron term is dominant. Indeed, in the total SOC matrix element (SOCME) of the valence shell, for 6d elements, such as Pt, the two-electron terms reduces the contributions of the one-electron term by approximately 10%. [START_REF] Marian | Spin-Orbit Coupling and Intersystem Crossing in Molecules Wiley Interdiscip[END_REF] In a one-center approximation, the SOC operator can be expressed as:

Ĥeff SO = 1 2m 2 e c 2 l i Z ef f I r 3 iI li • ŝi ( 4
)
where li is the angular momentum and ŝi is the spin operator of electron i. Herein, the screening is taken into account by replacing the nuclear charge Z I by an effective nuclear charge Z ef f I that is fitted to atomic data. [START_REF] Koseki | MCSCF/6-31G (d, p) Calculations of One-Electron Spin-Orbit Coupling Constants in Diatomic Molecules[END_REF] For each nucleus one can write a spin-orbit coupling constant, ζ = Z ef f I 2m 2 e c 2 r 3 , which is proportional to the effective nuclear charge and inversely proportional to the mean cubic radial distribution (r -3 ) of the electron. This constant is therefore responsible for the so-called heavy-atom effect, whereby SOC is expected to increase with the atomic number. However, it is important to stress that one must also consider the integral over the two states involved:

[ ĤSO ] ij = ψ S i | ĤSO |ψ T j .
(

) 5
This integral, sometimes referred to as the internal effect, depends principally on the character of the states involved, as outlined qualitatively by El-Sayed, [START_REF] El-Sayed | Spin Orbit Coupling and the Radiationless Processes in Nitrogen Heterocyclics[END_REF] and described in detail in section .

In quantum chemical studies of heavy-element compounds, where the inclusion of scalar and magnetic relativistic effects is mandatory, frequently effective core potentials (ECPs) are employed instead of the operators in Equations 3 and 4. The spin-orbit effective core potentials (SOECPs) are defined as the difference between the ECPs for the j = l + 1/2 and j = l -1/2 shells derived from two-or four-component atomic calculations. In fitting this difference to a linear combination of a few Gaussian basis functions, the original 1/r 3 iIdependence has been dropped and the action of the SOC operator has been relocated to the valence region. [START_REF] Ermler | Spin-Orbit Coupling and Other Relativistic Effects in Atoms and Molecules[END_REF][START_REF] Cao | Pseudopotentials and Model Potentials[END_REF][START_REF] Dolg | Relativistic Pseudopotentials: Their Development and Scope of Applications[END_REF] These atomic SOC operators can therefore be written as 95

ĤECP SO = lmax l=1 2∆V l (r) 2l + 1 l • ŝ +l m l =-l |l, m l l, m l | (6)
where l max is the largest angular momentum quantum number for which the SOECP is defined on this atom.

To date the calculation of SOCMEs has been implemented across many different quantum chemistry methods including time-dependent Hartree-Fock theory, [START_REF] Walker | Calculation of Spin-Orbit Coupling Constants in Diatomic Molecules from Hartree-Fock Wave Functions[END_REF] multi-configurational self-consistent field methods, [START_REF] Koseki | MCSCF/6-31G (d, p) Calculations of One-Electron Spin-Orbit Coupling Constants in Diatomic Molecules[END_REF][START_REF] Malmqvist | The Restricted Active Space (RAS) State iInteraction Approach With Spin-Orbit Coupling[END_REF] multi-reference configuration interaction [START_REF] Sjøvoll | A Determinantal Approach to Spin-Orbit Configuration Interaction[END_REF][START_REF] Buenker | Comparison of Spin-Orbit Configuration Interaction Methods Employing Relativistic Effective Core Potentials for the Calculation of Zero-Field Splittings of Heavy Atoms with a 2Po Ground State[END_REF][100][101][102] and multireference perturbation theory approaches, 103 algebraic diagrammatic construction based methods,, 104 equation-of-motion coupled-cluster theory [105][106][107] and semi-empirical approaches. 108 Within the framework of time-dependent density functional theory(TDDFT), Wang and

Ziegler 109 developed an approach based upon an approximation of the relativistic formalism. For calculating SOC within the framework of linear-response time-dependent density functional theory (LR-TDDFT), based on a perturbative correction of the non-relativistic collinear TDDFT, similar to the work of Russo and Chiodo,110,111 Tavernelli and co-workers 112 adopted an approach based upon a set of auxiliary many-electron wavefunctions, previously used to calculate nonadiabatic coupling vectors. 113,114 Most recently a simplification has been proposed which involves evaluation of SOC directly from the linear response amplitudes. 115,116 Nonadiabatic Coupling

The BO approximation 117,118 represents the cornerstone of quantum theory and is used throughout quantum chemistry and condensed matter physics to decouple the motion of nuclei and electrons. However, as is well established, 119 the highly non-equilibrium electronic distribution generated upon excitation drives rapid nuclear motion leading to the breakdown of this approximation. This makes it impossible to neglect the nonadiabatic coupling terms present in the time-dependent Schrödinger equation.

The first-order nonadiabatic coupling operators are non-local derivative operators which are discussed in detail in ref. 119 They depend inversely on the mass of the system, and also the derivative coupling vector expressed:

F ij = ψ i |∇ψ j = ψ i |∇ Ĥ0 |ψ j E j -E i (7)
where ψ i are the electronic basis functions that are solutions of the time-independent electronic Schrödinger equation and E i the corresponding energies. Calculating this term is possible with a range of single 120 and multi-reference 121-125 electronic structure methods.

Methods for calculating the coupling within the framework of TDDFT have also been demonstrated. 113,114,[126][127][128] However, within the adiabatic basis used by quantum chemistry methods, the nonadiabatic coupling diverges near intersections of the two coupled states. 129 This is illustrated in the second expression of Equation 7. The nonadiabatic coupling clearly exhibits an inverse dependence on the energy gap between surfaces. As this gap narrows, the coupling increases, but, if two surfaces become degenerate, the coupling becomes infinite. This creates a significant challenge for computational methods seeking to study the nonadiabatic dynamics.

To alleviate these issues, it is possible to transform the adiabatic states into the so-called (quasi)-diabatic representation. The adiabatic picture provides a set of energy-ordered PESs and nonlocal couplings elements via nuclear momentum-like operators. In contrast, in the diabatic picture, the PESs is related to an electronic configuration and the couplings are provided by local multiplicative Q-dependent potential-like operators. Importantly, the surfaces in the diabatic picture are smooth because they are constructed so that the singular coupling terms vanish, 130 although it should be stressed that coupling can only be made rigorously zero for diatomics. 131 Consequently, they can often be described by a low-order

Taylor expansion, as exploited below.

For a given set of adiabatic potential energy surfaces, diabatisation methods can be broadly classified into three groups: property-based, by ansatz and wave-function based. [132][133][134] Property based methods 132,[135][136][137][138][139][140][141][142] approximate the diabatic states using observables such as transition dipole moment and change in molecular dipole moment. By ansatz methods define the diabatic states at a certain geometry, Q 0 and then perform a Taylor expansion around this point. 143 The expansion is parametrised through fitting which is inexpensive, however the disadvantage is that in the absence of high molecular symmetry there might be some ambiguity in the diabatisation because only adiabatic electronic energies are used and no information about the electronic wavefunctions is utilised. Finally, wavefunction methods explicitly calculate the coupling elements and then transform them into states which minimise the coupling and preserve the character of the diabats. [144][145][146] Recent developments have also included hybrid methods, 147 and important in the context of the present review SOC-diabatisation procedures based upon model space diabatization (MSD) have been proposed. 148,149 Spin-Vibronic Coupling

The nonadiabatic coupling introduced in the previous section highlights the importance of specific vibrational degrees of freedom, along which this coupling is activated. The SOC discussed above in section is limited to a specific nuclear configuration, but as outlined in the introduction it is often not constant and also depends on the vibrational motion. In addition, in many cases it is insufficient to consider scenarios in which the states are coupled by vibronic or spin-orbit coupling, and one should account for the possibility for the states to be mixed by vibronic and spin-orbit coupling simultaneously. In ref. [START_REF] Albrecht | Vibronic Spin-Orbit Perturbations and the Assignment of the Lowest Triplet State of Benzene[END_REF] Albrecht outlined the four mechanisms for mixing the singlet and triplet states:

I Direct spin-orbit coupling, ĤSO , driven by the electronic character of the states and discussed in section .

II Vibrational spin-orbit, e.g. (∂ ĤSO /∂Q α )Q α , in which the size of the SOCME depends on the motion along a particular nuclear degree of freedom, Q α . This is written to first order, but of course higher-orders can be included if necessary.

III Spin-vibronic, TN ĤSO , spin-orbit coupling with vibronic coupling in the singlet manifold IV Spin-vibronic, ĤSO TN , spin-orbit coupling with vibronic coupling in the triplet manifold

The first, direct SOC has no dependence on the nuclear coordinates, while the latter three do and therefore it is these terms that make up the spin-vibronic coupling. Of these terms, only I and II can be present at first-order perturbation theory, while the terms involving a mixture of spin-orbit and nonadiabatic coupling appear only at second-order perturbation theory. Adopting each of these components, the full spin-orbit interaction, up to second order, can be written:

ĤSO = Ψ S 1 | ĤSO |Ψ T 1 + α ∂ Ψ S 1 | ĤSO |Ψ T 1 ∂Q α Q α + α β 1 2 ∂ 2 Ψ S 1 | ĤSO |Ψ T 1 ∂Q α ∂Q β Q α Q β + Ψ S 1 | ĤSO |Ψ T 2 Ψ T 2 | TN |Ψ T 1 E T 2 -E S 1 + Ψ S 1 | TN |Ψ S 2 Ψ S 2 | ĤSO |Ψ T 1 E S 2 -E T 1 ( 8
)
where Ψ represents the molecular wavefunction for a particular state, T N is the nonadiabatic coupling operator and H SO is the spin-orbit operator. Although the vibrational spin-orbit (terms 2 and 3 in Equation 8) are formally different from the interaction terms involving nonadiabatic coupling (terms 4 and 5 in Equation 8), in practice these terms can hardly be told apart, and which terms dominate will depend on the system under study.

In terms of calculating these couplings, the vibrational spin-orbit terms, which are described using first-order perturbation theory may be accounted for by calculating the vibrational dependence of the SOCME, as shown in refs. 150,151 in the framework of perturbation theory or ref. 152 in the framework of quantum dynamics. The latter terms are normally a little harder to incorporate, but can be routinely simulated using both excited state MD or quantum dynamics. [START_REF] Franco De Carvalho | Nonadiabatic Dynamics with Intersystem Crossings: A Time-Dependent Density Functional Theory Implementation[END_REF][153][154][155] Qualitative Rules for Intersystem Crossing

If the coupling between the two (initial and final) states of interest is small compared to their adiabatic energy difference, the rate of population transfer (k) between the two states can be described using perturbation theory, which to first order is Fermi's golden rule approximation written:

k = 2π h f | Ψ f | Ĥif |Ψ i | 2 δ(E i -E f ), (9)
where Ψ i and Ψ f are the total wavefunctions (electronic and vibrational) of the initial and final states and Ĥif is the Hamiltonian describing the coupling between them, which in the case of ISC would be the ĤSO . The δ function ensures the conservation of the molecular energy for the nonradiative transition. Assuming the Condon approximation, i.e direct SOC which has no dependency upon the vibrational degrees of freedom, the electronic and vibrational contributions are completely separable and it is possible for Equation 9to be rewritten as:

k = 2π h f | ψ f | Ĥif |ψ i | 2 k | ν f k |ν ia | 2 δ(E ia -E f k ). ( 10
)
This form highlights the two components, purely electronic SOC and vibrational density of states, which are crucial in determining the rate of ISC within a direct spin-orbit mechanism. The importance of these terms has led to them being widely studied leading to the development of qualitative rules making it possible to estimate the rate of ISC, which are now discussed.

El-Sayed Rules

As shown in Equation 3, the SOCME scale as Z 4 ef f /n 3 , where Z ef f is the effective nuclear charge and n is the principal quantum number of the valence electron. This is known as the heavy atom effect and therefore naively, one may expect that spin-orbit interactions increase as one moves down the periodic table. However, while this rule of thumb can work in many general cases for understanding the rate of ISC between two states, this breaks down if the two states are close in energy. In these cases, often a small coupling matrix element suffices to induce an efficient transition between the states. In addition, it does not capture any influence of the character of the states involved in the ISC process. The ISC rate dependence on the nature of the electronic transitions is described by El-Sayed's rules. [START_REF] El-Sayed | Spin Orbit Coupling and the Radiationless Processes in Nitrogen Heterocyclics[END_REF] In his work, El-Sayed highlighted that to achieve effective SOC, any change in spin must be accompanied by a corresponding change in angular momentum, so that total angular momentum is conserved. This original work was discussed in the context of organic chromophores considering the typical case of a transition from the lowest excited singlet state to the triplet manifold. El-Sayed rules predicted that 1 (ππ * ) ; 3 (nπ * ) ISC would be faster than 1 (ππ * ) ; 3 (ππ * ) ISC, as the latter involves the same orbitals. Therefore the requirement to conserve total angular momentum cannot be obeyed because there is no change in angular momentum associated with the transition. In contrast, at the lower D 2 symmetry, this is no longer possible and therefore within the one-electron one-centre approximation SOC is forbidden.

D 2d D 2 d xz$ d yz$ π * 2$ π * 1$ π * 2$ π * 1$ d xz$ d yz$ S1/T 1 = 1 4 (d xz ! ⇡ ⇤ 1 ) + 1 4 (d xz ! ⇡ ⇤ 2 )+ 1 4 (d yz ! ⇡ ⇤ 1 ) + 1 4 (d yz ! ⇡ ⇤ 2 ) S1/T 1 = 1 2 (d xz ! ⇡ ⇤ 1 ) + 1 2 (d xz ! ⇡ ⇤ 2 ) h T 1 | ĤSO | S1 i = hd xz ⇡ ⇤ 1 | ĤSO |d yz ⇡ ⇤ 1 i+ hd xz ⇡ ⇤ 2 | ĤSO |d yz ⇡ ⇤ 2 i
For the prototypical Bis-diimine Cu(I) complex, [Cu(dmp) 2 ] + (dmp = 2,9-dimethyl-1,10phenanthroline), S 1 -T 1 SOC at D 2d symmetry is ∼300 cm -1 , this falls to 30 cm -1 at D 2 symmetry.

Originally presented in the context of organic chromophores, El-Sayed's rules are generally applicable. Indeed, they can be used to explain why, despite the expected heavy atom effect, that in some cases the SOCME between metal-to-ligand charge transfer states of Au(I) metal organic complexes are smaller than the one for analogous Cu(I) metal organic complexes. This can be understood by assuming a one-electron one-center approximation, which as described in section is expected to be dominant for heavier elements. For SOCME between two metal to ligand charge transfer (MLCT) states involving dπ * character, the SOCME can be approximated as:

ψ T 1 | ĤSO |ψ S1 = d i π * s | ĤSO |d j π * t ( 11
)
Since the SOC constant will be significantly larger for the metal than the lighter elements e.g. C, H, O, and N, the SOCME will be largest when the d electrons contribute, which means i = j and s = t. Within the one-electron approximation, if both i = j and s = t, then SOCME would be zero. This is shown schematically in Figure 2. In the case of Au(I)

and Cu(I) complexes, because the crystal field splitting of Au(I) complexes is larger than for the corresponding Cu(I) compounds, the d electrons generally make a smaller contribution to the low-lying excited state leading to a smaller SOC. 158 Similar observations, which are essentially extensions of El-Sayed's rules have been made for charge-transfer states of organic chromophores. 159,160 However, it is important to stress that the derivation of El-Sayed's rules is based upon pure electronic states of a single character at a fixed nuclear configuration. Mixing between states of different character will lead to the breakdown of this rule and vibronic interactions, discussed in Section , can significantly accelerate El-Sayed forbidden ISC. 150,151 The Energy Gap Law

El-Sayed's rules address the purely electronic aspects of ISC, however Equation 10illustrates the importance of the vibrational terms, more commonly referred to as the vibrational density of states. In the Condon approximation, the vibrational component of the rate expression is the overlap between the initially populated and final vibrational density of states. This latter term is referred to as the Franck-Condon weighted density of states (FCWD). Assuming the high temperature limit, i.e. all modes relevant in transfer are below kT , the FCWD can be expressed: 161

F CW D = 1 √ 4πλk B T exp - (∆E + λ) 2 4λk B T (12)
In Equation 12, λ denotes the reorganisation energy. In electron-transfer theory, the reorganization energy describes intramolecular and solvent reorientation energy. For ISC, λ also should include solvent (or environment) reorganization energy, but the intramolecular reorganization energy corresponds to the energy variation in the initial singlet excited state when switching from the singlet equilibrium geometry to the triplet equilibrium geometry as depicted in Figure 3. ∆E is the driving force, i.e. the adiabatic energy difference between the initial and final states at their respective minima. In the former, the displacement of final state minima (i.e. structural change) with respect to the initial state is small (Figure 3 left) leading to a small reorganisation energy. This is to say that the potentials are nested. In this weak coupling limit the transition probability depends exponentially on the adiabatic energy difference (∆E). This relationship is commonly known as the energy gap law. Importantly, it is often forgotten that this rule applies only for a pair of nested states. In contrast, the strong coupling case is characterised by a large displacement of one potential minimum along a collective nuclear coordinate, {Q}, and so an intersection of the PESs can be expected. In this case, the rate of ISC will exhibit a Gaussian dependency on the parameter ∆E+λ. In this regime, occasionally an inverse relationship between the transition probability and ∆E is observed, i.e., the transition probability increases with increasing energy gap. 163

Energy {Q} ∆E S 1 -T 1 S 1 T 1 λ=0 (a) {Q} S 1 T 1 ∆E S 1 -T 1 λ (b)

Qualitative Estimation of Spin-Vibronic Coupling

Neither El-Sayed's nor the energy gap law discussed in the previous sections consider the coupled electronic, spin and nuclear dynamics which underpin the spin-vibronic mechanism.

In section , three terms which compose spin-vibronic coupling mechanisms were outlined.

The first (spin-vibrational) term is present at first order. Assuming a smoothly varying potential, the likely contribution of spin-vibronic coupling via these mechanisms can be estimated by determining the gradient of the SOC matrix elements and of the energy gap between the two coupled states as a function of the important reaction coordinates. While requiring additional calculations, this is not especially time-consuming.

A typical example for spin-vibrational coupling in the absence of a close-by intermediate electronic state is the ISC in free-base porphyrin following photoexcitation. 164 All low-lying electronic states of porphyrin are ππ * states and ISC is thus El-Sayed forbidden. Porphyrin derivatives are nevertheless used as singlet oxygen sensitisers in photodynamic therapy (PDT). At the planar S 1 equilibrium geometry, the S 1 -T 1 energy separation is ≈ 2400 cm -1 while their direct SOCME T 1 | ĤSO |S 1 is smaller than 0.05 cm -1 . Out-of-plane vibrations leave the energy gap between the S 1 and T 1 states nearly unchanged, but as shown in Figure 4 increase their mutual SOC substantially by mixing some σπ * character into the electronic wavefunctions. In this way, the ISC rate constant is enhanced by two orders of magnitude to k ISC ≈ 10 7 s -1 , which makes ISC competitive with fluorescence in this molecule. The remaining terms in Equation 8do not appear in first-order perturbation theory, and therefore a second-order expansion must be applied. This is written:

k = 2π h Ψ f | Ĥsoc |Ψ n Ψ n | Ĥvib |Ψ i E i -E n 2 δ(E f -E i ) (13)
whereby the initial (Ψ i ) and final (Ψ f ) states are coupled via an intermediate (Ψ n ) state.

In order to develop generalised considerations, which like the energy gap rule, can be as a rule of thumb to estimate spin-vibronic contributions, we recast Equation 13in the form of a semiclassical (Marcus-like) approach. 165,166 k ∼ 2π

4πh 2 λ if k b T |V inf | 2 [E n -E i ] 2 + λ in /2 × exp - [∆E if + λ if ] 2 4λ if k B T (14)
Here, V inf represents the second-order coupling matrix elements. This is equivalent to the purely electronic components of the coupling terms (Condon approximation). In the weak coupling limit, λ ∼ 0, if applicable for all three states involved, this equation will exhibit an exponential dependence upon the gap between the initial and final state and quadratically on the energy gap between the initial and intermediate states. This is shown in Figure 5a.

Theoretical Methods

Describing the rates of population transfer largely fall into two categories, i) time (in)dependent perturbation theory (e.g. Fermi's golden rule) which calculate the rates at a particular nuclear geometry or ii) nuclear dynamical methods, which explicitly consider the nuclear motion over a PES. The latter, which provides a solution to the time-dependent Schrödinger equation, should yield a more precise picture of rates and corresponding dynamics. However, such approaches can be challenging for complex molecular systems with a large number of degrees of freedom.

Independent of the particular method used for addressing the nuclear degrees of freedom, the probabilities for nonradiative transitions will be very sensitive to the relative energies and to the geometric displacements of the involved PES. It is therefore mandatory to use either highly accurate electronic structure methods for their construction or to perform extensive benchmarking to accurately assess the limitations of the model developed. Experience has shown that the electronic SOCME are often less sensitive to the level of electron correlation treatment than the energetic splittings. 103,115 In the following subsections the main approaches for describing the rate of spin-vibronic processes are described.

Perturbation Theory Approaches

The common starting point for approaches based upon perturbation theory is to approximate the initial and final states potentials typically with a harmonic oscillator model with vibrational frequencies Ω i and Ω f , respectively. The mass-weighted normal coordinates of the final state, Q f , can be related to the ones of the initial state, Q i , by a Duschinsky transformation 167

Q f = JQ i + D ( 15
)
where J is the Duschinsky matrix that accounts for the normal coordinate rotations while the displacement vector D expresses the geometry change between the two potentials. In molecules with common symmetry elements in the initial and final states, the Duschinsky matrix is block diagonal and the displacement vector has nonzero elements only for totally symmetric modes. Although the eigenfunctions of a harmonic oscillator form a complete set, the linear transformation proposed by Dushinsky is only an approximation. This is a consequence of the center-of-mass and Eckart conditions for the transformation from 3N

space-fixed to 3N -6 internal coordinates of non-linear molecules (3N -5 for linear ones). Due to these conditions, two electronic states with different equilibrium geometries and different normal modes give rise to two different internal coordinate systems. An axis-switching matrix, originally introduced by Watson and Hougen, 168 can be used for the transformation of the Eckart systems and for an approximate factorisation of the rotational coordinates from the vibrational subspaces Q i and Q f . 169,170 The zeroth-order axis-switching matrix T 0 which connects the equilibrium geometries of the two states has to be totally symmetric with respect to the largest common point group of these geometries. An alternative method for the determination of Duschinsky matrices and normal-mode-projected displacements for large floppy molecules using (redundant) internal coordinates has been presented by Reimers. 171

In the harmonic oscillator model, the vibrational wavefunctions are factorised further into a product of lower-dimensionality (in most cases one-dimensional) oscillators carrying n quanta each.

v ia = α v naα iα (Q iα ) (16) v f k = β v n kβ f β (Q f β ) (17)
To evaluate the integrals over vibrational wavefunctions of the initial and final states, Eq. ( 15)

is used to transform one set of vibrational functions into the basis of the other. In the following, we choose the normal coordinates and harmonic oscillator functions of the inital electronic state as the common basis and drop the index i in the denomination of the normal coordinates again.

To include vibrational SOC terms in the calculation of ISC rate constants, the Hamiltonian is typically expanded about the equilibrium geometry Q 0 of the initial electronic state.

In addition to the direct SOC term

k FC ISC = 2π h ψ f | ĤSO | ψ i 2 Q 0 k | v f k | v ia | 2 δ(E ia -E f k ) (18)
two contributions to the ISC rate are obtained from the derivative couplings, a cross term that contains Franck-Condon-and Herzberg-Teller-like expressions

k FC/HT ISC = 4π h ψ f | ĤSO | ψ i Q 0 k v f k | v ia δ(E ia -E f k ) (19) α ∂ ψ f | ĤSO | ψ i ∂Q α Q 0 k v k | Q α | v a δ(E ia -E f k )  
and a term that stems exclusively from the derivative couplings

k HT ISC = 2π h   α ∂ ψ f | ĤSO | ψ i ∂Q α Q 0 k v f k | Q α | v ia δ(E ia -E f k ) (20) β ∂ ψ f | ĤSO | ψ i ∂Q β Q 0 k v f k | Q β | v ia δ(E ia -E f k )   .
Herein denotes the real part of the expression. Note that SOCMEs may have imaginary phase and that, depending on the relative phase of the coupling terms, the off-diagonal contributions to the rate may be positive or negative.

In principle, individual rates for the non-radiative transition from an initial singlet state to the three fine-structure levels of a triplet state could be determined. This would require to go beyond first-order perturbation theory and to determine the detailed composition of the triplet substates, followed by a diabatisation procedure to determine the individual singlet-triplet coupling. In practice, the individual ISC rates are added up because the energetic splittings of the fine-structure levels are very small in spatially nondegenerate states. This is equivalent to adding the transition rates determined for the three Cartesian components of ĤSO . If the initial state is a triplet state, the procedure how to handle the de facto degeneracy in the initial state is not so clear cut. If spin relaxation among the triplet sublevels is faster than reverse-ISC (rISC), then each triplet sublevel is statistically populated by one third of the triplet density only and the rate constant for the total rISC will have to be divided by 3 after summation over all possible channels.

To evaluate the rate constants defined in Equations 18-20, different approaches have been pursued to approximate the Dirac δ distributions in these expressions. They can roughly be divided into integration schemes in the energy domain and in the time domain and will be briefly reviewed in the following.

Evaluation of ISC Rates in the Energy Domain

Toniolo and Persico proposed to replace the Dirac δ function in Eq. 9 by a step function of finite width 2η. 172

k ISC ≈ π hη f for which |E i -E f |≤η |V f i | 2 (21)
In Condon approximation, the vibrational part of the rate is obtained by explicitly summing over the Franck-Condon factors (FCFs) of the initial state (typically the vibrationally cold state v i0 with zero quanta in all normal modes) with all final states v kf in the interval 2η.

Integrals of the type

v kf | Q α | v i0
, appearing in Equations 19 and 20, are products of FCFs in all but one normal coordinates and an integral over Q α which is readily reduced to a sum of FC integrals in harmonic approximation. The authors applied this approach successfully to study the temperature-independent predissociation dynamics in nitrosoalkanes. 173 Later, this scheme has been extensively used by the Marian group to determine rate constants for El-Sayed forbidden ISC. [START_REF] Rai-Constapel | Isolated and Solvated Thioxanthone: A Photophysical Study[END_REF]150,164,[174][175][176][177] In practical applications, the interval width 2η has to be chosen with great care. The vibrational density of states (VDOS) grows exponentially with the number of vibrational modes and the adiabatic energetic separation between the initial and final states. 178 For psoralen, an organic compound with 54 normal modes and an S 1 -T 1 energy separation of about 7100 cm -1 , a VDOS of ρ ≈ 10 10 cm in the triplet state at the energy of the initial state has been estimated, 150 i.e. approximately 10 10 vibrational states are located in an interval of width 1 cm -1 . To reduce the computational effort, the interval is typically chosen to be very narrow and the number of coupling modes and the maximum number of quanta allowed in each accepting mode is limited. It has become standard practice to vary these parameters to determine a confidence interval for the computed rate constants.

While integration schemes in the energy domain give detailed insight into which vibrational modes promote the nonradiative transition, they scale very unfavourably with the size of the problem. Pre-screening of the FCFs, 179,180 sum rules 181 and mode selection 150 can be used to alleviate the computational demands and to extend the applicability of this approach. The fundamental difficulties nevertheless persist: With increasing number of normal modes and increasing adiabatic electronic energy difference, the VDOS grows to an extent that the computation of rate constants for nonradiative transitions, even for vibrationally cold molecules and in Condon approximation, becomes prohibitive in the energy domain.

Time-Dependent Methods

A way to avoid the explicit calculation of FC integrals, without resorting to high-temperature limits or estimated values for the VDOS, is to transform Fermi's golden rule into the Heisenberg picture. 130 To this end, the Dirac δ function is expressed as the Fourier integral 22) and substituted into the rate equations. Making use of the properties of conjugation and the resolution of the identity

δ(E i -E f ) = 1 2π +∞ -∞ e i(E i -E f )t/h dt (
1 = ∞ k | ψ f v f k ψ f v f k | yields k ISC = 1 h +∞ -∞ ψ i v ia | ĤSO (0) ĤSO (t) | ψ i v ia dt (23)
where ĤSO (t) = e i Ĥ0 t/h ĤSO e -i Ĥ0 t/h is a spin-orbit operator in the Heisenberg picture. Expression 23 is not necessarily limited to the harmonic oscillator model, but the evaluation of the autocorrelation function becomes particularly easy in this case.

Making use of Mehler's formula for Hermite polynomials,

k ISC = 1 h ψ f | ĤSO | ψ i 2 +∞ -∞ G(t)e i(E i -E f +0.5 tr(Ω i ))t/h dt . ( 24
)
To speed up the numerical integration, a Gaussian damping function was introduced. Like the interval width 2η in Eq. 21, the width of the damping function is a technical parameter which has to be chosen with care and ought to be varied to make sure that the calculated rate constants are converged. Furthermore, Etinski derived a short-time expression which gives the rate constant in closed form without any need for numerical integration. The timedependent integration schemes were later extended to include thermal effects 185 as well as vibrational spin-orbit coupling. Technically, the time-dependent approach can even be (and has been) used to determine rate constants for non-radiative transitions between pairs of singlet and triplet states with very large adiabatic energy gap, for example the ISC from the first-excited triplet state to the electronic ground state (T 1 ; S 0 ). 187,188 It should be kept in mind, however, that the harmonic oscillator approximation underlying the analytic expressions may not be well suited to describe the highly excited vibrational states of S 0 at the energy of the T 1 state.

Summarising, the time-dependent perturbational approaches allow the computation of ISC rate constants for molecules with many vibrational degrees of freedom and substantial singlet-triplet splittings with reasonable computational expense. They are difficult to apply if many electronic states are involved in the nonradiative process simultaneously. As shown in the following, they are easily extended to take account of thermal effects.

Including Temperature

In some cases, such as thermally activated delayed fluorescence (TADF), the singlet-triplet crossing occurs from an adiabatically lower-lying electronic state to a higher-lying one. As nonradiative transitions proceed under energy conservation, these can only be achieved if the initial state is vibrationally excited (Fig. 6), be it thermally or kinetically. Temperature can

p(E ia ) = 1 Z exp (-E ia /k B T ) ( 25
)
where Z = ∞ a exp (-E ia /k B T ) is the canonical partition function, k B is the Boltzmann constant and T is the absolute temperature. In Condon approximation, the rate constant for an incoherent population transfer at temperature T is then given by

k FC,T ISC = 2π Zh ψ f | ĤSO | ψ i 2 a k exp (-E ia /k B T ) | v f k | v ia | 2 δ(E ia -E f k ) (26)
and the derivative coupling terms in Eqs. 19 and 20 are modified accordingly.

Because of the high computational demands caused by the explicit summation over all pairs of vibrational states, inclusion of temperature effects is not feasible in the energy-based integration schemes. In the framework of the time-dependent approaches, formulae for the temprature dependence of rate constants were presented for both, direct and vibrational SOC, by Peng et al. 187 and by Etinski et al. 151,185 Quantum Dynamics Methods

As described in the introduction, an alternative approach is to explicitly simulate the excited state dynamics over a manifold of spin-vibronically coupled levels. Here all of the couplings between the excited states considered can be included into the Hamiltonian and therefore this approach can naturally incorporate all types of spin-vibronic interactions. The biggest challenge for these methods, as described below, is to calculate the multi-dimensional PES also the timescales accessible within these simulations are typically limited to few ps domain.

Multi-Configurational Time-Dependent Hartree Approach

Traditionally excited state dynamics have been studied with grid-based quantum dynamics. These methodologies are based upon solving the time-dependent Schrödinger equation (TDSE) for the nuclei, thus taking into account all quantum effects. The simplest way of solving the time-dependent Schödinger equation is to expand the nuclear wavefunction into a time-independent basis set, with time-dependent coefficients:

Ψ(x 1 , x 2 , ..., x f , t) = n 1 j 1 =1 ... n f j f =1 C j 1 ...j f (t) f k=1 χ (k) j k (x k ) (27)
where χ j is an orthonormal basis, such as the eigenfunctions of the harmonic oscillator and

x k is the nuclear coordinate. The analogous approach to Equation 27 in electronic structure theory is full configuration interaction (full CI). Consequently, while it rigorously describes the motion of a nuclear wavepacket, like full CI, it suffers from a severe computational expense, scaling exponentially with the number of degrees of freedom. This makes it difficult to describe nuclear dynamics in systems containing more than 5-6 degrees of freedom.

A particularly powerful approach which reduces this problem and which has become extensively used is the Multi-Configurational Time-Dependent Hartree (MCTDH) method. 189

In this approach the nuclear wavefunction ansatz takes the form:

Ψ(Q 1 , ..., Q f , t) = n 1 j 1 =1
...

n f j f =1 A j 1 ...j f (t) f k=1 ϕ (k) j k (Q k , t) (28)
where The ansatz for the MCTDH nuclear wavefunction appears rather similar to standard nuclear wavepacket approach (Equation 27), however crucially the basis functions are now time dependent. This means that fewer basis functions are required to converge these variational calculations as they adapt to provide the best possible basis for the description of the evolving wavepacket. In addition, the coordinate for each set of SPFs, Q k , can be a composite coordinate of one or more system coordinates. Thus the basis functions are d-dimensional, where d is the number of system coordinates within the specific SPF, usually between 1-4.

Q 1 , • • • , Q f are
This reduces the effective number of degrees of freedom for the purpose of the simulations.

The memory required by the standard method is proportional to N f , where N is the number of grid points for each f degree of freedom. In contrast, the memory needed by the MCTDH method scales as:

memory = f nN + n f (29)
where the first term is due to the (single-mode) single-particle function (SPF) representation, and the second term the wavefunction coefficient vector A. As n < N , often by a factor of five or more, the MCTDH method needs much less memory than the standard method, so allowing larger systems to be treated. Indeed the standard implementation of MCTDH can treat, depending on the exact details of the calculations, ∼50 nuclear degrees of freedom.

Importantly, the full strength of the MCTDH class of methods for solving the timedependent Schrödinger equation is revealed in its so-called Multi-Layer variant (ML-MCTDH), 190,191 which has treated systems containing over 1000 degrees of freedom. The mode combinations contained within each SPF discussed above is a very powerful way of reducing the effective number of degrees of freedom. However past 3-4 combinations, the SPFs become very complicated and unfeasible to propagate. But, it has already been discussed that the MCTDH approach is an efficient approach for propagating complicated wavefunctions. Therefore, in the ML-MCTDH, the complicated SPFs arising from many mode combinations are themselves propagated by the MCTDH equations of motion. It can be seen as using the MCTDH method to propagate the complicated SPFs of an underlying MCTDH simulation. This can in principle be repeated for multiple layers where the biggest challenge is the complexity of the underlying methods. A detailed discussion on all of these aspects, beyond the remit of the present chapter can be found in ref. 192 Including Temperature

The standard MCTDH wavefunction approach outlined in the previous section, describes the evolution of a particular well-defined initial state, and therefore calculations are effectively performed at 0 K or given some initial, but specific vibrational excitation. However, given the importance of the thermal energy in some cases, it can be important to explicitly address the temperature of the system. This is most applicable for energetically uphill reaction dynamics. In the case of ultrafast deactivation, it is stressed that even if an experiment is carried out at room temperature, a significant fraction of the initial excitation energy is often rapidly transferred into the vibrational degrees of freedom (vibronic coupling), given that only 0.1 eV is ∼1160 K, the temperature of the nuclei can very quickly exceed room temperature, making the initially temperature of the system a minor perturbation to the dynamics.

In the most rigorous approach, the effect of temperature may be included by adopting a density matrix approach. In density operator form, the 0 K state is expressed:

ρ = |Ψ Ψ| (30)
i.e. it is a pure state. However a system at finite temperature is an incoherent mixture of very many thermally excited states, |Ψ n and therefore the correct description is in the form of a density matrix:

ρ = n p n |Ψ n Ψ n | (31)
where p n is a probability, not an amplitude of finding a system in state Ψ n . So in addition to a probability of finding a particle described by a wavefunction at a specific location, there is now also a probability for being in a different state. 130

The equation of motion for the density matrix follows naturally from the definition of ρ and the time-dependent Schrödinger equation leading to the well known Liouville-von Neumann equation:

ih ∂ρ ∂t = [H, ρ] (32)
This applies to a so called closed quantum system, i.e. the Hamiltonian is not in contact with a dissipative environment. When used to describe an open quantum system, the equation has an additional term used to describe dissipation or population decay, however this was not considered for the examples discussed below.

In common with the multi-configuration wavefunction expansion used in the MCTDH scheme for wavefunctions, a similar expansion can be used in the case of density operators, in this case expressed:

ρ(Q 1 , ..., Q f , Q 1 , ..., Q f , t) = n 1 j 1 =1
...

n f j f =1 B j 1 ...j f (t) f k=1 σ (k) j k (Q k , Q k , t) (33)
where B j 1 ...j f denotes the MCTDH coefficients and σ presents the single-particle density operators (SPDO's) analogous to the single-particle functions of the conventional scheme.

The implementation of this approach is described in detailed in refs. 193,194 It is important to bear in mind that compared to the wavefunction approaches described in the previous section, the numerical treatment of density operators is more difficult since the dimensionality of the system formally doubles. 130 To overcome this, a number of approximate approaches have been proposed which retain the computationally less intensive wavefunction formalism. 195 An example is simply to sample various initial conditions of the vibrational levels for each normal mode according to the Boltzmann distribution.

Spin-Vibronic Model Hamiltonian

MCTDH and its multi-layer variant are able to simulate the dynamics of sizeable molecular systems and therefore, the largest challenge is the description of multidimensional PES. For excited states this is especially challenging owing to the breakdown of the BO approximation and the subsequent presence of nonadiabatic coupling. For quantum dynamics, as described in section , the diabatic picture is the method of choice as it provides smooth PES, that can often be described by a low-order Taylor expansion. However, the output of most quantum chemistry is in the adiabatic basis and so obtaining a diabatic representation is a crucial step.

One approach for obtaining an appropriate PES and the method of choice for all spin-vibronic studies presented below is the vibronic coupling model. 143 Its extension, the spinvibronic Hamiltonian matrix (H SO-vib ) is expressed as the sum of the non-relativistic vibronic model Hamiltonian matrix (H vib ), and the spin-orbit, (H SO ) Hamiltonian matrix:

H SO-vib = H vib + H SO (34)
The H SO is comprised of the SOC off-diagonal terms, which can either be Q-dependent 154 or independent. [START_REF] Eng | Spin-Vibronic Quantum Dynamics for Ultrafast Excited-State Processes[END_REF] These are incorporated in a spin-diabatic manner, 153 which is very convenient as one can use the usual SOC-free quantum chemistry methods and introduce the SOCs only to run the dynamics. It is important to note that rigorously, each triplet state should be described by the three m s levels (m s =-1,0,1) and consequently, each triplet state adds an additional three states to the simulation. An approximation to contract the three sub-levels can be made and the SOC be treated as the square root of the sum of the squares of each component. This clearly reduces the computational cost. However, the use of the absolute (real spherical) SOC representation is not straightforward when many singlet and triplet states are coupled and should be rigorously tested, especially if SOC is large.

The multistate vibronic interactions within the spin-free electronic excited states are contained within H vib . The starting point for H vib is a Taylor series expansion of the PES around a reference nuclear geometry Q 0 , usually the FC point using dimensionless (massfrequency scaled) normal-mode coordinates:

H vib = (T N + V 0 )1 + W (0) + W (1) + . . . (35)
Here T N is the kinetic energy operator and V 0 is the ground state potential. V 0 is not restricted to a particular form but is often defined as harmonic with vibrational frequencies ω i along dimensionless normal coordinates Q i . W describes the multistate vibronic coupling matrix within the spin-free representation. 143 Truncation at first order, as shown here, is referred to the linear vibronic coupling (LVC) model, while truncation at second order is referred to the quadratic vibronic coupling (QVC) model. Both have been largely used and have provided good agreement with experimental observations. The advantage of the LVC is that it can be relatively straightforward to obtain, but the disadvantage is that it can become inaccurate for larger displacements away from Q 0 when anharmonic effects become more important. The disadvantage of the QVC model, is that the number of expansion coefficients needed to describe the model can become quite numerous and require additional quantum chemistry data.

The zeroth-order coupling matrix contains the adiabatic state energies at Q 0 . The adiabatic potential surfaces are equal to the diabatic surfaces at this point, so W (0) is diagonal and is expressed as,

W (0) ij = α Φ i (Q 0 )| Ĥel |Φ j (Q 0 ) ( 36
)
where Ĥel is the standard clamped nucleus electronic Hamiltonian and Φ are the diabatic electronic functions. The first-order linear coupling matrix elements are written:

W (1) ij = α Φ i (Q 0 )| ∂ Ĥel ∂Q α |Φ j (Q 0 ) Q α ( 37
)
where the on-diagonal and off-diagonal terms are written:

W (1) ii = α κ (i) α Q α (38a) W (1) ij = α λ (i,j) α Q α (38b)
κ and λ are the expansion coefficients corresponding to the on-and off-diagonal matrix elements. The on-diagonal elements are the forces acting within an electronic surface and are responsible for structural changes of excited-state potentials compared to the ground state. They are therefore responsible for the reorganisation energy; they are called tuning modes. The off-diagonal elements are the nonadiabatic couplings responsible for transferring wavepacket population between different excited states; they are called coupling modes. 143

For the case of the LVC, in the limit of two interacting states, the coupling constants can be obtained from the following expressions 143 evaluated at Q 0 .

κ (i) α = ∂V i ∂Q α Q 0 (39a) λ (i,j) α = 1 8 ∂ 2 ∂Q 2 α (|V j -V i | 2 ) 1/2 Q 0 (39b)
where V i is the adiabatic PES of the spin-free state i. These show that in this model, only the gradients and the Hessians at Franck-Condon geometry are needed to obtain the model Hamiltonian, meaning that the number of quantum chemistry calculations required is quite small. Second-order terms, W [START_REF] Kasha | Energy transfer mechanisms and the molecular exciton model for molecular aggregates[END_REF] are expressed in a similar way. The off-diagonal terms at second order are rarely used, however the on-diagonal terms can be very important. The quadratic terms (i.e. Q α = Q β ) are responsible for changes in frequency of the excited state potential compared to the ground state, while the bilinear terms (i.e. Q α = Q β ) are responsible for mixing the normal modes in the excited state, i.e. the so called Duschinsky rotation effect (DRE). 152,196 For the spin-vibronic model which goes beyond the LVC model, the expansion coefficient of the potential are usually obtained using step wise fitting procedures. 152,196,197 This takes advantage of the fact that while transformations from the adiabatic to the diabatic representation are normally difficult to perform, the opposite way (i.e. diabatic to adiabatic) is relatively simple. 197 Consequently, starting from an initial guess, the diabatic potential can be refined using least squares fit to the computed adiabatic potential by means of standard quantum chemical methods. At each iteration of the fit, the diabatic Hamiltonian is transformed into the adiabatic basis to assess the quality of the fit. This approach is often referred to as diabatisation by ansatz. While in the preceding section the diabatisation is basically performed on the spin-free states and the SOC then added, it is possible to go beyond this assumption by including the SOC directly in the construction of diabatic states. 149 A more rigorous and quite involved treatment to include the nuclear coordinate dependence of the SOC has been proposed by Poluyanov and Domcke,198 and succesfully applied to several problems [199][200][201][202][203] implying intricate Jahn-Teller and SO effects. Very recently, a related approach was developed by Weike and Eisfeld 204 and applied to the general a 1 + e Jahn-Teller case. Finally, Zeng 148 proposes an extension of the model space diabatisation technique 205 to include SOC. Robust and accurate diabatisation techniques are a cornerstone for quantum dynamical treatment of spin-vibronic problems, and developments will allow for a better theoretical description of ISC.

An important consideration when setting up a model Hamiltonian for large polyatomic molecules is which nuclear degrees of freedom are required in the model to ensure an accurate description of the dynamics of interest. In addition, it is clear that for large systems the number of relevant terms, especially at second order, can rapidly become very large making it challenging to refine all of them. The former can be addressed by identifying important excited state structures, such as energy minima and conical intersections, and expressing them as a function of the normal mode displacements from Q 0 82,83 or using trajectory approaches combined with post-analysis. 206 This helps to clearly identify the modes responsible for significant structural changes in the excited state, but will not necessarily identify nonadiabatic coupling modes as they do not always induce structural changes in the excited states. Here additional help is provided using, if present, molecular symmetry. Indeed, a linear vibronic coupling matrix element will only be non-zero if the direct product of the vibrational mode symmetry Γ α and the irreducible representations of the states Γ i and Γ j contains the totally symmetric representation A:

Γ i ⊗ Γ j ⊗ Γ α ⊃ A . ( 40
)
Therefore off diagonal elements (nonadiabatic coupling elements) will only be non-zero if the product of the two state symmetries gives the symmetry of the specific mode. Without symmetry, accurate quantum dynamics are still possible, but the development of the Hamiltonian is more involved.

Finally, it is emphasised that these model Hamiltonians confine nuclear motion within a reduced number of vibrational modes. It is therefore crucial that the most important modes to the dynamics are identified. Indeed, it is often possible to identify a subset of important vibrational degrees of freedom in any excited state dynamics process. This can usually be achieved using the methods described above. However such models will not capture accurately effects such as vibrational relaxation and intramolecular vibrational redistribution.

Therefore simulations longer that a few picoseconds must carefully consider the influence of this effect on the interpretation of the dynamics.

On-the-fly Dynamics Methods

Grid-based quantum dynamics suffer from two limitations: i) Exponential scaling associated with the use of a direct product grid 207 and ii) the requirement to pre-compute the PES meaning that such simulations are limited to only a few degrees of freedoms or models.

Motivated primarily by the desire to overcome these limitations, a number of approaches based upon trajectories have been developed. These range from fully quantum approaches, which explicitly address the nuclear motion according to the TDSE 208 to mixed quantumclassical approaches, such as Tully's Trajectory Surface Hopping (TSH). 209,210 The appeal of methods based upon trajectories is that it becomes easier to apply them within a framework of direct dynamics, where the electronic properties of the system are calculated on-the-fly at each time step as and when it is required. This removes the need to calculating model PES before the simulations and may allow the system to evolve into the fully unconstrained configurational space. Moreover, these methods can easily be combined with a Quantum Mechanics / Molecular Mechanics (QM/MM) scheme which treats explicitly the solute/solvent interactions. 211

Trajectory Surface Hopping

One of the most commonly used approaches is Tully's TSH. 210 This approach discretises the nuclear wavepacket into a swarm of independent classical trajectories, which are subsequently propagated classically according to Newton's equations of motion. The effects of nonadiabatic coupling, forbidden within the framework of classical nuclear mechanics, are included by means of trajectory hops between different PES and are controlled by the nonadiabatic couplings within a stochastic algorithm. Convergence of these simulations is achieved by sampling a number of initial conditions and requires many trajectories to be run, usually on the order of 10 3 because of the slow convergence associated with the underlying classical equations of motion. In addition, it is stressed that the classical foundation of TSH means that such simulations will not necessarily converge onto the correct quantum solution. 212 In particular, quantum effects such a tunneling or interferences are not taken into account.

Excited-state dynamics including both nonadiabatic and spin-orbit couplings have a long history within surface hopping. 213 However, as mentioned above the main advantage of these approaches is the ability to calculate the potential and all relevant couplings onthe-fly allowing simulations to be performed in full nuclear configuration space. Here the group of González et al. 214 has developed an approach of TSH which incorporates spin-orbit and nonadiabatic couplings, using it extensively to study ISC in DNA bases and analogous structures. [START_REF] Richter | Ultrafast Intersystem Crossing Dynamics in Uracil Unravelled by Ab Initio Molecular Dynamics[END_REF][START_REF] Mai | Intersystem Crossing Pathways in the Noncanonical Nucleobase 2-Thiouracil: A time-Dependent Picture[END_REF][215][216][217][218] (see section ). Other groups have also used surface hopping in a similar manner either explicitly [START_REF] Martínez-Fernández | Competing Ultrafast Intersystem Crossing and Internal Conversion: A Time Resolved Picture for the Deactivation of 6-Thioguanine[END_REF][START_REF] Cui | Generalized Trajectory Surface-Hopping Method for Internal Conversion and Intersystem Crossing[END_REF] or by post processing the dynamics. [START_REF] Tavernelli | Nonadiabatic Molecular Dynamics with Solvent Effects: A LR-TDDFT QM/MM Study of Ruthenium (II) tris(bipyridine) in Water[END_REF]206 Atkins et al. recently used TSH including all relevant spin-vibronic couplings, to study the ultrafast dynamics of [Ru(bpy) 3 ] 2+ in the gas phase. 219 The authors found ISC within 30 fs (Figure 7), in good agreement with experimental observations 220 and reported that ISC competes with an internal conversion relaxation process within the singlet manifold to the lowest singlet state. They also reported, consistent with the spin-vibronic mechanism, that ISC is not only driven by the high density of states and the large SOC of the system but also by the vibrational motion. However, this was achieved by freezing the vibrational motion.

By doing this, the authors are also preventing the structural changes in the triplet state, taking the wavepacket away from the crossing point between the two states. Consequently, this creates an equilibrium between the singlet and triplet states at a fixed geometry, as shown in Most present implementations of GWP approaches consider only dynamics within states of a single multiplicity, however there is no formal limitation to include SOC. Given the recent progress made in these areas, many examples can be expected in the near future. Indeed, Curchod et al. 155 have recently extended the formalism of the most established approach, ab initio multiple spawning to include SOC as well as nonadiabatic couplings and used it to study the excited state dynamics of thioformaldehyde, seen in Figure 8. This showed that after excitation into the first excited electronic state (S 1 ), ISC occurs principally to the T 2 state. This is consistent with El-Sayed rules, as these states have different character, nπ * and ππ * , respectively, while the T 1 state which is barely populated has nπ * character. While not discussed within ref.

, 155 Figure 8 shows a clear involvement of the C=S bond during ISC, however whether this points to a spin-vibronic contribution to the ISC process or is simply related to dynamics in the T 2 state is unclear.

While the present example has a fairly modest size, this work sets the foundation for these approaches to be applied to larger systems. This is enhanced by the recent developments in quantum chemistry accelerated by graphical processing units (GPUs) 229 which when coupled to on-the-fly dynamics can make larger problems more tractable. [230][231][232]

Excited State Dynamics of Sulfur Dioxide: A Case Study

A number of recent studies have reported upon the ISC dynamics of SO 2 , including both spinorbit and nonadiabatic couplings. [START_REF] Mai | Non-adiabatic and Intersystem Crossing Dynamics in SO2. II. The Role of Triplet States in the Bound State Dynamics Studied by Surface-Hopping Simulations[END_REF][START_REF] Franco De Carvalho | Nonadiabatic Dynamics with Intersystem Crossings: A Time-Dependent Density Functional Theory Implementation[END_REF][START_REF] Lévêque | Communication: Theoretical Prediction of the Importance of the 3B2 state in the Dynamics of Sulfur Dioxide[END_REF] This provides an interesting point of comparison between quantum and trajectory-based dynamics.

Experimental and theoretical interest in the excited state dynamics of SO 2 is due mainly to its fundamental role in atmospheric photochemistry. Two low-lying singlet states, the absorbing one 1 B 1 and the 1 A 2 and three triplet states states showing clearly that a spin-vibronic mechanism is operated as illustrated in (Figure 10). Interestingly two recent nonadiabatic molecular surface hopping dynamics based on MR-CIS PESs including arbitrary couplings [START_REF] Mai | Non-adiabatic and Intersystem Crossing Dynamics in SO2. II. The Role of Triplet States in the Bound State Dynamics Studied by Surface-Hopping Simulations[END_REF] and on TD-DFT [START_REF] Franco De Carvalho | Nonadiabatic Dynamics with Intersystem Crossings: A Time-Dependent Density Functional Theory Implementation[END_REF] have also been performed. The latter study focuses on the comparison between gas-and liquid-phase dynamics. These two studies based on different electronic structure methods converge to comparable time-scales of the global ISC process, namely about 540 fs [START_REF] Mai | Non-adiabatic and Intersystem Crossing Dynamics in SO2. II. The Role of Triplet States in the Bound State Dynamics Studied by Surface-Hopping Simulations[END_REF] and about 250 fs. [START_REF] Franco De Carvalho | Nonadiabatic Dynamics with Intersystem Crossings: A Time-Dependent Density Functional Theory Implementation[END_REF] Both simulations overestimate the population of the triplet states with 50% in the 3 B 2 state 62 and 60% in the needed to validate the theoretical findings and to understand the photochemistry of this simple triatomic but complex molecule.

( 3 B 1 , 3 B 2 , 3 A 2 )

Experimental Observation of Spin-Vibronic Dynamics

From an experimental perspective, directly observing spin-vibronic dynamics calls for techniques that provide sensitivity to the spin and electronic states and vibrational motion of the system under study. Achieving such simultaneous detail within one technique is challenging and consequently a combination of approaches, usually supplemented with theory, is often used. In the following subsections, information about spin-vibronic dynamics obtained using different spectroscopic techniques is discussed. They also exhibit a strong solvent dependence. 238,242 To investigate this for PtPOP-BF 2 , Hofbeck et al. Impulsive Vibrational Spectroscopy (BB-IVS). 254 The principle motivation for time-domain implementations of vibrational spectroscopy is the ability to record the entire spectrum of molecular vibrations, but also simultaneously achieving higher spectral and temporal resolution than would be possible spontaneous by Raman scattering alone. Consequently, at present direct evidence of spin-vibronic coupling using these approaches has not been reported, although they have been used to elucidate the important modes in nonadiabatic dynamics, such as the singlet fission mechanism 258 or the photoisomerisaiton of retinal in channel rhodopsin 259 and therefore there is no technical barrier to understanding the spin-vibronic mechanisms.

Transient Optical Absorption and

Besides probing spin-vibronic dynamics, the ability of IR radiation to directly excite vibrational modes means that once vibrational roles are understood they can in principle be manipulated. Weinstein and co-workers have recently used an infrared pulse between the pump and probe pulses to control electron transfer. [260][261][262] Based on this knowledge and understanding of the vibronic pathways, the possibility of achieving similar control by chemical design opens new research directions for the future.

Multidimensional Electronic Spectroscopy

Linear spectroscopy is inherently one-dimensional, and provides an absorption spectrum as a function of excitation energy. As mentioned above, one of the limitations of optical transient absorption spectroscopy is the number of overlapping bands as systems become more complicated. Taking a cue from the correlation between spins routinely used in NMR, multidimensional infrared and optical spectroscopies are becoming increasingly common. [263][264][265] The former provides information about the coupling between vibrational degrees of freedom whereas 2D electronic spectroscopy is able to reveal couplings between electronic degrees of freedom. 266

Recently, Carberry et al. 267 used 2-dimensional electronic spectroscopy (2D-ES) to probe the excited state decay of [IrBr 6 ] 2-and measured the couplings between all of the ligand to metal charge transfer (LMCT) states. Indeed, Figure 13, shows the magnitude 2D spectrum, 6 ps after excitation. It exhibits a number of off-diagonal peaks, as predicted by mapping the linear absorption transitions onto the excitation and emission axes (dashed horizontal and vertical lines) and is consistent with 6 spin-orbit coupled LMCT states. Exploiting the dynamics of the off-diagonal elements, the authors were able to show that vibronic contributions, measurable for dynamics on surfaces of a single multiplicity, 268 play very little role in this case, and that the dominant source of coupling is SOC. Although this is not unexpected, given the elements involved, it does outline the potential for resolving spinvibronic dynamics and potentially disentangling the two components within the approach.

2D-ES has also been used to probe singlet fission 269 and reveals the crucial role of vibrational degrees of freedom in this conversion from singlet to triplet states. However, it is stressed that in singlet fission, the two generated spin-triplet excitons are initially correlated to form an overall spin-singlet state. This spin-allowed process differs from the spin-vibronic mechanism discussed here within the context of ISC.

Time-resolved X-ray Spectroscopy

The desire to resolve structural dynamics in excited-state deactivation mechanisms, which is important in the context of spin-vibronic dynamics, has led to a significant research effort that aimed at exploiting short wavelength probe pulses. This has led to the development of time-resolved diffraction methods using X-rays 270-272 or electrons 273,274 and core level spectroscopies using either X-rays [275][276][277][278] or electrons. 279,280 Among these, X-ray spectroscopy is particularly appealing owing to its ability to deliver geometric, electronic and spin state information about the system under study.

For the implementation of time-resolved X-ray spectroscopy, third generation light sources (i.e. storage rings that generate synchrotron radiation) are most suited because of their wide tunability, stability and high photon flux. However, the physics of storage rings limits the Xray pulse duration to 50-100 ps, making direct observation of vibrational motion in real time challenging. This limitation has been overcome by the laser-electron slicing scheme [281][282][283][284] and the so-called low-alpha modes, 285,286 albeit at the sacrifice of beam intensity, making experiments very challenging. Recent advances in technological developments, such as Xray Free Electron Lasers (X-FELs) and High Harmonic Generation (HHG), 287,288 has led to a paradigm shift in the capability of X-ray Spectroscopy to deliver simultaneously high temporal and spectral resolution on an extremely broad range of samples in a wide array of different environments. 275,278 Recently X-FELs have been exploited by Zhang et al. In terms of future perspectives for time-resolved X-ray Spectroscopy in the context of spin-vibronic dynamics, it is noted that the X-ray flux of X-FELs is typically 10-11 orders of magnitude higher than the slicing source at 3 rd generation light sources. This is offering new opportunities in photon-greedy second-order X-ray spectroscopies, such as resonant Xray emission spectroscopy (RXES), also referred to as resonant inelastic X-ray scattering (RIXS) [290][291][292] to be performed with femtosecond resolution. RXES, displays its spectral information in 2 dimensions (Figure 14), absorption and emission energy, which separate overlapping bands and can provide more dynamical insight. By measuring pump-probe RXES signal, sensitivity of the spin and electronic states and vibrational motion of the system under study could be achieved in a synergistic way. This represents an exciting route towards unravelling new insights into spin-vibronic dynamics. 293,294 Recently, Wernet et al. 289 used time-resolved RIXS to probe the ligand exchange dynamics of Fe(CO) 5 in solution and in particular the role of states of different spin-multiplicity.

Figure 14 shows a scheme of the experimental set and the measured spectra. The authors were able to observe that after photo-excitation, Fe(CO) 4 is formed in an excited singlet state that undergoes ISC to the triplet ground state or combines with a CO or solvent molecule to regenerate a penta-coordinated Fe species on a sub-picosecond timescale. While this present study does not offer direct relevance in the context of the spin-vibronic mechanism, it illustrates the power of time-resolved RIXS and motivates future studies in the area.

Spin-Vibronic Intersystem Crossing in Transition Metal

Complexes

Owing to their strong absorption in the UV-visible region of the spectrum and long-lived excited states, transition metal complexes have become a central component for a variety of applications, such as photocatalysts, 295 dye-sensitised solar cells (DSSCs) 296,297 and organic light emitting diodes (OLEDs). 298 The rich electronic flexibility associated with their ability to bind various ligands and to link polymers, wires, proteins and DNA open the route to a number of functions, such as luminescent or conformational probes, 299,300 diagnostic or therapeutic tools, 301-303 photoswitches 304,305 and long-range electron transfer trigger. 306,307 Their extensive use for photophysical, photochemical and photobiological applications has provided a strong driving force for a large research effort directed to their fundamental photophysics [308][309][310][311] in order to control these functions or elucidate rational material design.

This work has strongly highlighted the complexity of the dynamics that occurs on the ultrafast timescales, [START_REF] Chergui | On the Interplay Between Charge, Spin and Structural Dynamics in Transition Metal Complexes[END_REF] but also the breakdown of long held expectations. Indeed, the concepts of IC, driven by vibronic coupling and ISC, controlled by SOC, become rather artificial in the case of polyatomic molecules characterised by a high density of electronic excited states. Within the basis of spin-orbit and vibronically-coupled BO states, a realistic picture is provided by a spin-vibronic mechanism as illustrated by recent work reported in the proceeding subsections.

Jahn-Teller and Spin-Orbit Coupling Effects in Transition-Metal Trifluorides

One of the most well studied results of vibronic coupling is the Jahn-Teller effect. The interplay between Jahn-Teller (JT) and SOC effects has been explored in high-spin doublydegenerate electronic states of first-row transition metal trifluorides, the vibronic spectra of which have been computed from first principles. [START_REF] Mondal | Infrared Absorption Spectra of Jahn-Teller Systems: Application to the Transition-Metal Trifluorides MnF3 and NiF3[END_REF]312 Whereas JT vibronic coupling is quenched by strong SOC effects in vibronic spectra of electronic states of even number multiplicities, the vibronic structure is conserved in the spectra of states characterised by odd number multiplicities. This has been demonstrated for MnF 3 in its 5 E ground state and have been studied considering SOC together with vibrational strongly JT active normal modes including linear and up-to sixth order JT couplings, JT stretching normal modes including up-to fourth order JT couplings within a two-state four-mode model in the case of fluorescence spectroscopy. The experimental assignment of the refined spin-vibronic spectra in these high-spin states systems is still a challenge and quantum dynamics based on accurate electronic structure data (here from CASSCF calculations) [START_REF] Mondal | Infrared Absorption Spectra of Jahn-Teller Systems: Application to the Transition-Metal Trifluorides MnF3 and NiF3[END_REF]312 NiF 3 in its 4 E excited state is characterized by strong JT and spin-orbit couplings (Table 1). Interestingly and as illustrated by the internal infrared spectra computed without and with SOC, the strong original non-adiabatic effects (Figure 16) are dramatically reduced by SOC and the structure of the spectrum becomes much more simple.

Strictly speaking these pioneering studies are not related to ISC processes because they

Fe(II) Complexes

Fe(II) is a system with six d-electrons and, consequently, Fe(II) complexes exhibit either a high spin 5 T 2 ground state in a weak ligand field or a low spin 1 A 1 ground state in a strong ligand field. A transition between the two configurations, thereby controlling the magnetic properties of the complex, can be achieved by changes in temperature or by light. The latter is usually referred to as light-induced excited spin-state trapping (LIESST). [313][314][315][316] The high density of electronically excited states in many Fe(II) complexes, which derives from the simultaneous presence of metal-to-ligand charge transfer (MLCT) and metalcentered (MC) states means that the excited-state dynamics of LIESST are usually extremely rich and ultrafast. Consequently, this process has received significant attention, 14,317-321 a lot of which has been focused upon the prototypical LIESST complex, Iron(II)-tris-bipyridine ([Fe(bpy) 3 ] 2+ ), 284,322? -324 shown in Figure 17. These studies have shown that after optical excitation into the 1 MLCT band, [Fe(bpy) 3 ] 2+ relaxes into the non-emissive quintet MC ( 5 T 2 ) state, therefore undergoing two spin forbidden transitions, within 200 fs, [START_REF] Consani | Vibrational Coherences and Relaxation in the High-Spin State of Aqueous [Fe II (bpy) 3 ] 2+ Angew[END_REF][START_REF] Mccusker | Sub picosecond 1MLCT-5T2 Intersystem Crossing of Low-Spin Polypyridyl Ferrous Complexes[END_REF][START_REF] Bressler | Femtosecond XANES Study of the Light-Induced Spin Crossover Dynamics in an Iron(II)[END_REF] Figure 17. This decay is significantly faster than the usual time scales expected for a single ISC event, let alone two (singlet→quintet), and consequently direct observation of the intermediate states and description of the full mechanism was a significant challenge. Using the time-dependent perturbation approach outlined in section , Sousa et al. 325 characterise the excited-state decay mechanism including all possible intermediate electronic states along the decay pathway. Given the high density of excited states and strong coupling, it can be argued because the coupling between excited states is likely to be much greater than the energy gap between them, that perturbation theory is not strictly valid. However, the calculations gave a total deactivation time in good agreement with the experimentally determined rate and indicate that the complex can reach the final high-spin state by different deactivation channels, both of which include intermediate MC triplet states ( 3 MC) and require spin-vibronic interactions.

These mechanisms were later used to explain the femtosecond X-ray emission study of Zhang et al. [START_REF] Zhang | Tracking Excited State Charge and Spin Dynamics in Iron Coordination Complexes[END_REF] The authors reported that at 50 fs after photoexctiation, the transient XES spectrum, which is particularly sensitive to the spin state of the absorbing iron atom, 275,278 could only be described invoking the intermediate 3 MC. This conclusion was subsequently challenged by Aubock et al. [START_REF] Auböck | Sub-50-fs Photoinduced Spin Crossover in [Fe (bpy) 3] 2+[END_REF] who used pump-probe optical transient absorption spectroscopy to argue that the quintet state is impulsively populated in < 50 fs, the same time measured for the depopulation of the MLCT manifold. The authors did not exclude the contribution of an intermediate triplet state, but claimed its lifetime must be <20 fs.

Most recently, Lemke et al. [START_REF] Lemke | Coherent Structural Trapping Through Wavepacket Dispersion During Photoinduced Spin State Switching[END_REF] used ultrafast X-ray absorption spectroscopy reporting that the transient spectra were consistent with the short-lived 3 MC state which is populated within 50 fs and decays within 120 fs (Figure 18), similar to the conclusion of Zhang et al. [START_REF] Zhang | Tracking Excited State Charge and Spin Dynamics in Iron Coordination Complexes[END_REF] They were also able to probe not only the position, but the width of the wavepacket as predicted in ref. While the conclusions from these recent experimental works are slightly different, the most important point in the context of the present review is that all of these results illustrate that the deactivation process is significantly non-BO involving presence of spin-orbit and nonadiabatic coupling, which in the presence of a high density of electronically excited states leads to a heavily mixed spin-vibronic manifold of excited states. 255 Indeed, this relaxation process is one of the most stark examples of strongly inter-winded spin, electronic and structural dynamics occurring in a molecular system.

The spin-crossover phenomenon makes Fe(II) complexes very appealing for single-molecule magnets, but the photoexcited decay route makes [Fe(bpy) 3 ] 2+ and related complexes unsuitable for applications seeking to exploit the MLCT properties, as the MC states transfer the photoexcited electrons away from outer ligand-based regions of the complex, preventing charge transfer and/or injection. 326 Recently, motivated by the desire to achieve photosensitisers using earth abundant elements, 297 there have been many attempts to destabilise the MC states to achieve longer-lived MLCT states. Liu et al. 327 These simulations, which provide good agreement with the experimental decay timescales, rationalise the strong differences between the dynamics of these two complexes and highlight the interplay between the spin and nonadiabatic (spin-vibronic) coupling which influences the rate of decay from the MLCT to the MC states. However, the most important factor in the development of these complexes is the relative energetic position of the 1,3 MC and MLCT states, especially close to the FC geometry, which largely determines the lifetime of the MLCT states.

Cu(I) Complexes

For potential applications in OLEDs, [329][330][331] and light emitting electrochemical cells (LEECs), 332,333 or as chemo-and biosensors, 334,335 it is widely acknowledged that complexes containing 2 nd and 3 rd row transition metal ions presently form the most stable and versatile emitters. However, these metals are unappealing for commercial applications due to their high cost and low abundance. Consequently, one of the most attractive methods is to simply remove the metal centred states by adopting a complex based upon d 10 metal ions, such as Cu(I), Ag(I), Au(I), Zn(II) and Cd(II). 336 Among the most popular are those containing Cu(I) ions. 337 Amongst the most extensively studied is Cu(I) complexed with phenanthroline ligands (see ref. 338 and references therein) because they have many photophyscial properties which are similar to the ruthenium polypyridines. 157,[339][340][341][342] One of the first time-resolved spectroscopic studies of [Cu(dmp) 2 ] + was performed by Chen et al , 343 who used time-resolved absorption spectroscopy to probe the dynamics following photoexcitation of the optically bright S 3 state. 344 They reported two principal dynamical processes occurring in 500-700 fs and 10-20 ps. Based upon the ultrafast ISC occurring in [Ru(bpy) 3 ] 2+ , 257 they assigned the first to ultrafast ISC and the second to the structural relaxation. At the same time Nozaki and co-workers 156 reported that the 10-20 ps relaxation was due to ISC, which is relatively slow in [Cu(dmp) 2 ] + due to a strong effect of the structural rearrangements on the magnitude of the SOC matrix elements. Indeed, at the FC geometry the coupling between S 1 and T 1 was estimated to be ∼300 cm -1 , but at the distorted excited-state geometry, expected to be most important on the timescale of tens of picosecond, it is only ∼30 cm -1 (See Figure 2).

Later, Tahara and co-workers 346 probed the excited state dynamics using femtosecond fluorescence up-conversion. Following photoexcitation at 420 nm they found three principal time-components. An initial strong fluorescence, which decayed with a time constant of ∼45 fs, was attributed to the decay of the initially populated excited state via IC. This was followed by two time components of 660 fs and 7.4 ps, assigned to the PJT distortion and ISC, respectively. Later Chen and co-workers 347 reported similar time-components, 80 fs, 510 fs and 10-15 ps. They concluded, in agreement with Tahara that the longer timescale was ISC, and assigned the shortest time constant (80 fs) to IC and the PJT distortion.

Finally Tahara and co-workers 348 used transient absorption spectroscopy to investigate the dynamics following excitation at 550 nm. This transition corresponds to the lowest singlet MLCT state (S 1 ), which is dipole forbidden but gains intensity through vibronic coupling with the main absorption band. They observed distinct wavepacket dynamics corresponding to the PJT distortion, and were able to obtain a low-frequency vibrational spectrum of the important modes.

All of these latter studies [346][347][348][349] have ruled out ultrafast ISC. However, the analysis of these experiments only considered SOC between the S 1 and T 1 states, and the SOC at a minimum of the S 1 geometry. As pointed out by Zgierski,350 there are four closely spaced and vibronically coupled triplet states in close proximity to S 1 , and therefore the S 1 -T 1 coupling is not the only relaxation channel into the triplet states. Indeed it was shown in ref. A recent quantum dynamics study using a model spin-vibronic Hamiltonian 154,344 found that after photoexcitation a component of ultrafast intersystem crossing, shown in the population kinetics of Figure 20a, occurs at the same time as the structural JT distortion. The mechanism for ISC is shown to be a dynamic effect, in the sense that it arises from the system traversing the PJT coordinate where the singlet and triplet states become degenerate, leading to efficient crossing. It is enhanced by the strong vibronic coupling within the triplet manifold and the small energy gap between the states. This contribution of ultrafast ISC does depend on the rate of vibrational cooling leading to a wavepacket relaxed in the minimum of the S 1 state at which geometry the SOC is significantly weaker and the crossing is only between the S 1 and T 1 state. However, as demonstrated this occurs in about 1 ps, 351 after most of the ISC shown in Figure 20a has occurred. In addition, these conclusions were recently supported using excited-state molecular dynamics simulations, explicitly including the effect of the solvent in a QM/MM fashion. 206 This illustrates again the entwined nature of electronic, spin and structural relaxation associated with the spin-vibronic mechanism which suggests that both processes are intimately coupled in all molecules in this series, making decoupling them into different mechanism steps an approximation. 352 In the context of these Cu(I) complexes, this spin-vibronic dynamics leading to an ultrafast component of ISC is actually detrimental. Indeed, Huang et al. 353 recently demonstrated that when attached to a TiO 2 , charge injection for a related Cu(I)phenanthroline complex, from the 1 MLCT is two orders of magnitude faster than from the 3 MLCT. Consequently for efficient charge injection one needs to restrict ISC.

The research of Cu(I) complexes has largely focused upon the four-coordinate systems.

However, unlike four-coordinate tetrahedral Cu(I) compounds, the three-coordinate geometry eliminates the possibility of flattening distortion in the excited state. 354 However, reports on luminescent linear Cu(I) complexes are rare. [355][356][357][358][359][360] Particularly interesting with regard to spin-vibronic coupling are linear Cu(I) complexes involving the standard N-heterocylic carbene IPR (1,3-bis(2,6-diisopropylphenyl)imidazol-2-ylidene) as σ-donor and various pyridine derivatives as π-chromophore ligands that were reported to be highly efficient blue to blue-green phosphorescence emitters. 360 For each of these complexes, the luminescence quantum yield was found to depend on the excitation wavelength. The highest quantum yield was measured for excitation wavelengths close to the origin of the S 1 band. Difference densities obtained from DFT/MRCI calculations (Fig. 21) show the S 1 state of the pyridyl and methylpyridyl complexes to be an MLCT state resulting mainly from a d σ → π pyridyl excitation. The corresponding triplet state exhibits a very similar electronic structure and therefore the direct SOC between these states is small. In second order, their interaction is mediated by another close-lying triplet state of mixed MLCT and ligand-centered (LC) type, wherein the MLCT configuration corresponds to a d π → π pyridyl excitation. As the dominant 1 MLCT and 3 MLCT configurations differ from the leading term of the 3 MLCT/LC wavefunction by a d σ → d π single excitation, their mutual spin-orbit interaction is expected to be substantial. And indeed, due to the large SOCME (267 cm -1 ) and the close ener-getic proximity of the states, the vibrationally relaxed S 1 state is predicted to convert very rapidly within 16 ps to the 3 MLCT/LC state, leading to complete suppression of prompt fluorescence. The 3 MLCT/LC state can further interact via SOC of comparable magnitude (266 cm -1 ) and/or derivative coupling with the 3 MLCT state, establishing a temperaturedependent equilibrium so that TADF becomes conceivable in addition to phosphorescence.

Photoexcitation with shorter wavelengths may reach locally excited triplet states of the diisopropylphenyl substituents which do not phosphoresce and hence diminish the luminescence quantum yield. and rISC to proceed with rate constants of the order of 10 9 s -1 at room temperature.

Re(I) α-diimine Complexes

Rhenium(I) tricarbonyl α -diimine complexes [Re(X)(CO) 3 (L)] n+ with X being a halogen, an alkyl group or a pyridyl ligand and L a bidentate-diimine ligand or two monodentate pyridyl ligands, play a central role in the photochemistry and photophysics of 3 rd row transition metal complexes since their discovery in the 1970s. 362 Indeed, thanks to their synthetic flexibility, tunable photo activity and rich photophysics and photochemistry, these molecules are easily linked to bio systems or incorporated in polymers, wires and supramolecular assemblies finding various applications in the fields of photocatalysis, light-emitting devices, electron/energy transfer processes, non-linear optics and cancer treatments. [363][364][365][366][367][368][369] The control of their excited-state activity is essentially driven by the character of the lowest long-lived triplet state, MLCT, IL, MC or LLCT, easily tunable by the nature of X and L and accessible via ultrafast vibrational decay. 370 In order to investigate in detail the role of SOC in this class of complexes, pioneering time-resolved spectroscopic experiments and theoretical studies have been performed on a number of complexes. [START_REF] Cannizzo | Femtosecond Fluorescence and Intersystem Crossing in Rhenium (I) Carbonyl-Bipyridine Complexes[END_REF]364,371,372,[372][373][374][375][376][377][378][379][380][381][382] Surprisingly, fs-ps ultra-fast luminescent decay, measured in this family of Re(I) complexes, does not follow the heavy atom SOC effects, all the complexes investigated so far exhibiting the same particularities, namely an ISC process characterized by three time domains τ 1 ∼ 85-150 fs, τ 2 ∼ 0.3-1.7 ps and τ 3 ∼ 40 ns -7.5 µ s and three domains of energy, namely 500-550 nm, 550-600 nm and >610 nm whatever the ligands X and L are as illustrated by fs fluorescence up-conversion, transient absorption and time-resolved infra-red experiments. [START_REF] Cannizzo | Femtosecond Fluorescence and Intersystem Crossing in Rhenium (I) Carbonyl-Bipyridine Complexes[END_REF]373,383 After careful theoretical investigations of the static absorption and emission properties of a series of [Re(X)(CO) 3 (L)] n+ (X = Cl, Br, I; L= 2,2'-bipyridine, n=0; X= imidazole or pyridine; L = 1,10-phenanthroline, n= +1) of C s symmetry, depicted in Figure 22,372,[384][385][386][387][388] a model Hamiltonian for solving a multi-state multimode problem including vibronic and SOC within the LVC approximation and the assumption of harmonic potentials has been developed (see section 3.2.3). The simulation of the ultrafast dynamics by wavepacket propagations using MCTDH 192 is based on DFT and its TDDFT electronic structure data. [START_REF] Eng | Spin-Vibronic Quantum Dynamics for Ultrafast Excited-State Processes[END_REF][START_REF] Fumanal | Ultrafast Excited-State Decays in [Re (CO) 3 (N, N)(L)] n+: Nonadiabatic Quantum Dynamics[END_REF][START_REF] Harabuchi | Exploring the Mechanism of Ultrafast Intersystem Crossing in Rhenium (I) Carbonyl Bipyridine Halide Complexes: Key Vibrational Modes and Spin-Vibronic Quantum Dynamics[END_REF]385 X = imidazole L = phenanthroline X = pyridine L = phenanthroline X = Cl, Br, I L = bipyridine As illustrated for the imidazole-substituted complex Figure 22 similar experimental luminescent decay characteristics together with comparable chemical properties can lead to a very different spin-vibronic mechanism paving the way to laser pulse control in transition metal complexes. The two lowest 1 MLCT, the three lowest 3 MLCT and the 3 IL phen states of [Re(imidazole)(CO) 3 (phen)] + , the electronic densities of which are depicted in Figure 26, have been included in the simulation because potentially populated by ultrafast decay from the absorbing state S 2 . This leads to six electronic states coupled vibronically and by SOC within a fifteen modes model generating fourteen adiabatic spin-orbit states. [START_REF] Fumanal | Ultrafast Excited-State Decays in [Re (CO) 3 (N, N)(L)] n+: Nonadiabatic Quantum Dynamics[END_REF] The 3 IL phen (T 3 ) state is stabilized as compared to the 3 IL state localized on the bpy ligand in the halide complexes. pulses but also chemically or by environment effects.

Spin-Vibronic ISC in Small Molecules and Organic Chromophores

Small Aromatic Molecules

The earliest examples of studies characterising the spin-vibronic nature of spin-flip transitions were observed in the phosphorescence properties of small aromatic hydrocarbons, such as benzene. [START_REF] Albrecht | Vibronic Spin-Orbit Perturbations and the Assignment of the Lowest Triplet State of Benzene[END_REF][START_REF] Lim | Vibronic Interaction between (n, π*) and (π, π*) States and Spin Orbit Coupling in Nitrogen Heterocyclics[END_REF][START_REF] Lim | Vibronic Spin-Orbit Interactions in Heteroaromatic Molecules. II. Phosphorescence of Quinoxaline and Other Diazanaphthalenes[END_REF][START_REF] Henry | Spin-Orbit Coupling in Aromatic Hydrocarbons. Calculation of the Radiative Triplet Lifetimes of Naphthalene, Anthracene, and Phenanthrene[END_REF][START_REF] Henry | Spin-Orbit Coupling in Aromatic Hydrocarbons. Analysis of Nonradiative Transitions between Singlet and Triplet States in Benzene and Naphthalene[END_REF] Indeed in aromatic hydrocarbons, the triplet-ground state transition is often forbidden under the Condon approximation and therefore vibrations must be involved for phosphorescence to be activated. Benzene is a classic example of a photochemically rich molecule, with a number of competing pathways that can result after photoexcitation. 389,390 Extending these studies into the femtosecond time domain, Fielding and co-workers used time-resolved photoelectron spectroscopy (TRPES) to show the involvement of ISC competing with the IC in the photoexcited decay of benzene on a sub-picosecond timescale (Figure 29a). [START_REF] Parker | Ultrafast Dynamics of the S1 Excited State of Benzene[END_REF][START_REF] Minns | Competing Iltrafast Intersystem Crossing and Internal Conversion in the "Channel 3" Region of Benzene[END_REF]152,196 Although this still remains a minor channel, any ultrafast ISC in a simple hydrocarbon is surprising due to the very weak SOCME. Despite the small magnitude of the SOCME, 391 supporting quantum dynamics simulations [START_REF] Parker | Ultrafast Dynamics of the S1 Excited State of Benzene[END_REF][START_REF] Minns | Competing Iltrafast Intersystem Crossing and Internal Conversion in the "Channel 3" Region of Benzene[END_REF]152,196 demonstrated that ultrafast ISC occurs via spinvibronic effect activated by the prefulvene mode (Figure 29b) where weakly-coupled (S 1 , T 1 and T 2 ) states are near degenerate, leading to efficient crossing.

Another small aromatic molecule, which has been widely studied due to its potential application in optoelectronic is thiophene. 392 Just like many other small hetroaromatic molecules, thiophene exhibits ultrafast deactivation of the S 1 state and is non-fluorescent. [393][394][395][396] These previous studied have indicated that a ring opening mechanism is responsible for the internal conversion from the excited to the ground state. Recently, Schnappinger et al. 397 studied the relaxation of thiophene including both singlet and triplet states. They reported that after internal conversion, at the ring opening conical interactions, there are two possible pathways. One returns the system to the closed ring ground state, while the other leads to a flat potential, along which the SOCME increases and the singlet-triplet energy gaps nearly vanish. This leads to efficient intersystem crossing promoted by this vibrational motion in a mechanism which is very similar to as described above for benzene.

Deactivation of DNA bases by ISC

In living organisms exposed to UV light, DNA is being continuously damaged and repaired.

Interaction with UV light generates an excited electronic state in DNA which will decay via either photophysical or photochemical pathways. 398,399 Photophysical relaxation is characterised by nonradiative decay from an excited state to the ground state, resulting in repopulation of the ground electronic state. Consequently, these pathways have been widely

studied for understanding intrinsic photoprotection mechanisms. Research into these photoprotection mechanisms dynamics is dominated by femtosecond time-resolved spectroscopic studies of single nucleobases in either the gas [400][401][402][403][404][405][406] or solution phase, 407-411 often supported by theoretical work. 215,406,[412][413][414][415][416][417][418][419][420][421] Despite extensive studies, the precise role of ISC in photostability is not well established.

To address this, the González group have extensively used excited state molecular dynamics to probe the competition between IC and ISC in DNA bases. [START_REF] Richter | Ultrafast Intersystem Crossing Dynamics in Uracil Unravelled by Ab Initio Molecular Dynamics[END_REF][START_REF] Mai | Intersystem Crossing Pathways in the Noncanonical Nucleobase 2-Thiouracil: A time-Dependent Picture[END_REF][START_REF] Mai | The DNA Nucleobase Thymine in Motion-Intersystem Crossing Simulated with Surface Hopping[END_REF][215][216][217]422 Recently, using nonadiabatic dynamics simulations at the multi-state complete active space second-order perturbation theory (MS-CASPT2) level of theory, Mai et al 72 studied the deactivation mechanism of 2-thiouracil. As show in Figure 30, the authors report two sub-picosecond ISC channels promoted by out of the ring plane motions of the S atom which leads to near degeneracy of singlet and triplet states and sizeable of SOCME ∼150 cm -1

For the related natural DNA base cytosine, excited state molecular dynamics simula- tions 215,216 employed potentials, non adiabatic and spin-orbit couplings determined at the state-averaged complete-active-space self-consistent-field (CASSCF) level. On the basis of these simulations, the authors reported a decay of the S 1 population via ISC to the T 2 state to be ultrafast. At least for vibrationally cold S 1 populations, these results are at variance with the experimental findings on supersonic jet-cooled cytosine by Lobsiger et al. 423,424 A recent combination of experimental and theoretical studies implies that the main deactivation channel for S 1 populations with an excess energy below 550 cm -1 is IC to the electronic ground state and that the S 1 ;T 1 ISC is moderately fast (nanosecond time scale), and not ultrafast. 406 The slight out-of-plane distortion of the S 1 minimum geometry of cytosine and the concomitant small admixture of 1 n O π * character in the S 1 wavefunction makes the S 1 ;T 1 ( 1 ππ * ; 3 ππ * ) ISC nearly as fast as an El-Sayed allowed transition.

The presumable reason for these contradictory findings is the electronic structure of the S 1 state. The CASSCF wavefunction is reported to have predominantly n O π * character at the S 1 minimum whereas the spectral signature clearly identifies S 1 to be a slightly out-of-plane distorted 1 ππ * state. Higher-level correlated methods, such as CASPT2 and spin-component scaled coupled cluster with approximate treatment of doubles (SCS-CC2), place the 1 ππ * state adiabatically below the 1 n O π * state in cytosine. 406,415 This underpins the importance of basing photodynamical studies of heteroaromatic compounds on reliable correlated electronic structure methods. Using the perturbation theory methods described in section based on CC2 potentials to determine ISC rate constants of the photoexcited pyrimidine bases uracil and thymine, Etinski et al. 425 predicted ISC to play an important role in the electronic relaxation of the initially exicted 1 ππ * state in the gas phase. This was confirmed later experimentally. 426

The triplet quantum yield of uracil may be even enhanced if a nitrogen atom is substituted for the CH group in 6-position. 427 Energy profiles and SOCMEs computed along linearly interpolated reaction paths (Figure 31) point towards strong nonadiabatic and spin-vibronic coupling in 6-azauracil. Absorption of UV light excites the S 2 ( 1 ππ * ) state. Essentially, two different decay pathways were proposed. 163,428 The first one involves the T The reduced triplet quantum yield is traced back to the substantial blue shift of the n O π * states relative to the ππ * excited states in polar protic environments, making their direct or indirect participation in spin-vibronic decay mechanisms of the photoexcited 1 ππ * states less probable. 425,431 Similar blue shifts are encountered in many aromatic carbonyl compounds (see also section ). In the xanthone family of dyes, the polarity and proticity of the solvent environment can be applied to actively control the triplet quantum yield. [START_REF] Rai-Constapel | Chimeric Behavior of Excited Thioxanthone in Protic Solvents: II. Theory[END_REF]432 Heteroaromatic Photosensitisers for Photodynamic Therapy PDT uses photosensitisers, i.e. photo-addressable molecules with little or no toxicity, to initiate programmed cell death (apoptosis). PDT is applied to cure tissue related diseases such as skin cancer, vitiligo, or psoriasis, but may also be used to kill microbial cells, e.g. in blood samples. Type I photosensitizers trigger radical reactions and are not of interest in the context of the present review. Type II photosensitizers either undergo photochemical reactions in the triplet state or transfer spin and excitation energy to molecular oxygen to form singlet dioxygen which in turn initiates apoptose. [433][434][435] (PhSens)

hν → 1 (PhSens) * ISC ; 3 (PhSens) * 3 (PhSens) * + O 2 ( 3 Σ - g ) → (PhSens) + O * 2 ( 1 ∆ g )
Prerequisites for a potent photosensitiser are the existence of a long-lived triplet excited state and its efficient population via ISC. Three prototypical type II photosensitisers are displayed in Fig. 32. All of them are heteroaromatic compounds with a 436 In the longer wavelength regime, porphyrin or phenothiazine derivatives are used. 433,437 Some of the porphyrin derivatives, e.g. protoporphyrin IX, can be synthesised by the patient's body after administration of 5-aminolevulinic acid. 438 Psoralens are light-sensitive compounds naturally occurring, e.g., in cow parsnip. Pho-toexcited psoralens have been found to perform [2 + 2]-cycloadditions with thymine bases at the pyrone and the furan sides, linking two DNA strands covalently together and thus preventing replication of diseased cells. 436 In addition, singlet oxygen formation is observed.

Theoretical studies by Tatchen et al. 150,439,440 have shown that the ISC pathways involving nπ * states are energetically not accessible in aqueous solution and that the rate constants for the 1 ππ * ; 3 ππ * transitions are small (< 10 5 s -1 ) in Condon approximation. Out-of-plane vibrational distortions enhance the spin-orbit coupling (Figure 33) to an extent that the S 1 ;T 1 ISC outcompetes the fluorescence by two orders of magnitude and triplet formation becomes the dominant process in aqueous solution. ] excitations. Although each of these terms exhibits a large transition dipole moment with the electronic ground state, the probability for a radiative transition is very low in Condon approximation because the individual contributions to the electric dipole transition moment nearly cancel. In-plane deformation vibrations lift this cancellation and make the Q-band transitions vibronically allowed. 151,164,181,443,444 As a consequence, the Q bands of porphyrins exhibit a very characteristic intensity pattern with a weak 0-0 peak and substantially more intense vibronic transitions. 445,446 Although the spin-orbit interaction between the S 1 ( 1 B 2u )

q 1 [pm] <T 1x |H SO |S 1 > <T 1y |H SO |S 1 > <T 1z |H SO |S 1 > <T 2x |H SO |S 1 > <T 2y |H SO |S 1 > <T 2z |H SO |S 1 > (c)
and T 1 ( 3 B 3u ) states is formally allowed in D 2h symmetry, it is very weak at the S 1 minimum because both states have ππ * character. The squared matrix element which enters the perturbation expression 18 has a value of merely 0.004 cm -2 which may be compared to an energy separation of about 2400 cm -1 at the DFT/MRCI level. 164 In-plane deformations do not change the SOCME markedly, but out-of-plane vibrational distortions mix some σπ * character into the electronic wavefunctions thereby enhancing the spin-orbit interaction (Figure 4(b)). Not surprisingly, the S 1 ;T 1 ISC rate constant is found to be temperature dependent with calculated values of ∼ 1 × 10 7 s -1 at 0 K and ∼ 9 × 10 7 s -1 at room temperature. 151

Thionine (3,7-diamino-phenothiazin-5-ium cation) is a photosensitiser which belongs to the phenothiazine (PTZ) family of dyes. While the PTZ core is a ubiquitous donor in TADF donor-acceptor compounds (see also section ), its cationic diamino-substituted derivatives are efficient singlet oxygen sensitisers absorbing in the 590-660 nm wavelength regime. Although only two ππ * excited triplet states are located adiabatically below the first excited singlet state S 1 (π H π * L ), 447 thionine exhibits a sizeable triplet quantum yield of 0.55 in dilute water solutions. Based on computed potential energy curves and ISC rate constants in Herzberg-Teller approximation, it was suggested that the triplet formation in water proceeds

primarily through the S 1 (π H π * L ) ;T 2 (π H-1 π * L ) ;T 1 (π H π * L ) channel. 176 The T 2 (π H-1 π * L )
state is seen to cross the S 1 (π H π * L ) potential close to the minimum of the latter so that even small out-of-plane distortions are sufficient to speed up the El-Sayed forbidden nonradiative transition substantially. Also in this case, the spin-vibronic transition is markedly enhanced by thermal effects. 151

Thermally Activated Delayed Fluorescence (TADF)

TADF, also known as E-type delayed fluorescence (DF), 448 has recently emerged as a very attractive approach for harvesting the 75% of triplet excited states generated by electrical excitation in organic light emitting diodes (OLEDs). 449,450 One of the key components of TADF is the rate of the conversion from the triplet state into the singlets, as long lifetimes in the triplet state lead to excited state quenching mechanisms such as triplet-triplet annihilation, reducing efficiency. 451 This calls for a detailed understanding of the factors affecting TADF.

Due to the importance of charge transfer states, 450 which are used to minimise the energy gap between the low-lying singlet and triplet states, it was initially assumed that the states involved in the rISC were the singlet ( 1 CT) and triplet ( 3 CT) charge transfer. However, SOC between these intramolecular CT states is formally zero (See section ). This means that other electronic states are likely to be involved. Using D-A and D-A-D molecular TADF systems [452][453][454] Monkman, Bryce and co-workers demonstrated that two of the excited states involved could be independently tuned and therefore they must be of different character.

Further insight into the mechanism was provided by Ward et al. 455 who showed that different D-A-D molecules with very similar energy gaps (∆E S 1 -T 1 ) exhibit large variations in k rISC .

They found that by sterically hindering the motion of D and A group, the emission could be switched from TADF to phosphorescence. This indicates a mechanism which goes beyond the Condon approximation and is dynamic in nature, in the sense that it depends on molecular vibrations.

These observations were recently rationalised by Gibson et al. 456 using QD to study the rISC of a D-A molecule composed of a phenothiazine donor and a dibenzothiophene-S,Sdioxide acceptor (PTZ-DBTO2), shown in Figure 34a. In this work, as shown schematically in Figure 34b, the authors showed that for PTZ-DBTO2, three states were crucial to de-scribe the rISC mechanism. These were the 1 CT, [START_REF] Kasha | Characterization of electronic transitions in complex molecules[END_REF] CT and a local ππ * ( 3 LE) excited state on the donor unit. The rates obtained were in good agreement with those reported experimentally 457 (Figure 34c black trace), but most importantly the authors show that the non-adiababtic coupling between the two triplet states was critical for describing the rISC.

Indeed, if this was removed the rate become close to zero (Figure 34c dark blue trace).

Importantly, the critical role of vibronic coupling between the two triplet states for rISC between the 3 LE; 1 CT states cannot be described within first-order perturbation theory approach and therefore must involve spin-vibronic coupling. In addition, for PTZ-DBTO2 in low polarity environments as shown in Figure 34b, a direct second-order coupling for

3 LE; 1 CT would require population transfer between the two CT states. However, as shown in Figure 34c the coupling between these two states has an insignificant role. Consequently, the relevant terms are:

k rIC = 2π h | Ψ3 CT |H vib |Ψ3 LE | 2 δ(E3 CT -E3 LE ) (41)
and

k rISC = 2π h Ψ1 CT | ĤSO |Ψ3 LE Ψ3 LE |H vib |Ψ3 CT E3 CT -E3 LE 2 δ(E1 CT -E3 LE ) (42)
Firstly, the vibronic coupling between 3 LE and 3 CT promotes, on a timescale much faster than the rISC, an equilibrium between the two states. Obviously, the position of this equilibrium depends both on the size of the nonadiabatic coupling and the energy gap. Subsequently, the second-order term, Equation 42, couples the 3 CT and the 1 CT, using the 3 LE as an intermediate state. This latter second-order term is very efficient because of the good vibrational overlap between the almost degenerate initial and final states, 3 CT and 1 CT, respectively. Importantly, this explains recent experimental results which demonstrated that steric hindrance of D-A dihedral angle switches the main pathway from TADF to phosphorescence 455 .This steric hindrance is equivalent to removing the vibronic coupling term, which is shown herein to be strongest along modes exhibiting a distortion of the D-A dihedral angle. S3, Green: No HFI, Blue: No Vibronic Coupling.

The temperature dependent rate of a thermally activated process is given by the Arrhenius equation provided that the equilibration is faster than the radiative or nonradiative deactivation of the electronically excited molecule. The exponential decrease in the rate with activation energy, which this imposes, strongly promotes processes with small activation barriers. Experimentally to elucidate the gap, there are two approaches adopted. The first is obtained by fitting the integrated DF emission as a function of temperature and therefore gives a thermal activation derived from the Arrhenius equation. 450 The second is defined as the difference between the onset of the fluorescence to phosphorescence signals and is therefore the optical gap. Importantly, it has been repeatedly found that the gap extracted from these two approaches are different, sometimes by a much as 0.35 eV. 452 Indeed, larger differences were generally found for the higher performing molecules. Adachi et al. 459 proposed that this difference is due to conformational changes occurring during the rather long transient lifetime of the triplet excited state. Here, coordinate dependent vibronic coupling, on-diagonal rather than nonadiabatic, alters the adiabatic energy gap between the T 1 and S 1 states. This corresponds to the strong coupling limit in the pioneering paper of Englman and Jortner. 162 However recently, Gibson et al. 458 proposed that the nonadiabatic coupling associated with the spin-vibronic mechanism could also play a significant role in this. Indeed, as shown in Figure 35, the nonadiabatic coupling forms an equilibrium between the two lowest triplet states, T 1 ( 3 LE) and T 2 ( 3 CT), which leads to significant population transfer, even without thermal activation. This has the effect of lowering the activation barrier for TADF because, according to second-order perturbation theory, it becomes dominated by the S 1 -T 2 energy gap, rather than the S 1 -T 1 energy gap.

The mechanism proposed above is not limited to three excited states. Indeed Marian and co-workers recently studied 3-(9,9-dimethylacridin-10(9H)-yl)-9H-xanthen-9-one (ACRXTN) 460 and 10-Phenyl-10H,10'H-spiro[acridine-9,9'-anthracen]-10'-one (ACRSA). 149 In both of these cases, the important excited states involve CT, ππ * and nπ * states. Indeed in the case of the latter, the authors reported k rISC ∼10 9 s -1 , one of the largest rates observed. This is discussed in more detail in section . The ACRXTN TADF emitter is composed of an acridine donor and xanthone acceptor.

The xanthone unit is well-known for its complicated photophysics, [461][462][463][464][465] and this is further complicated by the addition on the acridine introducing CT states. This emitter has been applied in OLEDs exhibiting low intensity roll-off with increasing currently denstiy 466 and as an assistant dopant in purely organic OLEDs. 467 The former observation suggests a high rate of rISC, minimising the lifetime of the triplet states.

The important molecular orbitals (MOs) of ACRXTN are shown in Fig. 36 and can be grouped as following:

1. MOs that are localised on the xanthone moiety: n O (Figure 36 High-level quantum chemistry calculations using DFT/MRCI-R 468,469 show that consequently, the electronic transitions involving these orbitals can be characterised as local xanthone excitations (LEX) and CT transitions. Importantly, the involvement of states with different character means that TDDFT can struggle to correctly predict the relative ordering of these states, and therefore care should always be taken when calculating the energetics of states with these methods. At the excited state geometry of the CT states, the geometry and wave function characteristics are almost identical at the singlet and triplet CT minima, corresponding to nested potentials separated by 0.06 eV and therefore the weak coupling proposed by Jortner and Englman 162 applies. However, as expected the electronic SOCME between these two states is effectively zero, and therefore it seems unlikely this can explain the fast equilibration of the singlet and triplet populations. To overcome this, vibronic coupling to close-by LE states is required. Figure 37 shows the energetic proximity of the five lowest electronically excited states ( 1 CT, 3 CT, T ππ * ,T nπ * , S nπ * ) along a linearly interpolated path connecting the 1 CT minimum (R = 0.0) and the minimum of the S nπ * state. This, as sketched in the inset in Figure 37 and the structural changes are dominated by the elongation of the C=O bond.

This clearly shows that this motion drives the potential energy surfaces of the low-lying excited states toward various intersections. Indeed, at the intersection substantial spin-orbit interactions are observed between the 1,3 CT and T ππ * . Both of these strongly promote a spin-vibronic mechanism, which explains the rapid rISC, that cannot be explained with the Condon approximation at the CT-optimised geometry.

It is interesting to note that Gibson et al. used ρMCTDH in their studies. While this approach rigorously includes the temperature effect it is very computationally expensive because the numerical treatment of density operators is more difficult since the dimensionality of the system formally doubles. 130 In contrast in ref. 149 Lyskov et al. exploited the equilibrium which will be formed between singlet and triplet states in a closed quantum system, i.e. one with emission and non-radiative decay not incorporated. This equilibrium between the singlet and triplet manifolds represents the balance between the k ISC and k rISC pathways and it becomes possible to retrieve a rate equation:

[S all ] = k rISC 3(k rISC + k ISC ) (1 -exp -(k rISC +k ISC )t ) (43)
with an equilibrium triplet concentration of k rISC /3(k rISC + k ISC ) at t → ∞. Fitting this equation to the population kinetics can be used to determine k ISC and k rISC .

Finally, it is emphasized that, as discussed in section , the mechanisms presented above and proposed by in ref. 460 and 149,456 are formally different, the former invoking vibrational spin-orbit, while the latter using derivative coupling. In practice these terms can hardly be told apart, and these studies highlight the importance of not only considering the energetic alignment of the states but also the vibrational motion responsible for coupling and mixing these excited states.

Environment Effect on Spin-Vibronic Dynamics

The involvement of multiple electronic states of different electronic character means that the spin-vibronic mechanism is likely to be strongly affected by the influence of the local embedding environment of the photoexcited molecule. Indeed, excited states exhibiting a larger excited-state dipole moment will be more stabilised by the electrostatic field generated by the solvent. These solvent-induced shifts will mean that the intramolecular potential surfaces are strongly displaced and their crossing points modified and therefore can suppress, induce and/or accelerate the spin-vibronic mechanism. This effect, in the context of the spin-vibronic mechanism, was first reported by Lim et al. [START_REF] Lim | Vibronic Spin-Orbit Interactions in Heteroaromatic Molecules. II. Phosphorescence of Quinoxaline and Other Diazanaphthalenes[END_REF]470 for the solvent dependent phosphorescence lifetime of several polycyclic monoazines.

Heteroaromatic Compounds

Using the perturbation theory approaches described above, Marian and co-workers calculated the rates of ISC in a series of heteroaromatic compounds. [START_REF] Rai-Constapel | Isolated and Solvated Thioxanthone: A Photophysical Study[END_REF][START_REF] Rai-Constapel | Photophysics of Xanthone: A Quantum Chemical Perusal[END_REF][START_REF] Rai-Constapel | Chimeric Behavior of Excited Thioxanthone in Protic Solvents: II. Theory[END_REF][START_REF] Rai-Constapel | Solvent Tunable Photophysics of Acridone: A Quantum Chemical Perspective[END_REF][START_REF] Mundt | Thioxanthone in Apolar Solvents: Ultrafast Internal Conversion Precedes Fast Intersystem Crossing[END_REF]150,151,174,175,185,431 In all these systems, ISC is strongly influenced by the environment due to the simultaneous importance of both nπ * and ππ * states which are differently shifted by solvents of higher polarity.

Particularly interesting in this respect is the photophysics of flavins and aromatic carbonyl compounds. The authors used two approaches to account for solvation, firstly one based upon purely a continuum solvation model to mimic solute-solvent interactions in non-protic media, and a second for protic solvents which also incorporated explicit solvent molecules forming hydrogen bonds. One class of molecules, for which ISC can occur on the subpicosecond time scale and is thus competitive with IC, comprises the aromatic carbonyl compounds of the xanthone family (Figure 39). What makes the mechanistic picture so involved is the (at least) biphasic decay and the strong solvent dependence of the kinetic processes, the observation of recurrences and inverse energy gap law behaviour in some cases and the general lack of experimental knowledge on the location and characteristics of optically dark states that could mediate El-Sayed forbidden transitions. [START_REF] Mundt | Thioxanthone in Apolar Solvents: Ultrafast Internal Conversion Precedes Fast Intersystem Crossing[END_REF]432,[473][474][475][476][477][478][479][480][481] A particularly remarkable feature in the photopysics of the aromatic carbonyl compounds, besides the strong solvent dependence of their radiative and nonradiative relaxation processes, is the T 2 ;S 1 rISC that outcompetes the T 2 ;T 1 IC in certain solute-solvent combinations. 432,477,482 Quantum chemical studies can therefore help elucidating the mechanisms of ISC in these molecules and to understand their solvent dependence. and successfully predict the occurrence of DF in thioxanthone in methanol solution. [START_REF] Rai-Constapel | Isolated and Solvated Thioxanthone: A Photophysical Study[END_REF]432 The computed ISC rate constants made it possible to discriminate between alternative relaxation pathways in various solvents (methanol, tetrafluoroethanol, cyclohexane), thus aiding to unravel the complex kinetic schemes of the photorelaxation processes in this molecule. [START_REF] Rai-Constapel | Chimeric Behavior of Excited Thioxanthone in Protic Solvents: II. Theory[END_REF][START_REF] Mundt | Thioxanthone in Apolar Solvents: Ultrafast Internal Conversion Precedes Fast Intersystem Crossing[END_REF]432 The recent theoretical prediction 54 that the photoexcited acridone may also show DF in acetonitrile solution (Fig. 41) still awaits confirmation.

Transition Metal Systems

The effect of the solvent also plays a significant role for transition metal systems, which often possess states of various different characters, e.g. metal centred (MC) or charge transfer (LMCT, MLCT, etc.). Indeed, for the diplatinum complexes discussed in section 4. The absorption spectra of linear two-coordinate copper complexes of cyclic (alkyl)(amino)carbenes (CAAC)CuX (X = halide) show negative solvatochromism, characterised by a blue shift in more polar solvents, and their solid-state emission is significantly blue shifted with respect to the solution spectra at room temperature. 355,357,358,484 Similar time-and environment-dependent shifts of the luminescence have been observed for (CAAC)-Au-Cz and (CAAC)-Cu-Cz complexes (Cz=carbazolate).

Environment Effects on Metal-Free TADF Emitters

A number of studies both in solution 488 and the solid state 453,[489][490][491][492][493][494][495] have demonstrated the effect of the local embedding environment on the photophysics of TADF molecules. This can be easily understood from the theory work 149,456,460 discussed in section . Indeed, because TADF is not simply a cyclic process between the singlet and triplet CT manifolds, but involves coupling to other states of different character and therefore different dipole moments this means that the relative shift of the states as a function of polarity will be different and crucially this will alter the gap between these states and is likely to significantly change the efficiency of TADF. Recently, the effect of the polarity has been simulated for ACRSA. 149 The authors used a spin-vibronic Hamiltonian and quantum nuclear dynamics to study the rate of ISC using a

Hamiltonian which depended on solvation. In this case, the effect of polarity was described in an implicit manner, with a polarisable continuum model. 496 i.e. the Condon approximation. Indeed, the often strongly nonadiabatic nature of electronic excited states means that the spin, electronic and vibrational dynamics of the system cannot be described independently. Molecular excited states therefore cannot be well represented by a simple ladder of states, as depicted in a usual Jablonski diagram. Instead, the interplay amongst spin, electronic and nuclear dynamics leads to mixing and consequently to rather complicated spin-vibronic levels. This applies to molecules both with and without the presence of heavy elements.

Probing and understanding these effects in increasingly complicated systems calls for the use of new experimental techniques and more importantly for an increasing synergy between both complementary experimental techniques, and high-level theory that can underpin the experimental interpretation. New experimental procedures, such as multi-dimensional elec-mentary to more established techniques such as transient optical absorption, time-resolved fluorescence spectroscopy and time-resolved vibrational spectroscopy. Indeed these are in principle capable of not only probing the correlated spin-vibronic dynamics, but also of directly probing the spin-vibronic couplings.

In the area of theory, there has been much progress including the ability to calculate the important coupling elements across a broad spectrum of quantum chemistry techniques incorporating an almost full selection of trade-off between accuracy and computational expense. It is possible to calculate the rate of spin-vibronic processes using either perturbation theory approaches or by explicit propagation of the nuclear equations of motion. For the former, spin-vibronic effects up to second order can be included alongside temperature and the effects of an environment. Compared to approaches based upon dynamics, the computational efficiency of these approaches means that it can be routinely applied to larger chemical systems.

In terms of approaches based upon nuclear dynamics, the power of grid-based quantum dynamics, which exploit model spin-vibronic Hamiltonians, has been repeatedly demonstrated. At the same time there has also been significant development in on-the-fly approaches, which now offer the ability to perform simulations on full dimensional potential energy surfaces and to include atomistic descriptions of the solvent environment. When comparing the two methods it is important to keep in mind that the calculation of the potentials for quantum dynamics are usually based upon interpolated models along a reduced number of collective vibrational modes. In contrast, the on-the-fly simulations require computations of the energies, forces, and nonadiabatic couplings at each time step of the dynamics. This implies a very different computational load for the two approaches. In this case, one has to be careful about the level of quantum chemistry used for the on-the-fly simulations, as generally many more quantum chemistry calculations are required, which can make the simulations prohibitive. While this is also true for grid based quantum dynamics on models, for the latter one can also manipulate the parameters of the model to understand the physics behind the spin-vibronic mechanism. This is much harder for on-the-fly approaches.

In addition in terms of mechanistic understanding of spin-vibronic dynamics, although these on-the-fly approaches offer the ability to perform dynamics within full nuclear configuration space, for complex systems, direct analysis of the results can become quite cumbersome due to high dimensionality, especially in terms of determining the important nuclear motions and their effect. This places a greater emphasis on the quality of the post-processing of the simulations. Most importantly, it is emphasised that in many cases grid based and on-the-fly approaches offer completely contrasting approximations and therefore there is a huge merit in increasing the synergy of these approaches and use the on-the-fly approaches to build high-level first principles models. 206,498 The criticality of spin-vibronic coupling within functional materials, such as TADF, demonstrated herein represents an important subset of a much broader research question, namely how do we efficiently exploit the excited-state properties of molecules? This is the core focus of a significant research effort aimed at a wide range of applications, such as organic and dye-sensitised solar cells, photocatalysis, OLEDs and fluorescent imaging. At present, the design of new molecules that act as the key components of these devices has largely focused upon large-scale synthetic programs aimed at developing and then exploiting structure-property relationships. However, while this can lead to incremental progress and simple design rules that can be used to manipulate a specific property, such as emission or absorption wavelength, it is usually unsuccessful in simultaneously controlling all of the required photophysical properties. This is, at least in part, due to the close interplay amongst spin, electronic and nuclear dynamics leading to mixing and rather complicated spin-vibronic levels. Consequently, to ensure continued progress in applied research fields, design procedures that go beyond the Born-Oppenheimer approximation and which explicitly take spin-vibronic couplings into account are required.

The design of new molecules can be achieved computationally, using for example virtual high-throughput screening and has the potential to significantly speed-up material discovery.

Recently, Gomez et al. 499 developed a virtual screening method for TADF emitters (although this can be more generally applied) with the capacity to explore the chemical space of 1.6 million molecules. However, at the heart of these approaches are a number of criteria for specific molecular properties and therefore as for experimental design, a detailed understanding of the key components of spin-vibronic must be achieved to optimise the design of these so called descriptors. Calculating couplings, both spin-orbit and nonadiabatic is obviously significantly much more computationally expensive and therefore high-throughput design adopting this approach is unlikely. We therefore recommend that such approaches could be included as an additional medium throughput refining step in such procedure.

Figure 1 :

 1 Figure1: Schematic representation of a potential energy curve and relevant couplings as a function of an arbitrary nuclear coordinate {Q}. Tn denotes nonadiabatic coupling between states of the same multiplicity and ĤSO denotes spin-orbit coupling. In the case that ĤSO between the S 1 and T 1 states was zero, decay could occur via a more complicated, secondorder channel, S 1 →T 2 →T 1 , involving both spin-orbit and nonadiabatic coupling. This is an example of the spin-vibronic mechanism as described in this review.

Figure 2 :

 2 Figure 2: Schematic representation of the origin of SOCME between a singlet and triplet MLCT state, of two different molecular symmetries. Based on the SOCME of the Cu(I)-Phenanthroline complexes. 156,157 The arrows are indicative of possible one-electron transitions. At D 2d symmetry the singlet and triplet states are composed of four different transitions and therefore the SOCME are formed from mixing of the configurations with different d contributions, but same π * character.In contrast, at the lower D 2 symmetry, this is no longer possible and therefore within the one-electron one-centre approximation SOC is forbidden. For the prototypical Bis-diimine Cu(I) complex, [Cu(dmp) 2 ] + (dmp = 2,9-dimethyl-1,10phenanthroline), S 1 -T 1 SOC at D 2d symmetry is ∼300 cm -1 , this falls to 30 cm -1 at D 2 symmetry.

Figure 3 :

 3 Figure 3: Schematic representation of the (a) weak and (b) strong coupling limits.

Figure 4 :

 4 Figure 4: Example for the enhancement of ISC by vibrational spin-orbit coupling: 164 (a) Out-of-plane vibrational modes of free-base porphyrin promoting SOC between 3 (ππ * ) and 1 (ππ * ) states. (b) Circles: Variation of T 1 | ĤSO |S 1 along Q 15 ; Squares: Variation of T 2 | ĤSO |S 1 along Q 19 ; Triangles: Variation of T 2 | ĤSO |S 0 along Q 30 .

Figure 6 :

 6 Figure 6: Thermal activation of the T 1 state vibrational levels (schematic) facilitating reverse intersystem crossing

  the nuclear coordinates, A j 1 ...j f are the MCTDH expansion coefficients and ϕ (k) j k are the n k expansion functions for each degree of freedom k known as single particle functions (SPFs).

Figure 7 :

 7 Figure 7: (a) Time-resolved normalized singlet (bluish) and triplet (brownish) populations over 30 fs. For the triplet states, the different ms states are summed as one. Highlighted states are the S1 in blue, the S8 in green, the S9 in light blue, and the T1 in red. (b) Normalised total population of all of the singlets (pale blue) and all of the triplets (red) summed over all of the trajectories. (c) Normalized total population of the singlets and triplets summed over all trajectories when the initial geometries are frozen (case I). (d) Normalized total population of the singlets and triplets summed over all trajectories run from the FC geometry allowing geometrical changes (case II). (e) Normalized total population of the singlets and triplets summed over all trajectories run from the frozen FC geometry (case III).Figure reproduced from ref.219

Figure 7 .

 7 Figure 7. This equilibrium will reflect the mixing, driven by the magnitude of the coupling and size of the energy gap at that geometry. While this does not exclude a spin-vibronic mechanism in [Ru(bpy) 3 ] 2+ , it does show that identifying refined spin-vibronic effects in transition metal complexes by means of trajectories-based methods remains a challenge because of the high density of excited states. This emphasises the importance of post-processing of on-the-fly simulations. Indeed, for complex systems, direct analysis of the results can become quite cumbersome due to high dimensionality, especially in terms of determining the important nuclear motions. Consequently, there is great merit in adopting excited state dynamics based upon model Hamiltonians, which only include the most important nuclear degrees of freedom. Indeed, excited state dynamics, especially at early times, are usually dominated by a subset of full nuclear configuration space.206

Figure 8 :

 8 Figure 8: Dynamics of thioformaldehyde after photoexcitation to S 1 . Upper panel: C=S bond length for all basis functions. Basis functions are associated with the S 1 (light gray), T 1 (red), or T 2 (blue) electronic state. Lower panel: Population of the two triplet states during the dynamics (light area indicates the standard error). The total number of basis functions is given in orange. Figure reproduced from ref.155

  may contribute to the ultrafast dynamics (< 1 ps) and to the lowest part of the absorption spectrum characterised by three bands in the UV domain, corresponding to one forbidden and two allowed transitions. The vibrational structure of the band observed between 3.8 -5.1 eV exhibits a complicated vibrational structure hardly deciphered experimentally. Moreover only the 3 B 1 state has been put in evidence by means of phosphorescence experiments.233 A pioneering study based on full-dimensional quantum dynamics and MRCI electronic structure data by Xie et al.[START_REF] Xie | Ab initio Determination of Potential Energy Surfaces for the First Two UV Absorption Bands of SO2[END_REF] including the two lowest singlet ( 1 B 1 and 1 A 2 ) and the lowest triplet ( 3 B 1 ) states has been revisited by Leveque et al.[START_REF] Lévêque | Communication: Theoretical Prediction of the Importance of the 3B2 state in the Dynamics of Sulfur Dioxide[END_REF] pointing to the importance of the 3 B 2 state in a complicated mechanism combining conical intersections and SOC (Figure9). The overall ISC process has been carefully analysed on the basis of the time evolution (0-200 fs) of the populations of the excited states including the individual spin-orbit components of the triplet

Figure 9 :

 9 Figure 9: One-D cuts of the A" PES associated to the low-lying singlet and triplet excited states of SO 2 as function of the bending angle φ. Figure reproduced with permission from ref. 80

Figure 10 :

 10 Figure 10: Short time-scale mechanism of singlet to triplet states ISC processes in SO 2 adapted from ref.80

  Time-resolved Fluorescence Spectroscopy Two of the most commonly used techniques to resolve excited state dynamics are transient optical absorption spectroscopy and time-resolved emission (fluorescence or phosphorescence)spectroscopy. For the former, the pump pulse excites the sample into an excited state, while the probe pulse measures the change in absorption as a function of time. Transient absorption spectra track the excited-state dynamics through a number of dominant spectral features including i) the ground-state bleach, i.e. a loss of signal at the ground-state absorption due to a fraction of the sample becoming promoted into the excited state, ii) stimulated emission originating from the probe pulse driving the excited-state population back to the ground state and iii) excited-state absorption (ESA), arising from the absorption of excited-state species promoting them to higher-lying excited states. It is important to stress that for ESA the final states are not often known and even for relatively simple molecules can result in broad and unresolved bands. An alternative approach, which avoids these overlapping bands, is time-resolved emission spectroscopy, which detects the light emitted as a function of time after the pump pulse. This has the advantage of not involving higher lying unknown excited states. It is therefore in principle simpler than transient absorption but is limited to bright states, i.e. those that give out light.

Figure 11 :

 11 Figure 11: Energy level diagram and relaxation mechanisms of the lowest states of Pt(POP-BF 2 ). Figure reproduced from ref. 234

  Figure 12: (a) Representative transient absorption spectra of PtPOP at 11 (open circles) and 100 ps (closed circles) after excitation at 370 nm into the 1 A 2u absorption band (dashed vertical arrow), together with a static absorption spectrum (black curve). (b) Time traces at fixed wavelengths. The upper-right inset zooms into the initial 3.5 ps time window, showing the wavepacket dynamics of the Pt-Pt stretch. (c) Simplified potential energy scheme of the ground state and 1,3 A 2u lowest-excited states of PtPOP. Figure reproduced from ref. 238

FSRS

  has been used to study the ultrafast ISC dynamics of an Fe(II) spin crossover system 255 and the nature of ISC in [Ru(bpy) 3 ] 2+ . 256 Indeed, before the development of FSRS, it was not possible to probe ISC processes on this time-scale with vibrational spectroscopy because of its ultrafast nature and the problems associated with Heisenberg's uncertainty principle. From ref., 256 the authors concluded, consistent with other experiments, 257 that ISC occurs within ∼ 100 fs and were able to identify key modes during the dynamics, such as C=C stretching modes, and assigned solvent-mediated ISC process consistent with TSH simulations.[START_REF] Tavernelli | Nonadiabatic Molecular Dynamics with Solvent Effects: A LR-TDDFT QM/MM Study of Ruthenium (II) tris(bipyridine) in Water[END_REF] However it is not clear whether ISC is dominated by the rate of solvation, spin conversion or vibrational relaxation in the triplet metal-to-ligand-charge-transfer ( 3 MLCT) state or a convolution of all three since they are expected to occur on similar time scales.

Figure 13 :

 13 Figure 13: The magnitude of the 2D spectrum 6 ps after excitation. The off-diagonal peaks among all six LMCT transitions are induced by SOC. Figure reproduced from ref. 267

  [START_REF] Zhang | Tracking Excited State Charge and Spin Dynamics in Iron Coordination Complexes[END_REF] and Lemke et al.[START_REF] Lemke | Coherent Structural Trapping Through Wavepacket Dispersion During Photoinduced Spin State Switching[END_REF] who used X-ray emission and absorption, respectively, to probe the evolution of the spin-state of [Fe(bpy) 3 ] 2+ during light-induced spin crossover dynamics, identifying an intermediate triplet state, discussed in more detail below.

Figure 14 :

 14 Figure 14: Scheme with optical-laser pump and soft X-ray probe after the pump-probe time delay ∆t. The intensity of RIXS is measured at the Fe L 3 -absorption edge with a dispersive grating spectrometer. b) Electron configuration of ground-state Fe(CO) 5 with single-electron transitions of X-ray probe and laser-pump processes (orbital assignments according to Fe 2p and 3d or ligand 2p character and according to symmetry along the Fe-CO bonds; the asterisk marks antibonding orbitals). RIXS at the Fe L 3 -absorption edge with 2p→d * σ excitation involves scattering to final d 7 π d * σ ligand-field excited states. Optical d π → 2π * excitation triggers dissociation. c) Measured Fe L 3 -RIXS intensities (encoded in colour) versus energy transfer and incident photon energy. Top: ground-state Fe(CO) 5 (negative delays, probe before pump). Middle and bottom: difference intensities for delay intervals of 0-700 fs and 0.7-3.5 ps, respectively, isolating transients by subtracting scaled intensities of unpumped Fe(CO) 5 from the measured intensities.Figure reproduced from ref. 289

  help at pointing to the combined JT/SOC effects on the adiabatic PES and consequently on the spectral characteristics. A strong JT effect as in MnF 3 or CoF 3 in their ( 5 E ) state will give rise to different characteristics depending on the SOC as illustrated in Figure 15. In the case of MnF 3 (Figure 15 left) the SO splitting is completely quenched by the strong JT effect and the weak SOC leads to an increase in density together with an enhancement of the double-hump shape of the spectrum as compared to the spectrum without SOC. Because of the stronger SOC effect in CoF 3 (Figure 15 right) a significant difference is observed in terms of shape and structure between the spectra simulated without and with SOC.

Figure 15 :

 15 Figure 15: Potential energy curves of the 5 E states of MnF 3 (top left) and CoF 3 (top right) along the bending normal mode with SOC and associated vibronic fluorescence spectra, without (a) and with (b) SOC with the permission of ref.312

Figure 16 :

 16 Figure 16: Internal infrared vibronic spectrum of NiF 3 in its 4 E excited state without (a) and with (b) SOC at T = 0K. Reproduced from Ref.85

Figure 17 :

 17 Figure 17: Schematic of the potential energy surfaces, proposed decay pathways, and time scales involved in the excited-state deactivation of [Fe(bpy) 3 ] 2+ . Figure reproduced from ref. 78

  293

Figure 18 :

 18 Figure 18: Schematic representation of the structural trapping during the light-induced spin-state conversion in [Fe(bpy) 3 ] 2+ from LS to HS state along the Fe-N distance reaction coordinate. The photoexcited MLCT (manifold) decays, through the 3 T state (t 5 2g e 1 g L 0 ), towards the HS state within (120(10) fs) and a large fraction of energy is dissipated. It expands and coherently oscillates (breathing mode, 265?fs period) around the HS equilibrium structure while losing energy. The wave packet disperses at 330?fs time constant and vibrationally cools inside the HS state potential within 1.6?ps. (b) Schematic time evolution of the wave packet in the HS potential based on the simulated distribution model. Figure reproduced from ref.29

  photoexcited decay mechanism of [Fe(bmip) 2 ] 2+ 78 and a tert-butyl substituent analogue.[START_REF] Pápai | Effect of tert-Butyl Functionalization on the Photoexcited Decay of a Fe (II)-N-Heterocyclic Carbene Complex[END_REF]

Figure 19 :

 19 Figure 19: Schematic potential curves and ultrafast relaxation processes of (a) [Cu(phen) 2 ] + , (b) [Cu(dmp) 2 ] + , and (c) [Cu(dpp) 2 ] + . phen = 1,10-phenanthroline, dmp = 2,9-dimethyl-1,10-phenanthroline and dpp = 2,9-diphenyl-1,10-phenanthroline. Figure reproduced from ref. 345

  344 that along a flattening distortion mode, identified to be important in the wavepacket dynamics of ref.348 the T 2 , T 3 and T 4 states all intersect the S 1 state.

Figure 20 : 1 .

 201 Figure 20: (a) Relative diabatic state populations of S 1 (blue), S 2 (green), S 3 (red), the triplet (T 1-4 ) states (purple), for 1 ps following photoexcitation. The black dashed line is the biexponential fit of the S 3 diabatic state populations. (b) Schematic representation of the key dynamical steps occurring within the ultrafast nonadiabatic dynamics of [Cu(dmp) 2 ] + . (i) Internal conversion from S 3 to S 2 . (ii) Vibrational relaxation from S F C 1 to S JT 1 . (iii) Ultrafast ISC from S F C 1

  .

  154

Figure 21 :

 21 Figure 21: Difference densities of the low-lying singlet and triplet excitations in the linear IPR-Cu(I)-methylpyridyl complex. Figures reproduced from ref.360

Figure 22 :

 22 Figure 22: Schematic representation of [Re(X)(CO) 3 (L)] n+ (X = Cl, Br, I; L= 2,2'-bipyridine, n=0; X= imidazole or pyridine; L = 1,10-phenanthroline, n= +1)

Figure 23 :Figure 24 :

 2324 Figure 23: Electronic densities of the five lowest singlet and triplet states of [Re(X)(CO) 3 (L)] (X = Br and L= 2,2'-bipyridine).

Figure 25 :

 25 Figure 25: Spin-vibronic mechanism of the ultra-fast luminescent decay observed in [Re(X)(CO) 3 (L)] n+ (X = Cl, Br and L = 2,2'-bipyridine, n=0) left and [Re(X)(CO) 3 (L)] n+ (X = imidazole and L = 1,10-phenanthroline, n= +1), red arrows: vibronic; black arrows: SOC.

Figure 26 :Figure 27 : 1

 26271 Figure 26: Electronic densities of the six lowest singlet and triplet states of [Re(X)(CO) 3 (L)] + (X = imidazole and L = 1,10-phenanthroline, n= +1)

Figure 28 :

 28 Figure 28: Time-evolution of the diabatic populations of the low-lying singlet and triplet excited states of [Re(X)(CO) 3 (L)] + (X = imidazole and L= 1,10-phenanthroline adapted from 82

Figure 29 :

 29 Figure 29: (a) Experimental integrated photoelectron yields as a function of delay for a probe wavelength 243 nm. Inset: Partial integrated photoelectron yields. (b) Cut through the potential energy surfaces of benzene along the prefulvene mode: The vector from the Franck-Condon point to the S 1 /S 0 conical intersection. Figure reproduced from ref.[START_REF] Parker | Ultrafast Dynamics of the S1 Excited State of Benzene[END_REF][START_REF] Minns | Competing Iltrafast Intersystem Crossing and Internal Conversion in the "Channel 3" Region of Benzene[END_REF]

Figure 30 :

 30 Figure 30: (a) Excited-state populations from 44 trajectories (thin lines) and global fits (thick lines) to the populations. (b) Net population transfer (numbers of trajectories) among the electronic states. (c) Fitted time constants (in femtoseconds) from the global fit. The T 3 population was absorbed into the T 2 population in all panels. The thickness of the lines in panels b and c relates to the amount of transferred population. Figure reproduced from ref. 72

Figure 31 :

 31 Figure 31: CC2 linearly interpolated paths between the S 2 , S 1 and T 2 minima illustrating IC (a) and ISC (b) decay pathways in 6-azauracil. Photoexcitation primarily populates the S 2 state (blue squares). Figures reproduced from ref.163

  ( 3 nπ * ) state as a transient. Due to the non-BO nature of these states, a strong variation of the 1 S 2 | ĤSO | 3 T 2 with the nuclear coordinates is observed. The second one involes a conical intersection of S 2 with S 1 ( 1 nπ * ) which acts as a doorway state from which ISC to the T 1 ( 3 ππ * ) state proceeds. Remarkably, an inverse energy gap behaviour for the S 1 ;T 1 ISC was predicted by theory: 163 A blue shift of the S 1 potential energy surface by about 0.2 eV places the intersection between the S 1 and T 1 potentials close to the S 1 minimum, thus increasing the overlap of the vibrational wavefunctions and consequently speeding up the ISC from τ ISC = 125 ps in the gas phase to τ ISC = 30 ps in acetonitrile solution. Recent femtosecond transient absorption spectroscopy measurements confirm the predominance of the second pathway: The S 2 ;S 1 IC is found to proceed at the subpicosend time scale whereas a time constant of 5.2 ± 0.1 ps is measured for the S 1 ;T 1 ISC in acetonitrile. The experiments also reveal a clear dependence of the S 1 ;T 1 rate constant on the polarity of the solvent,showing a faster formation of the T 1 state in more polar solvents. In contrast, aqueous solution markedly diminishes ISC in uracil and the other pyrimidine nucleic acid bases.429,430

Figure 32 :

 32 Figure 32: Prototypical type II photosensitisers

  150

Figure 33 :

 33 Figure 33: (Out-of-plane normal mode q 1 (a), potential energy curves (b) and SOCMEs (c) of psoralen plotted against q 1 . Figures reproduced from ref.150

Figure 34 :

 34 Figure 34: (a) Relative populations of the 3 LE state associated with ISC after excitation into the 1 CT state. (b) The relative populations of the 1 CT state associated with rISC after initially populating the 3 LE state. (c) Population kinetics of the 3 CT state during the rISC dynamics after initial population of the 3 LE state, i.e (b). Inset is a zoom into the early time kinetics. Black: Full model Hamiltonian described in TableS3, Green: No HFI, Blue: No Vibronic Coupling.

  ∆E S1-T1 = 0.10 eV

Figure 35 :

 35 Figure35: Energy barrier for TADF, determined from the temperature variation of k rISC plotted against 1000/T (K -1 ) for two models which have ∆E S 1 -T 1 = 0.06 and 0.1 eV. In the case of the latter, the magnitude of nonadiabatic coupling has been doubled to enable these simulations to exhibit sufficient population transfer to S 1 at lower temperatures (225 K). Figure reproced from ref.458

Figure 36 :

 36 Figure 36: Frontier orbitals of ACRXTN at the optimised ground-state geometry.

2 .

 2 MOs that are localised on the acridine substituent: π H (Figure36(c))

Figure 37 :

 37 Figure 37: Left:DFT/MRCI-R energy profiles along a linearly interpolated path connecting the singlet CT minimum (R = 0.0) and the minimum of the S(n O π * ) state (R = 1.0). Solid lines, singlets; dashed lines, triplets; triangles, CT states; circles, n O π * L states; squares, ππ * states with leading π H-1 π * L term; Right: Schematic of the proposed ISC and rISC mechanism in ACRXTN, reproduced from ref. 460

FlavinsFigure 38 :

 38 Figure 38: Schematic view of the potential energy surfaces of lumiflavin in (a) vacuum and (b) water solution. (c) Coordination of flavin mononucleotide in the LOV domain of the blue-light sensor YtvA of Bacillus subtilis and (d) hydrogen-bonding pattern of lumiflavin in water.

Figure 39 :

 39 Figure 39: Chemical structures of xanthone (X = O), thioxanthone (X =S), and acridone (X =NH)

3 π

 3 photophysics of these aromatic ketones. In cases where either the 1 π H π * L and 1 n O π * L states

Figure 40 :

 40 Figure 40: DFT/MRCI energy profiles of electronically excited thioxanthone in various solvents

Figure 41 :

 41 Figure 41: DFT/MRCI energy profiles and feasible ISC channels of electronically excited acridone in acetonitrile solution. Due to strong vibronic interaction, the El-Sayed forbidden transitions proceed on the subnanosecond timescale.

Figure 42 :

 42 Figure 42: Schematic of the solute-solvent interactions of molecules with negative solvatochromism. (a) The highly polar electronic ground state of the solute molecule is stabilised by the polar environment. (b) The dipolar solvent-solute interactions are unfavourable for the less polar singlet or triplet excited state at short time delays after excitation or if the solvent reorientation is hindered in the solid state. (c) Solvent reorientation stabilises the excited state and leads to a red shift of the luminescence at long time delays.

Figure 43 :

 43 Figure 43: The temperature dependence of the intensity (black line) and CT onset energy (purple circles). The change in CT onset energy plateaus below the glass temperature, representative of the PEO film becoming rigid. The black dashed line represents the energy of the 3 LE at 2.58eV, with the peak in intensity apparent as the CT energy crosses resonance. b) Relative rate of rISC as a function of temperature for the D-A complex PTZ-DPTO2. The rates were extracted from the population of the 1CT state at 0.5 ns of dynamics simulations initiated from the lowest triplet state. Reproduced from ref. 492

Figure 44 :

 44 Figure 44: One-dimensional q 138 cut through excited PESs of ACRSA in different environments. Reproduced from ref. 149

Figure 44 shows

 44 Figure 44 shows the important 1 nπ * , 3 ππ * and CT states along the normal mode responsible for the structural change between the 1 nπ * and 3 ππ * states. Importantly, while the shape of the curves remains similar, as supported by the expansion parameters of the Hamiltonian given in ref., 149 the relative separation of the potentials changes dramatically due to the different dipole moments of the three excited states.

Figure 45 :

 45 Figure 45: Time-dependent population of the singlet excited states of ACRSA in vacuum, toluene, and acetonitrile following the triplet CT exciton formation. Reproduced from ref. 149

Figure 45

 45 Figure45shows the effect of this on the time-dependent population of the singlet excited states of ACRSA. As expected the population is fastest in low polarity solvents where the states are closest together, in agreement with finding of ref.492 Interestingly, in the case of ACRSA, the authors reported k rISC ∼10 9 s -1 , one of the largest rates observed, and which

Finally

  , we want to stress that, as shown by various examples discussed in this review, spin and vibronic coupling interplay are at work in both light organic compounds and molecular systems containing heavy-element. Internal conversion and intersystem crossing may act on a similar, ultrafast, time-scale calling for a balanced and intertwined theoretical description able to decipher the intricate mechanism of ultrafast excited-state processes, in conjunction with new experimental techniques. doctoral stay in the group of Per E. M. Siegbahn at Stockholm's University, Sweden, she became Assistant Professor at the University of Bonn where she completed her Habilitation in 1991. She headed the Computational Chemistry team at the Fraunhofer Scientific Computing and Algorithms Institute (SCAI) in Sankt Augustin, Germany, before joining the Heinrich-Heine-University Düsseldorf, Germany, as a Full Professor of Theoretical and Computational Chemistry in 2001. Since 2014 she is a regular member of the North-Rhine Westphalian Academy of Sciences and Arts. Her current research interests center on the development and application of theoretical and computational excited-state electronic structure methods, with a particular emphasis on spin-forbidden transitions.

  dependent DFT UDFT: unrestricted DFT LR-TDDFT: linear-response TDDFT MSD: model space diabatization FC: Franck-Condon FCWD: FC weighted density of states FCFs: FC factors VDOS: vibrational density of states DF: delayed fluorescence TADF: thermally activated DF TDSE: time-dependent Schrdinger equation CI: single particle functions ML-MCTDH: multi-layer MCTDH SPDO's: single-particle density operators LVC: linear vibronic coupling QVC: quadratic vibronic coupling DRE: Duschinsky rotation effects TSH: trajectory surface hopping QM/MM: quantum mechanics/molecular mechanics GWP: Gaussian wavepacket CCS: coupled-coherent states MCE: multi-configurational Ehrenfest vMCG: variational multi-configurational Gaussian wavepacket GPUs: graphical processing units ESA: excited state absorption 2D-IR: 2-dimensional infrared FSRS: femtosecond stimulated Raman scattering BB-IVS: broad band impulsive vibrational spectroscopy CT: charge transfer MLCT: metal-to-ligand CT LLCT: ligand-to-ligand CT LMCT: ligand-to-metal CT XLCT: X to ligand CT MC: metal-centered IL: intra-ligand LC: ligand-centered LE: local excited LEX: local xanthone excitations 2D-ES: 2-dimensional electronic spectroscopy X-FELS: X-ray free electron lasers HHG: high harmonic generation RXES: X-ray emission spectroscopy RIXS: Resonant Inelastic X-ray scattering DSSCs: dye-sensitized solar cells OLEDs: organic light-emitted diodes JT: Jahn-Teller PJT: pseudo-JT LIESST: light-induced excited spin-state trapping LEECs: light emitting electrochemical cells RASI: rotationally assisted spin-state inversion CMA : carbine-metalamide TRPES: time-resolved photoelectron spec-: multi-state CAS second order perturbation theory CC2: coupled cluster doubles SCS-CC2: spin component scaled CC2

( 120 )

 120 Fatehi, S.; Alguire, E.; Shao, Y.; Subotnik, J. E. Analytic Derivative Couplings Between Configuration-Interaction-Singles States with Built-in Electron-Translation Factors for Translational Invariance. J. Chem. Phys. 2011, 135, 234105 . (121) Dallos, M.; Lischka, H.; Shepard, R.; Yarkony, D. R.; Szalay, P. G. Analytic Evaluation of Nonadiabatic Coupling Terms at the MR-CI Level. II. Minima on the Crossing Seam: Formaldehyde and the Photodimerization of Ethylene. J. Chem. Phys. 2004, Lischka, H.; Dallos, M.; Szalay, P. G.; Yarkony, D. R.; Shepard, R. Analytic evaluation of nonadiabatic coupling terms at the MR-CI level. I. Formalism. J. Chem. Phys. 2004, 120, 7322-7329 . (123) Kuhlman, T. S.; Glover, W. J.; Mori, T.; Møller, K. B.; Martínez, T. J. Between 122 Ethylene and Polyenes-the Non-Adiabatic Dynamics of Cis-dienes. Faraday Discuss. 2012, 157, 193-212 . (124) Mori, T.; Glover, W. J.; Schuurman, M. S.; Martinez, T. J. Role of Rydberg States in the Photochemical Dynamics of Ethylene. J. Phys. Chem. A 2012, 116, 2808-2818 . (125) Park, J. W.; Shiozaki, T. Analytical Derivative Coupling for Multistate CASPT2 Theory. J. Chem. Theory Comput. 2017, 13, 3676-3683 . (126) Chernyak, V.; Mukamel, S. Density-Matrix Representation of Nonadiabatic Couplings in Time-Dependent Density Functional (TDDFT) Theories. J. Chem. Phys. 2000, Send, R.; Furche, F. First-Order Nonadiabatic Couplings from Time-Dependent Hybrid Density Functional Response Theory: Consistent Formalism, Implementation, and Performance J. Chem. Phys. 2010, 132, 044107 . (128) Ou, Q.; Fatehi, S.; Alguire, E.; Shao, Y.; Subotnik, J. E. Derivative Couplings Between TDDFT Excited States Obtained by Direct Differentiation in the Tamm-Dancoff Approximation. J. Chem. Phys. 2014, 141, 024114 . (129) Domcke, W.; Yarkony, D.R. and Horst, K.A. Conical intersections: Electronic Structure, Dynamics and Spectroscopy; World Scientific, 2004; Vol. 15 . (130) Tannor, D. J. Introduction to quantum mechanics; University Science Books, 2007 123 . (131) Mead, C. A.; Truhlar, D. G. Conditions for the Definition of a Strictly Diabatic Electronic Basis for Molecular Systems. J. Chem. Phys. 1982, 77, 6090-6098 . (132) Subotnik, J. E.; Yeganeh, S.; Cave, R. J.; Ratner, M. A. Constructing Diabatic States from Adiabatic states: Extending Generalized Mulliken-Hush to Multiple Charge Centers with Boys Localization. J. Chem. Phys. 2008, 129, 244101 . (133) Van Voorhis, T.; Kowalczyk, T.; Kaduk, B.; Wang, L.-P.; Cheng, C.-L.; Wu, Q. The Diabatic Picture of Electron Transfer, Reaction Barriers, and Molecular Dynamics. Ann. Rev. Phys. Chem. 2010, 61, 149-170 . (134) Subotnik, J. E.; Alguire, E. C.; Ou, Q.; Landry, B. R.; Fatehi, S. The Requisite Electronic Structure Theory to Describe Photoexcited Nonadiabatic Dynamics: Nonadiabatic Derivative Couplings and Diabatic Electronic Couplings. Acc. of Chem. Res. 2015, 48, 1340-1350 . (135) Werner, H.-J.; Meyer, W. MCSCF study of the Avoided Curve Crossing of the Two Lowest 1Σ+ states of LiF. J. Chem. Phys. 1981, 74, 5802-5807 . (136) Cave, R. J.; Newton, M. D. Generalization of the Mulliken-Hush Treatment for the Calculation of Electron Transfer Matrix Elements. Chem. Phys. Lett. 1996, 249, 15-19 . (137) Cave, R. J.; Newton, M. D. Calculation of Electronic Coupling Matrix Elements for Ground and Excited State Electron Transfer Reactions: Comparison of the General-124 ized Mulliken-Hush and Block Diagonalization Methods. J. Chem. Phys. 1997, 106, Subotnik, J. E.; Vura-Weis, J.; Sodt, A. J.; Ratner, M. A. Predicting Accurate Electronic Excitation Transfer Rates via Marcus Theory with Boys or Edmiston-Ruedenberg Localized Diabatization J. Phys. Chem. A 2010, 114, 8665-8675 . (139) Fatehi, S.; Alguire, E.; Subotnik, J. E. Derivative Couplings and Analytic Gradients for Diabatic States, with an Implementation for Boys-Localized Configuration-Interaction Singles. J. Chem. Phys. 2013, 139, 124112 . (140) Zhu, X.; Yarkony, D. R. On the Construction of Property Based Diabatizations: Diabolical Singular Points. J. Phys. Chem. A 2015, 119, 12383-12391 . (141) Hoyer, C. E.; Xu, X.; Ma, D.; Gagliardi, L.; Truhlar, D. G. Diabatization Based on the Dipole and Quadrupole: The DQ method. J. Chem. Phys. 2014, 141, 114104 . (142) Hoyer, C. E.; Parker, K.; Gagliardi, L.; Truhlar, D. G. The DQ and DQΦ Electronic Structure Diabatization Methods: Validation for General Applications. J. Chem. Phys. 2016, 144, 194101 . (143) Köppel, H.; Domcke, W.; Cederbaum, L. S. Multimode Molecular Dynamics Beyond the Born-Oppenheimer Approximation. Adv. in Chem. Phys. 1984, 57, 59-246 . 125 (144) Pacher, T.; Cederbaum, L.; Köppel, H. Approximately Diabatic States from Block Diagonalization of the Electronic Hamiltonian. J. Chem. Phys. 1988, 89, 7367-7381 . (145) Domcke, W.; Woywod, C. Direct Construction of Diabatic States in the CASSCF Approach. Application to the Conical Intersection of the 1A2 and 1B1 Excited States of Ozone. Chem. Phys. Lett. 1993, 216, 362-368 . (146) Domcke, W.; Woywod, C.; Stengle, M. Diabatic CASSCF Orbitals and Wavefunctions. Chem. Phys. Lett. 1994, 226, 257-262 . (147) Wittenbrink, N.; Venghaus, F.; Williams, D.; Eisfeld, W. A New Approach for the Development of Diabatic Potential Energy Surfaces: Hybrid Block-Diagonalization and Diabatization by Ansatz. J. Chem. Phys. 2016, 145, 184108 . (148) Zeng, T. A Diabatization Protocol that Includes Spin-Orbit Coupling J. Chem. Phys. 2017, 146, 144103 . (149) Lyskov, I.; Marian, C. M. Climbing up the Ladder: Intermediate Triplet States Promote the Reverse Intersystem Crossing in the Efficient TADF Emitter ACRSA. J. Phys. Chem. C 2017, 121, 21145-21153 . (150) Tatchen, J.; Gilka, N.; Marian, C. M. Intersystem Crossing Driven by Vibronic Spin-Orbit Coupling: A Case Study on Psoralen. Phys. Chem. Chem. Phys. 2007, 9, Perun, S.; Tatchen, J.; Marian, C. M. Singlet and Triplet Excited States and Intersystem Crossing in Free-Base Porphyrin: TDDFT and DFT/MRCI Study. ChemPhysChem 2008, 9, 282-292 . (165) Berkelbach, T. C.; Hybertsen, M. S.; Reichman, D. R. Microscopic Theory of Singlet Exciton Fission. II. Application to Pentacene Dimers and the Role of Superexchange. J. Chem. Phys. 2013, 138, 114103 . (166) Symalla, F.; Friederich, P.; Massé, A.; Meded, V.; Coehoorn, R.; Bobbert, P.; Wenzel, W. Charge Transport by Superexchange in Molecular Host-Guest Systems. Phys. Rev. Lett. 2016, 117, 276803 . (167) Duschinsky, F. The Importance of the Electron Spectrum in Multi Atomic Molecules Concerning the Frank-Condon Principle. Acta Physicochim. 1937, 7, 551-566 . (168) Hougen, J. T.; Watson, J. K. G. Anomalous Rotational Line Intensities in Electronic Transitions of Polyatomic Molecules: Axis-Switching Canad. J. Phys. 1965, 43, Özkan, I. Franck-Condon Principle for Polyatomic Molecules: Axis-Switching Effects and Transformation of Normal Coordinates J. Mol. Spectrosc. 1990, 139, 147-162 . (170) Tatchen, J. E. Spin-verbotene photophysikalische Prozesse in organischen Molekülen: Entwicklung quantenchemischer Methoden und Anwendung auf Psoralene. Ph.D. thesis, Heinrich Heine University Düsseldorf, Germany, 2006 . (171) Reimers, J. R. A Practical Method For the Use of Curvilinear Coordinates in Calculations of Normal-Mode-Projected Displacements and Duschinsky Rotation Matrices for Large Molecules. J. Chem. Phys. 2001, 115, 9103-9109 . (172) Toniolo, A.; Persico, M. Efficient Calculation of Franck-Condon Factors and Vibronic Couplings in Polyatomics. J. Comput. Chem. 2001, 22, 968-975 . (173) Toniolo, A.; Persico, M. A Theoretical Study of Spectroscopy and Predissociation Dynamics in Nitrosoalkanes. J. Chem. Phys. 2001, 115, 1817-1827 . (174) Salzmann, S.; Tatchen, J.; Marian, C. M. The Photophysics of Flavins: What Makes the Difference Between Gas Phase and Aqueous Solution? J. Photochem. Photobiol. A 2008, 198, 221-231 . (175) Salzmann, S.; Silva-Junior, M. R.; Thiel, W.; Marian, C. M. Influence of the LOV Domain on Low-Lying Excited States of Flavin: A Combined Quantum-Mechanics / Nolecular-Mechanics Investigation. J. Phys. Chem. B 2009, 113, 15610-15618 . (176) Rodriguez-Serrano, A.; Rai-Constapel, V.; Daza, M. C.; Doerr, M.; Marian, C. M. A Theoretical Study of Thionine: Spin-Orbit Coupling and Intersystem Crossing.

Table 1 :

 1 JT and Spin-Orbit coupling terms of MnF 3 , CoF 3 and NiF 3 in low-lying high-spin states together with the normal mode frequencies (in cm -1 ) from Ref.[START_REF] Mondal | Infrared Absorption Spectra of Jahn-Teller Systems: Application to the Transition-Metal Trifluorides MnF3 and NiF3[END_REF]312 MnF 3 MnF 3 NiF 3 CoF 3 CoF 3

		5 E	5 E	4 E	5 E	5 E
	Linear JT	764	311	548	548	264
	Quadratic JT	40	-17	26	28	-12
	Spin-Orbit	145	82	234	293	155
	Bending	176	176	198	184	193
	Stretching	769	824	-	780	799

  ). Solid lines, singlets; dashed lines, triplets; triangles, CT states; circles, n O π * L states; squares, ππ * states with leading π H-1 π * L term; Right: Schematic of the proposed ISC and rISC mechanism in ACRXTN, reproduced from ref. 460

  1, refs. 238,242 have both shown a strong solvent effect on the rate of ISC, for which the authors suggested that different solvents, most probably with higher polarity, stabilise an intermediate state providing a spin-vibronic channel from the lowest singlet into the lowest triplet state. Whereas the ultrafast fluorescence decay τ 1 of the Re(I) complexes discussed in section 5.4 is weakly affected by solvent effects, the intermediate time-scales τ 2 may vary from 340-470 fs in acetonitrile to 1.09-1.4 ps in DMF for the Cl and Br substituted complexes, respectively.483 This effect is hardly explained on the basis of the character and energetics of the involved excited states not really affected by the solvent effects. In contrast solvation dynamics, not included in the simulations discussed in the present review, should play a central role in this case. Some linear two-coordinate coinage metal complexes show untypical environment effects.

  In the solid state (frozen solutions, films or crystalline environment), solvent reorientation is sterically hindered. For this reason, delayed luminescence in the solid state is expected to be blue shifted with respect to luminescence in liquid solution, in agreement with the experimental findings. To model the emission at longer time delays (nanosecond to microsecond time scale) in liquid solution, adaptation of the solvent environment to the charge distribution of the solute's excited state has to be taken into account. In principle, this could be achieved by QM/MM simulations, but excitedstate QM/MM molecular dynamics for such long time intervals are computationally very demanding. Alternatively, one can make use of the corrected linear response polarizable continuum model (cLR-PCM) 486 or other state-specific solvent approaches 487 in quantum chemical calculations.

  [START_REF] Henry | Spin-Orbit Coupling in Aromatic Hydrocarbons. Analysis of Nonradiative Transitions between Singlet and Triplet States in Benzene and Naphthalene[END_REF] s -1 , one of the largest rates observed, and which is achieved because of the interaction between the low lying CT, ππ * and nπ * states. A polar environment such as acetonitrile destabilises nπ * energies and stabilises CT exciton energies, by that deteriorating the efficient rISC pathway. The rate constants k rISC ∼ 2 × 10 7 s -1 and k ISC ∼ 5 × 10 8 s -1 for ACRSA in that solvent 149 are not much different from corresponding values in the related spiro-acridine compound 10-phenyl-10H-spiro[acridine-9,9'-fluorene]-2'-7'-dicarbonitrile (ACRFLCN) which features a low-lying LE 3 (ππ * ) state but lacks low-lying nπ * states and for which a value of k ISC ∼ 7 × 10 7 s -1 was reported. 497Summary and OutlookIn this review, we have summarised the recent work shedding new insight onto the spinvibronic mechanism of intersystem crossing. Although the underpinning theory of this mechanism was established in the 1960's, it is only now, with the development of ultrafast spectroscopies and high-level theoretical tools, that the importance of this phenomenon in a wide variety of molecular systems is being established.

These experimental and theoretical studies have unequivocally demonstrated that the interpretation of ISC dynamics is often not possible using simple qualitative analysis based upon empirical rules, such as the energy gap law and El-Sayed's rule. In such situations one must go beyond the assumptions based upon ISC driven only by direct SOC between states of different multiplicity, in which the coupling remains unchanged by vibrational motion,

  . (14) Bertoni, R.; Cammarata, M.; Lorenc, M.; Matar, S. F.; Létard, J.-F.; Lemke, H. T.; Collet, E. Ultrafast Light-induced Spin-State Trapping Photophysics Investigated in Fe (phen) 2 (NCS) 2 Spin-crossover Crystal. Acc. Chem. Res. 2015, 48, 774-781 . (15) Ohkoshi, S.-i.; Tokoro, H. Photomagnetism in cyano-bridged bimetal assemblies. Acc. Chem. Res. 2012, 45, 1749-1758 . (16) Knoll, J. D.; Turro, C. Control and Utilization of Ruthenium and Rhodium Metal Complex Excited States for Photoactivated Cancer Therapy. Coord. Chem. Rev. 2015, Aliprandi, A.; Mauro, M.; De Cola, L. Controlling and imaging biomimetic selfassembly. Nat. Chem. 2016, 8, 10 . (19) Wang, C.; Qin, J.; Shen, X.; Riedel, R.; Harms, K.; Meggers, E. Asymmetric Radical-Radical Cross-Coupling through Visible-Light-Activated Iridium Catalysis. Angew. Born, M.; Huang, K. Dynamical theory of crystal lattices; Oxford university press, Worth, G.; Cederbaum, L. Beyond Born-Oppenheimer: Molecular Dynamics Through a Conical Intersection Ann. Rev. Phys. Chem. 2004, 55, 127-158 .

	.
	(117) 1998
	.
	(118) Born, M.; Oppenheimer, R. Zur Quantentheorie der Molekeln Annalen der Physik
	1927, 389, 457-484
	.
	(119)
	282, 110-126
	.
	128, 4365-4370
	.
	(18) Chem. Int. Ed. 2016, 55, 685-688
	.

(17) Blanco-Rodríguez, A.M.; Busby, M. ; Gradinaru, C. ; Crane, B.R. ; Di Bilio, A.J. ; Matousek, P. ; Towrie, M. ; Leigh, B.S. ; Richards, J.H.; Vlcek, A.; et al. Excited-State Dynamics of Structurally Characterized [ReI (CO) 3 (phen)(HisX)]+(X= 83, 109) Pseudomonas a eruginosa Azurins in Aqueous Solution. J. Am. Chem. Soc. 2006,

  Photochem. Photobiol. Sci. 2012, 11, 1860-1867 . (177) Rodriguez-Serrano, A.; Rai-Constapel, V.; Daza, M. C.; Doerr, M.; Marian, C. M. Internal Heavy Atom Effects in Phenothiazinium Dyes: Enhancement of Intersystem Crossing via Vibronic Spin-Orbit Coupling. Phys. Chem. Chem. Phys. 2015, 17, Haarhoff, P. The Density of Vibrational Energy Levels of Polyatomic Molecules. Mol. Dierksen, M.; Grimme, S. An Efficient Approach for the Calculation of Franck-Condon Integrals of Large Molecules J. Chem. Phys. 2005, 122, 244101 . (180) Jankowiak, H. C.; Stuber, J. L.; Berger, R. Vibronic Transitions in Large Molecular Systems: Rigorous Prescreening Conditions for Franck-Condon Factors. J. Chem. Santoro, F.; Lami, A.; Improta, R.; Bloino, J.; Barone, V. Effective Method for the Computation of Optical Spectra of Large Molecules at Finite Temperature Including the Duschinsky and Herzberg-Teller Effect: The Q x Band of Porphyrin as a Case Study. J. Chem. Phys. 2008, 128, 224311 . (182) von Mehler, F. G. Ueber die Entwicklung einer Function von beliebig vielen Variablen nach Laplaceschen Functionen höherer Ordnung. J. Reine Angew. Math. 1866, 66, Markham, J. J. Interaction of Normal modes with Electron Traps. Rev. Mod. Phys. Etinski, M.; Tatchen, J.; Marian, C. M. Time-Dependent Approaches for the Calculation of Intersystem Crossing Rates. J. Chem. Phys. 2011, 134, 154105-154114 . (185) Etinski, M.; Tatchen, J.; Marian, C. M. Thermal and Solvent Effects on the Triplet Formation in Cinnoline. Phys. Chem. Chem. Phys. 2014, 16, 4740-4751 . (186) Peng, Q.; Niu, Y.; Deng, C.; Shuai, Z. Vibration Correlation Function Formalism of Radiative and Non-Radiative Rates for Complex Molecules. Chem. Phys. 2010, 370, Peng, Q.; Niu, Y.; Shi, Q.; Gao, X.; Shuai, Z. Correlation Function Formalism for Triplet Excited State Decay: Combined Spin-orbit and Nonadiabatic Couplings.J. Banerjee, S.; Baiardi, A.; Bloino, J.; Barone, V. Temperature Dependence of Radiative and Nonradiative Rates from Time-Dependent Correlation Function Methods. J.

	(183) 1959, 31, 956-989
	.
	11350-11358 . (178) Phys. 1964, 7, 101-117 . (184) 215 -222 . (179) Phys. 2007, 127, 234101 . (187) Chem. Theory Comput. 2013, 9, 1132-1143 . (188) Chem. Theory Comput. 2016, 12, 774-786 . (181) 161-176 (189) Beck, M. H.; Jäckle, A.; Worth, G. A.; Meyer, H.-D. The Multiconfiguration Time-
	.
	131

Dependent Hartree Method: A Highly Efficient Algorithm for Propagating

Wavepack- ets. Phys. Rep. 2000, 324, 1-105 .

B 1 state 71 when reaching 1 ps as compared to the simulation based on full-dimensional quantum dynamics, namely 25% when the three triplets are included 80 and 30% when 3 B 2 is excluded.[START_REF] Xie | Ab initio Determination of Potential Energy Surfaces for the First Two UV Absorption Bands of SO2[END_REF] Other discrepancies between time scales and excited-state populations deduced from quantum dynamics and trajectory approaches cannot be rigorously analysed, mainly because on-the-fly dynamics are interpreted in terms of adiabatic states. However, as pointed out by Köppel et al.[START_REF] Lévêque | Communication: Theoretical Prediction of the Importance of the 3B2 state in the Dynamics of Sulfur Dioxide[END_REF] and put in evidence in SO 2 , strong interference effects between different singlet to triplet coupling channels, not taken into account in surface hopping trajectories, play a central role in the population transfers.To the best of our knowledge SO 2 is the only system for which full-dimensional quantum dynamics and non-adiabatic molecular dynamics, both including SOC applied to two singlets and three triplets have been performed. Further sophisticated experiments are certainly

E excited state, for CoF 3 in its 5 E and 5 E excited states, for NiF 3 in its 4 E excited state and for CoF 3 in its 5 E and 5 E excited states. These systems are particularly instructive because they show limiting cases of strong JT/ weak SOC, weak JT/weak SOC and strong JT/ strong SOC (Table1) that affect differently either the vibronic fluorescence spectra (MnF 3 and CoF 3 ) or the internal vibronic infrared spectra (MnF 3 and NiF 3 ). These systems

Acknowledgement

TJP acknowledges the EPSRC, Projects EP/N028511/1 and EP/P012388/1 and the Leverhulme Trust, Project RPG-2016-103, for funding. CM thanks the Deutsche Forschungsgemeinschaft (MA1051/17-1 and INST 208/704-1 FUGG) and the Anton Beetz-Stiftung for financial support. CD and EG thank ANR-10-LABX-0026 Chimie des Systmes Complexes and ANR-15-CE29-0027-01 DeNeTheor as well as the Frontier Research in Chemistry Foundation, Strasbourg.

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