Highly efficient surface hopping dynamics using a linear vibronic coupling model

Effective Electron-Vibration Coupling by Ab Initio Methods

The description of electron-phonon coupling in materials is complex, with varying definitions of coupling constants in the literature and different theoretical approaches available. This article analyzes different levels of theory to introduce and compute these coupling constants. Within the quasi-particle picture, we derive an effective linear-coupling Hamiltonian, describing the interaction of electronic quasi-particles with vibrations. This description allows a comparison between coupling constants computed using density functional theory and higher-level quasi-particle approaches by identifying the Kohn-Sham potential as an approximation to the frequency-independent part of the self-energy. We also investigate their dependence on the exchange-correlation (XC) functional. Despite significant deviations of the Kohn-Sham eigenvalues, which arise from different XC functionals, the resulting coupling constants are remarkably similar. A comparison to quasi-particle methods, such as the well-established G0W0 approach, reveals significant quasi-particle weight renormalization. Surprisingly, however, in nearly all the considered cases, the coupling constants computed in the DFT framework are excellent approximates of the ones in the quasi-particle framework, which is traced back to a significant cancellation of competing terms. Other quasi-particle methods, such as the Outer Valence Green's Function approach and the ΔSCF method, are also included in the comparison. Moreover, we investigate the coupling of vibrations to excitonic excitations and find, by comparison to time-dependent density functional theory and extended multiconfiguration quasi-degenerate second-order perturbation theory, that knowing the underlying electron- and hole-vibration couplings is sufficient to accurately determine the exciton-vibration coupling constants in the studied cases.

Approximate Hamiltonians from a Linear Vibronic Coupling Model for Solution-Phase Spin Dynamics

The linear vibronic coupling (LVC) model is an approach for approximating how a molecular Hamiltonian changes in response to small changes in molecular geometry. The LVC framework thus has the ability to approximate molecular Hamiltonians at low computational expense but with quality approaching multiconfigurational ab initio calculations, when the change in geometry compared to the reference calculation used to parametrize it is small. Here, we show how the LVC approach can be used to project approximate spin Hamiltonians of a solvated lanthanide complex along a room-temperature molecular dynamics trajectory. As expected, the LVC approximation is less accurate as the geometry diverges from that at which the model was parametrized. We examine the accuracy of the predicted Hamiltonians by performing time-dependent quantum simulations of the spin dynamics of the molecule, with reference to the dynamics obtained using spin Hamiltonians projected from ab initio calculations at each step. We find that quantitatively accurate behavior is obtained when LVC parametrizations are performed at least every 10 fs during the trajectory.

Tuning Electronic Relaxation of Nanorings Through Their Interlocking

Electronic and vibrational relaxation processes can be optimized and tuned by introducing alternative pathways that channel excess energy more efficiently. An ensemble of interacting molecular systems can help overcome the bottlenecks caused by large energy gaps between intermediate excited states involved in the relaxation process. By employing this strategy, catenanes composed of mechanically interlocked carbon nanostructures show great promise as new materials for achieving higher efficiencies in electronic devices. Herein, we perform nonadiabatic excited state molecular dynamics on different all-benzene catenanes. We observe that catenanes experience faster relaxations than individual units. Coupled catenanes present overlapping energy manifolds that include several electronic excited states spatially localized on the different moieties, increasing the density of states that ultimately improve the efficiency in the energy relaxation. This result suggests the use of catenanes as a viable strategy for tuning the internal conversion rates in a quest for their utilization for new optoelectronic applications.

Simulation of the non-adiabatic dynamics of an enone-Lewis acid complex in an explicit solvent

Unlocking the full potential of Lewis acid catalysis for photochemical transformations requires a comprehensive understanding of the ultrafast dynamics of substrate-Lewis acid complexes. In a previous article [Peschel et al., Angew. Chem. Int. Ed., 2021, 60, 10155], time-resolved spectroscopy supported by static calculations revealed that the Lewis acid remains attached during the relaxation of the model complex cyclohexenone-BF3. In contrast to the experimental observation, surface-hopping dynamics in the gas phase predicted ultrafast heterolytic dissociation. We attributed the discrepancy to missing solvent interactions. Thus, in this work, we present an interface between the SHARC and FermiONs++ program packages, which enables us to investigate the ultrafast dynamics of cyclohexenone-BF3 in an explicit solvent environment. Our simulations demonstrate that the solvent prevents the dissociation of the complex, leading to an intriguing dissociation-reassociation mechanism. Comparing the dynamics with and without triplet states highlights their role in the relaxation process and shows that the Lewis acid inhibits intersystem crossing. These findings provide a clear picture of the relaxation process, which may aid in designing future Lewis acid catalysts for photochemical applications. They underscore that an explicit solvent model is required to describe relaxation processes in weakly bound states, as energy transfer to the solvent is crucial for the system to reach its minimum geometries.

Transient-absorption spectroscopy of dendrimers via nonadiabatic excited-state dynamics simulations

The efficiency of light-harvesting and energy transfer in multi-chromophore ensembles underpins natural photosynthesis. Dendrimers are highly branched synthetic multi-chromophoric conjugated supra-molecules that mimic these natural processes. After photoexcitation, their repeated units participate in a number of intramolecular electronic energy relaxation and redistribution pathways that ultimately funnel to a sink. Here, a model four-branched dendrimer with a pyrene core is theoretically studied using nonadiabatic molecular dynamics simulations. We evaluate excited-state photoinduced dynamics of the dendrimer, and demonstrate on-the-fly simulations of its transient absorption pump-probe (TA-PP) spectra. We show how the evolutions of the simulated TA-PP spectra monitor in real time photoinduced energy relaxation and redistribution, and provide a detailed microscopic picture of the relevant energy-transfer pathways. To the best of our knowledge, this is the first of this kind of on-the-fly atomistic simulation of TA-PP signals reported for a large molecular system.

Resolving Photoinduced Femtosecond Three-Dimensional Solute-Solvent Dynamics through Surface Hopping Simulations

Photoinduced dynamics in solution is governed by mutual solute-solvent interactions, which give rise to phenomena like solvatochromism, the Stokes shift, dual fluorescence, or charge transfer. Understanding these phenomena requires simulating the solute's photoinduced dynamics and simultaneously resolving the three-dimensional solvent distribution dynamics. If using trajectory surface hopping (TSH) to this aim, thousands of trajectories are required to adequately sample the time-dependent three-dimensional solvent distribution functions, and thus resolve the solvent dynamics with sub-Ångstrom and femtosecond accuracy and sufficiently low noise levels. Unfortunately, simulating thousands of trajectories with TSH in the framework of hybrid quantum mechanical/molecular mechanical (QM/MM) can be prohibitively expensive when employing ab initio electronic structure methods. To tackle this challenge, we recently introduced a computationally efficient approach that combines efficient linear vibronic coupling models with molecular mechanics (LVC/MM) via electrostatic embedding [Polonius et al., JCTC 2023, 19, 7171-7186]. This method provides solvent-embedded, nonadiabatically coupled potential energy surfaces while scaling similarly to MM force fields. Here, we employ TSH with LVC/MM to unravel the photoinduced dynamics of two small thiocarbonyl compounds solvated in water. We describe how to estimate the number of trajectories required to produce nearly noise-free three-dimensional solvent distribution functions and present an analysis based on approximately 10,000 trajectories propagated for 3 ps. In the electronic ground state, both molecules exhibit in-plane hydrogen bonds to the sulfur atom. Shortly after excitation, these bonds are broken and reform perpendicular to the molecular plane on timescales that differ by an order of magnitude due to steric effects. We also show that the solvent relaxation dynamics is coupled to the electronic dynamics, including intersystem crossing. These findings are relevant to advance the understanding of the coupled solute-solvent dynamics of solvated photoexcited molecules, e.g., biologically relevant thio-nucleobases.

A Portrait of the Chromophore as a Young System-Quantum-Derived Force Field Unraveling Solvent Reorganization upon Optical Excitation of Cyclocurcumin Derivatives

The study of fast non-equilibrium solvent relaxation in organic chromophores is still challenging for molecular modeling and simulation approaches, and is often overlooked, even in the case of non-adiabatic dynamics simulations. Yet, especially in the case of photoswitches, the interaction with the environment can strongly modulate the photophysical outcomes. To unravel such a delicate interplay, in the present contribution we resorted to a mixed quantum-classical approach, based on quantum mechanically derived force fields. The main task is to rationalize the solvent reorganization pathways in chromophores derived from cyclocurcumin, which are suitable for light-activated chemotherapy to destabilize cellular lipid membranes. The accurate and reliable decryption delivered by the quantum-derived force fields points to important differences in the solvent's reorganization, in terms of both structure and time scale evolution.

Bifurcation of Excited-State Population Leads to Anti-Kasha Luminescence in a Disulfide-Decorated Organometallic Rhenium Photosensitizer

We report a rhenium diimine photosensitizer equipped with a peripheral disulfide unit on one of the bipyridine ligands, [Re(CO)3(bpy)(S-Sbpy4,4)]+ (1+, bpy = 2,2'-bipyridine, S-Sbpy4,4 = [1,2]dithiino[3,4-c:6,5-c']dipyridine), showing anti-Kasha luminescence. Steady-state and ultrafast time-resolved spectroscopies complemented by nonadiabatic dynamics simulations are used to disclose its excited-state dynamics. The calculations show that after intersystem crossing the complex evolves to two different triplet minima: a (S-Sbpy4,4)-ligand-centered excited state (3LC) lying at lower energy and a metal-to-(bpy)-ligand charge transfer (3MLCT) state at higher energy, with relative yields of 90% and 10%, respectively. The 3LC state involves local excitation of the disulfide group into the antibonding σ* orbital, leading to significant elongation of the S-S bond. Intriguingly, it is the higher-lying 3MLCT state, which is assigned to display luminescence with a lifetime of 270 ns: a signature of anti-Kasha behavior. This assignment is consistent with an energy barrier ≥ 0.6 eV or negligible electronic coupling, preventing reaction toward the 3LC state after the population is trapped in the 3MLCT state. This study represents a striking example on how elusive excited-state dynamics of transition-metal photosensitizers can be deciphered by synergistic experiments and state-of-the-art calculations. Disulfide functionalization lays the foundation of a new design strategy toward harnessing excess energy in a system for possible bimolecular electron or energy transfer reactivity.

Suppression of Dexter transfer by covalent encapsulation for efficient matrix-free narrowband deep blue hyperfluorescent OLEDs

Hyperfluorescence shows great promise for the next generation of commercially feasible blue organic light-emitting diodes, for which eliminating the Dexter transfer to terminal emitter triplet states is key to efficiency and stability. Current devices rely on high-gap matrices to prevent Dexter transfer, which unfortunately leads to overly complex devices from a fabrication standpoint. Here we introduce a molecular design where ultranarrowband blue emitters are covalently encapsulated by insulating alkylene straps. Organic light-emitting diodes with simple emissive layers consisting of pristine thermally activated delayed fluorescence hosts doped with encapsulated terminal emitters exhibit negligible external quantum efficiency drops compared with non-doped devices, enabling a maximum external quantum efficiency of 21.5%. To explain the high efficiency in the absence of high-gap matrices, we turn to transient absorption spectroscopy. It is directly observed that Dexter transfer from a pristine thermally activated delayed fluorescence sensitizer host can be substantially reduced by an encapsulated terminal emitter, opening the door to highly efficient 'matrix-free' blue hyperfluorescence.

Mechanistic photophysics of tellurium-substituted cytosine: Electronic structure calculations and nonadiabatic dynamics simulations

Previously, the MS-CASPT2 method was performed to study the static and qualitative photophysics of tellurium-substituted cytosine (TeC). To get quantitative information, we used our recently developed QTMF-FSSH dynamics method to simulate the excited-state decay of TeC. The CASSCF method was adopted to reduce the calculation costs, which was confirmed to provide reliable structures and energies as those of MS-CASPT2. A detailed structural analysis showed that only 5% trajectories will hop to the lower triplet or singlet state via the twisted (S2 /S1 /T2 )T intersection, while 67% trajectories will choose the planar intersections of (S2 /S1 /T3 /T2 /T1 )P and (S2 /S1 /T2 /T1 )P but subsequently become twisted in other electronic states. By contrast, ~28% trajectories will maintain in a plane throughout dynamics. Electronic population revealed that the S2 population will ultrafast transfer to the lower triplet or singlet state. Later, the TeC system will populate in the spin-mixed electronic states composed of S1 , T1 and T2 . At the end of 300 fs, most trajectories (~74%) will decay to the ground state and only 17.4% will survive in the triplet states. Our dynamics simulation verified that tellurium substitution will enhance the intersystem crossings, but the very short triplet lifetime (ca. 125 fs) will make TeC a less effective photosensitizer.

Spin-Vibronic Intersystem Crossing and Molecular Packing Effects in Heavy Atom Free Organic Phosphor

We present a detailed investigation into the excited state properties of a planar D3h symmetric azatriangulenetrione, HTANGO, which has received significant interest due to its high solid-state phosphorescence quantum yield and therefore potential as an organic room temperature phosphorescent (ORTP) dye. Using a model linear vibronic coupling Hamiltonian in combination with quantum dynamics simulations, we observe that intersystem crossing (ISC) in HTANGO occurs with a rate of ∼1010 s-1, comparable to benzophenone, an archetypal molecule for fast ISC in heavy metal free molecules. Our simulations demonstrate that the mechanism for fast ISC is associated with the high density of excited triplet states which lie in close proximity to the lowest singlet states, offering multiple channels into the triplet manifold facilitating rapid population transfer. Finally, to rationalize the solid-state emission properties, we use quantum chemistry to investigate the excited state surfaces of the HTANGO dimer, highlighting the influence and importance of the rotational alignment between the two HTANGO molecules in the solid state and how this contributes to high phosphorescence quantum yield.

Spin-Vibronic Coupling Controls the Intersystem Crossing of Iodine-Substituted BODIPY Triplet Chromophores

4,4-Difluoro-4-borata-3a-azonia-4a-aza-s-indacene (BODIPY) dyes are extensively used in various applications of their triplet states, ranging from photoredox catalysis, through triplet sensitization to photodynamic therapy. However, the rational design of BODIPY triplet chromophores by ab initio modelling is limited by their strong interactions of spin, electronic and vibrational dynamics. In particular, spin-vibronic coupling is often overlooked when estimating intersystem crossing (ISC) rates. In this study, a combined experimental and theoretical approach using spin-vibronic coupling to correctly describe ISC in BODIPY dyes was developed. For this purpose, seven π-extended BODIPY derivatives with iodine atoms in different positions were examined. It was found that the heavy-atom effect of iodine atoms is site specific, causing high triplet yields in only some positions. This site-specific ISC was explained by El-Sayed rules, so both the contribution and character of the molecular orbitals involved in the excitation must be considered when predicting the ISC rates. Overall, the rational design of BODIPY triplet chromophores requires using (i) the high-quality electronic structure theory, including both static and dynamical correlations; and (ii) the two-component wave function Hamiltonian, and rationalizing; and (iii) ISC based on the character of the molecular orbitals of heavy atoms involved in the excitation, expanding El-Sayed rules beyond their traditional applications.

Coronene: a model for ultrafast dynamics in graphene nanoflakes and PAHs

Assuming a delta pulse excitation, quantum wavepackets are propagated on the excited state manifold in the energy range from 3.4-5.0 eV for coronene and 2.4-3.5 eV for circumcoronene to study the time evolution of the states as well as their lifetimes. The full-dimensional (102 and 210 degrees of freedom for coronene and circumcoronene respectively) non-adiabatic dynamics simulated with the ML-MCTDH method on twelve coupled singlet electronic states show that the different absorption spectra are only due to electronic delocalisation effects that change the excited state energies, but the structural dynamics in both compounds are identical. Breathing and tilting motions drive the decay dynamics of the electronic states away from the Frank-Condon region independently of the size of the aromatic system. This promising result allows the use of coronene as a model system for the dynamics of larger polycyclic aromatic hydrocarbons (PAHs) and graphene one dimensional sheets or nanoflakes.

Vibrational Coherences of the Photoinduced Mixed-Valent Creutz-Taube Ion Revealed by Excited State Dynamics

A recent study of photoinduced mixed-valency in the one-electron reduced form (μ-pz)[RuII(NH3)5]24+ of the Creutz-Taube ion used transient absorption spectroscopy with vis-NIR broadband detection to uncover a mixed-valent excited state with a typical intervalence charge transfer band and a nanosecond lifetime [Pieslinger et al. Angew. Chem., Int. Ed. 2022, 61, e202211747]. Herein, we use excited state dynamics simulations with implicit solvation to elucidate the electronic and vibrational evolution in the first 10 ps after the optical excitation. A manifold of excited states with weak interaction between the metal centers is populated already at time zero due to the breakdown of the Condon approximation and dominates the population of electronic states at short time scales (<0.5 ps). A long-lived vibrational wave packet mostly confined to oscillations of the metal center-bridge distances is observed. The oscillations are traced to the electronic structure properties of states with weak metal-metal coupling. The long-lived mixed-valent excited state of the Creutz-Taube ion analogue is formed vibrationally cold and has a more compact geometry. While experimentally, intersystem crossing and vibrational relaxation were deduced to be completed within 1 ps, our analysis indicates that both processes might persist at longer times.

Twisting Aromaticity and Photoinduced Dynamics in Hexapole Helicenes

Curved aromatic molecules are attractive electronic materials, where an additional internal strain uniquely modifies their structure, aromaticity, dynamics, and optical properties. Helicenes are examples of such twisted conjugated systems. Herein, we analyze the photoinduced dynamics in different stereoisomers of a hexapole helicene by using nonadiabatic excited-state molecular dynamics simulations. We explore how changes in symmetry and structural distortion modulate the intramolecular energy redistribution. We find that distinct helical assembly leads to different rigid distorted structures that in turn impact the nonradiative energy relaxation and ultimately formation of the self-trapped exciton. Subsequently, the value of the twisting angles relative to the central triphenylene core structure controls the global molecular aromaticity and electronic localization during the internal conversion process. Our work sheds light on how the future synthesis of novel curved aromatic compounds can be directed to attain specific desired electronic properties through the modulation of their twisted aromaticity.

Galván I, Alavi A, Aleotti F, Aquilante F, Autschbach J, Avagliano D, Baiardi A, Bao JJ, Battaglia S, Birnoschi L, Blanco-González A, Bokarev SI, Broer R, Cacciari R, Calio PB, Carlson RK, Carvalho Couto R, Cerdán L, Chibotaru LF, Chilton NF, Church JR, Conti I, Coriani S, Cuéllar-Zuquin J, Daoud RE, Dattani N, Decleva P, de Graaf C, Delcey M, De Vico L, Dobrautz W, Dong SS, Feng R, Ferré N, Filatov(Gulak) M, Gagliardi L, Garavelli M, González L, Guan Y, Guo M, Hennefarth MR, Hermes MR, Hoyer CE, Huix-Rotllant M, Jaiswal VK, Kaiser A, Kaliakin DS, Khamesian M, King DS, Kochetov V, Krośnicki M, Kumaar AA, Larsson ED, Lehtola S, Lepetit MB, Lischka H, López Ríos P, Lundberg M, Ma D, Mai S, Marquetand P, Merritt ICD, Montorsi F, Mörchen M, Nenov A, Nguyen VHA, Nishimoto Y, Oakley MS, Olivucci M, Oppel M, Padula D, Pandharkar R, Phung QM, Plasser F, Raggi G, Rebolini E, Reiher M, Rivalta I, Roca-Sanjuán D, Romig T, Safari AA, Sánchez-Mansilla A, Sand AM, Schapiro I, Scott TR, Segarra-Martí J, Segatta F, Sergentu DC, Sharma P, Shepard R, Shu Y, Staab JK, Straatsma TP, Sørensen LK, Tenorio BNC, Truhlar DG, Ungur L, Vacher M, Veryazov V, Voß TA, Weser O, Wu D, Yang X, Yarkony D, Zhou C, Zobel JP, Lindh R. The OpenMolcas Web: A Community-Driven Approach to Advancing Computational Chemistry

The developments of the open-source OpenMolcas chemistry software environment since spring 2020 are described, with a focus on novel functionalities accessible in the stable branch of the package or via interfaces with other packages. These developments span a wide range of topics in computational chemistry and are presented in thematic sections: electronic structure theory, electronic spectroscopy simulations, analytic gradients and molecular structure optimizations, ab initio molecular dynamics, and other new features. This report offers an overview of the chemical phenomena and processes OpenMolcas can address, while showing that OpenMolcas is an attractive platform for state-of-the-art atomistic computer simulations.

LVC/MM: A Hybrid Linear Vibronic Coupling/Molecular Mechanics Model with Distributed Multipole-Based Electrostatic Embedding for Highly Efficient Surface Hopping Dynamics in Solution

We present a theoretical framework for a hybrid linear vibronic coupling model electrostatically embedded into a molecular mechanics environment, termed the linear vibronic coupling/molecular mechanics (LVC/MM) method, for the surface hopping including arbitrary coupling (SHARC) molecular dynamics package. Electrostatic embedding is realized through the computation of interactions between environment point charges and distributed multipole expansions (DMEs, up to quadrupoles) that represent each electronic state and transition densities in the diabatic basis. The DME parameters are obtained through a restrained electrostatic potential (RESP) fit, which we extended to yield higher-order multipoles. We also implemented in SHARC a scheme for achieving roto-translational invariance of LVC models as well as a general quantum mechanics/molecular mechanics (QM/MM) interface, an OpenMM interface, and restraining potentials for simulating liquid droplets. Using thioformaldehyde in water as a test case, we demonstrate that LVC/MM can accurately reproduce the solvation structure and energetics of rigid solutes, with errors on the order of 1-2 kcal/mol compared to a BP86/MM reference. The implementation in SHARC is shown to be very efficient, enabling the simulation of trajectories on the nanosecond time scale in a matter of days.

Spin-phonon coupling and magnetic relaxation in single-molecule magnets

Electron-phonon coupling is important in many physical phenomena, e.g. photosynthesis, catalysis and quantum information processing, but its impacts are difficult to grasp on the microscopic level. One area attracting wide interest is that of single-molecule magnets, which is motivated by searching for the ultimate limit in the miniaturisation of binary data storage media. The utility of a molecule to store magnetic information is quantified by the timescale of its magnetic reversal processes, also known as magnetic relaxation, which is limited by spin-phonon coupling. Several recent accomplishments of synthetic organometallic chemistry have led to the observation of molecular magnetic memory effects at temperatures above that of liquid nitrogen. These discoveries have highlighted how far chemical design strategies for maximising magnetic anisotropy have come, but have also highlighted the need to characterise the complex interplay between phonons and molecular spin states. The crucial step is to make a link between magnetic relaxation and chemical motifs, and so be able to produce design criteria to extend molecular magnetic memory. The basic physics associated with spin-phonon coupling and magnetic relaxation was outlined in the early 20th century using perturbation theory, and has more recently been recast in the form of a general open quantum systems formalism and tackled with different levels of approximations. It is the purpose of this Tutorial Review to introduce the topics of phonons, molecular spin-phonon coupling, and magnetic relaxation, and to outline the relevant theories in connection with both the traditional perturbative texts and the more modern open quantum systems methods.

Impact of the core on the inter-branch exciton exchange in dendrimers

Organic dendrimers with π conjugated systems are capable of capturing solar energy as a renewable source for human use. Nonetheless, further study regarding the relationship between the structure and the energy transfer mechanism in these types of molecules is still necessary. In this work, nonadiabatic excited state molecular dynamics (NEXMD) were carried out to study the intra- and inter-branch exciton migration in two tetra-branched dendrimers, C(dSSB)₄ and Ad(BuSSB)₄, which differ in their respective carbon and adamantane core. Both systems undergo a ladder decay mechanism between excited states, with back-and-forth transitions between S₁ and S₂. Despite presenting very similar absorption-emission spectra, differences in the photoinduced energy relaxation are observed. The size of the core impacts the inter-branch energy exchange and transient exciton localization/delocalization, which ultimately condition the relative energy relaxation rates, being faster in Ad(BuSSB)₄ with respect to C(dSSB)₄. Nevertheless, the photoinduced processes lead to a progressive final exciton-self-trapping in one of the branches of both dendrimers, which is a desirable feature in organic photovoltaic applications. Our results can inspire the design of more efficient dendrimers with the desired magnitude of inter-branch exciton exchange and localization/delocalization according to changes in their core.

New Gradient Correction Scheme for Electronically Nonadiabatic Dynamics Involving Multiple Spin States

It has been recommended that the best representation to use for trajectory surface hopping (TSH) calculations is the fully adiabatic basis in which the Hamiltonian is diagonal. Simulations of intersystem crossing processes with conventional TSH methods require an explicit computation of nonadiabatic coupling vectors (NACs) in the molecular-Coulomb-Hamiltonian (MCH) basis, also called the spin-orbit-free basis, in order to compute the gradient in the fully adiabatic basis (also called the diagonal representation). This explicit requirement destroys some of the advantages of the overlap-based algorithms and curvature-driven algorithms that can be used for the most efficient TSH calculations. Therefore, although these algorithms allow one to perform NAC-free simulations for internal conversion processes, one still requires NACs for intersystem crossing. Here, we show that how the NAC requirement is circumvented by a new computation scheme called the time-derivative-matrix scheme.

Nonadiabatic Forward Flux Sampling for Excited-State Rare Events

We present a rare event sampling scheme applicable to coupled electronic excited states. In particular, we extend the forward flux sampling (FFS) method for rare event sampling to a nonadiabatic version (NAFFS) that uses the trajectory surface hopping (TSH) method for nonadiabatic dynamics. NAFFS is applied to two dynamically relevant excited-state models that feature an avoided crossing and a conical intersection with tunable parameters. We investigate how nonadiabatic couplings, temperature, and reaction barriers affect transition rate constants in regimes that cannot be otherwise obtained with plain, traditional TSH. The comparison with reference brute-force TSH simulations for limiting cases of rareness shows that NAFFS can be several orders of magnitude cheaper than conventional TSH and thus represents a conceptually novel tool to extend excited-state dynamics to time scales that are able to capture rare nonadiabatic events.

Nonlinear Molecular Electronic Spectroscopy via MCTDH Quantum Dynamics: From Exact to Approximate Expressions

We present an accurate and efficient approach to computing the linear and nonlinear optical spectroscopy of a closed quantum system subject to impulsive interactions with an incident electromagnetic field. It incorporates the effect of ultrafast nonadiabatic dynamics by means of explicit numerical propagation of the nuclear wave packet. The fundamental expressions for the evaluation of first- and higher-order response functions are recast in a general form that can be used with any quantum dynamics code capable of computing the overlap of nuclear wave packets evolving in different states. Here we present the evaluation of these expressions with the multiconfiguration time-dependent Hartree (MCTDH) method. Application is made to pyrene, excited to its lowest bright excited state S₂ which exhibits a sub-100-fs nonadiabatic decay to a dark state S₁. The system is described by a linear vibronic coupling Hamiltonian, parametrized with multiconfiguration electronic structure methods. We show that the ultrafast nonadiabatic dynamics can have a remarkable effect on the spectral line shapes that goes beyond simple lifetime broadening. Furthermore, a widely employed approximate expression based on the time scale separation of dephasing and population relaxation is recast in the same theoretical framework. Application to pyrene shows the range of validity of such approximations.

Photodynamics of the Molecular Ruby [Cr(ddpd)2]3

The introduction of strong-field ligands can enable luminescence in first-row transition-metal complexes. In this way, earth-abundant near-infrared emitters can be obtained using early 3d metals. A prime example is the molecular ruby [Cr(ddpd)₂]³⁺ (ddpd = N,N'-dimethyl-N,N'-dipyridin-2-ylpyridine-2,6-diamine) that can achieve high phosphorescence quantum yields at room temperature in aqueous solution. To understand these remarkable properties, here, we simulate its photodynamics in water using trajectory surface hopping on linear vibronic coupling potentials parametrized from multiconfigurational CASSCF/CASPT2 calculations. We find that after excitation to the second absorption band, a relaxation cascade through metal-centered states occurs. After an initial back-and-forth intersystem crossing with higher-lying doublet states, the complex relaxes through a manifold of quartet metal-centered states to the low-lying doublet metal-centered states which are responsible for the experimentally observed emission. These electronic processes are driven by an elongation of the Cr-ligand bond lengths as well as the twisting motion of the trans-coordinated pyridine units in the ddpd ligands. The low-lying doublet states are reached within 1-2 ps and are close in geometry to the doublet minima, thus explaining the high phosphorescence quantum yield of the molecular ruby [Cr(ddpd)₂]³⁺.

Can range-separated functionals be optimally tuned to predict spectra and excited state dynamics in photoactive iron complexes?

Density functional theory is an efficient computational tool to investigate photophysical and photochemical processes in transition metal complexes, giving invaluable assistance in interpreting spectroscopic and catalytic experiments. Optimally tuned range-separated functionals are particularly promising, as they were created to address some of the fundamental deficiencies present in approximate exchange-correlation functionals. In this paper, we scrutinize the selection of optimally tuned parameters and its influence on the excited state dynamics, using the example of the iron complex [Fe(cpmp)₂]²⁺ with push-pull ligands. Various tuning strategies are contemplated based on pure self-consistent DFT protocols, as well as on the comparison with experimental spectra and multireference CASPT2 results. The two most promising sets of optimal parameters are then employed to carry out nonadiabatic surface-hopping dynamics simulations. Intriguingly, we find that the two sets lead to very different relaxation pathways and timescales. While the set of optimal parameters from one of the self-consistent DFT protocols predicts the formation of long-lived metal-to-ligand charge transfer triplet states, the set in better agreement with CASPT2 calculations leads to deactivation in the manifold of metal-centered states, in better agreement with the experimental reference data. These results showcase the complexity of iron-complex excited state landscapes and the difficulty of obtaining an unambiguous parametrization of long-range corrected functionals without experimental input.

Analytic Linear Vibronic Coupling Method for First-Principles Spin-Dynamics Calculations in Single-Molecule Magnets

Accurate modeling of vibronically driven magnetic relaxation from ab initio calculations is of paramount importance to the design of next-generation single-molecule magnets (SMMs). Previous theoretical studies have been relying on numerical differentiation to obtain spin-phonon couplings in the form of derivatives of spin Hamiltonian parameters. In this work, we introduce a novel approach to obtain these derivatives fully analytically by combining the linear vibronic coupling (LVC) approach with analytic complete active space self-consistent field derivatives and nonadiabatic couplings computed at the equilibrium geometry with a single electronic structure calculation. We apply our analytic approach to the computation of Orbach and Raman relaxation rates for a bis-cyclobutadienyl Dy(III) sandwich complex in the frozen-solution phase, where the solution environment is modeled by electrostatic multipole expansions, and benchmark our findings against results obtained using conventional numerical derivatives and a fully electronic description of the whole system. We demonstrate that our LVC approach exhibits high accuracy over a wide range of coupling strengths and enables significant computational savings due to its analytic, "single-shot" nature. Evidently, this offers great potential for advancing the simulation of a wide range of vibronic coupling phenomena in magnetism and spectroscopy, ultimately aiding the design of high-performance SMMs. Considering different environmental representations, we find that a point charge model shows the best agreement with the reference calculation, including the full electronic environment, but note that the main source of discrepancies observed in the magnetic relaxation rates originates from the approximate equilibrium electronic structure computed using the electrostatic environment models rather than from the couplings.

Simulations of molecular photodynamics in long timescales

Nonadiabatic dynamics simulations in the long timescale (much longer than 10 ps) are the next challenge in computational photochemistry. This paper delimits the scope of what we expect from methods to run such simulations: they should work in full nuclear dimensionality, be general enough to tackle any type of molecule and not require unrealistic computational resources. We examine the main methodological challenges we should venture to advance the field, including the computational costs of the electronic structure calculations, stability of the integration methods, accuracy of the nonadiabatic dynamics algorithms and software optimization. Based on simulations designed to shed light on each of these issues, we show how machine learning may be a crucial element for long time-scale dynamics, either as a surrogate for electronic structure calculations or aiding the parameterization of model Hamiltonians. We show that conventional methods for integrating classical equations should be adequate to extended simulations up to 1 ns and that surface hopping agrees semiquantitatively with wave packet propagation in the weak-coupling regime. We also describe our optimization of the Newton-X program to reduce computational overheads in data processing and storage. This article is part of the theme issue 'Chemistry without the Born-Oppenheimer approximation'.

Surface hopping dynamics in periodic solid-state materials with a linear vibronic coupling model

We report a surface hopping approach in which the implemented linear vibronic coupling Hamiltonian is constructed and the electronic wavefunction is propagated in the reciprocal space. The parameters of the linear vibronic coupling model, including onsite energies, phonon frequencies, and electron-phonon couplings, are calculated with density-functional theory and density-functional perturbation theory and interpolated in fine sampling points of the Brillouin zone with maximally localized Wannier functions. Using this approach, we studied the relaxation dynamics of the photo-excited hot carrier in a one-dimensional periodic carbon chain. The results show that the completeness of the number of Hilbert space k points and the number of phonon q points plays an important role in the hot carrier relaxation processes. By calculating the relaxation times of hot carriers under different reciprocal space sampling and extrapolating with the stretched-compressed exponential function, the relaxation times of hot electrons and holes in the quasi-continuous energy band are obtained. By considering the feedback effect in the hopping processes and analyzing the time-dependent phonon energy in different normal modes, we found that the long-wave longitudinal optical phonons play a major role in the relaxation dynamics of hot electrons and holes. We, therefore, provided herein an efficient and accurate approach for modeling the photophysical processes in periodic solid-state material systems.

Theoretical Challenges in Polaritonic Chemistry

Polaritonic chemistry exploits strong light-matter coupling between molecules and confined electromagnetic field modes to enable new chemical reactivities. In systems displaying this functionality, the choice of the cavity determines both the confinement of the electromagnetic field and the number of molecules that are involved in the process. While in wavelength-scale optical cavities the light-matter interaction is ruled by collective effects, plasmonic subwavelength nanocavities allow even single molecules to reach strong coupling. Due to these very distinct situations, a multiscale theoretical toolbox is then required to explore the rich phenomenology of polaritonic chemistry. Within this framework, each component of the system (molecules and electromagnetic modes) needs to be treated in sufficient detail to obtain reliable results. Starting from the very general aspects of light-molecule interactions in typical experimental setups, we underline the basic concepts that should be taken into account when operating in this new area of research. Building on these considerations, we then provide a map of the theoretical tools already available to tackle chemical applications of molecular polaritons at different scales. Throughout the discussion, we draw attention to both the successes and the challenges still ahead in the theoretical description of polaritonic chemistry.

Taming Disulfide Bonds with Laser Fields. Nonadiabatic Surface-Hopping Simulations in a Ruthenium Complex

Laser control of chemical reactions is a challenging field of research. In particular, the theoretical description of coupled electronic and nuclear motion in the presence of laser fields is not a trivial task and simulations are mostly restricted to small systems or molecules treated within reduced dimensionality. Here, we demonstrate how the excited state dynamics of [Ru(^S-Sbpy)(bpy)₂]²⁺ can be controlled using explicit laser fields in the context of fewest-switches surface hopping. In particular, the transient properties along the excited state dynamics leading to population of the T₁ minimum energy structure are exploited to define simple laser fields capable of slowing and even completely stopping the onset of S-S bond dissociation. The use of a linear vibronic coupling model to parametrize the potential energy surfaces showcases the strength of the surface-hopping methodology to study systems including explicit laser fields using many nuclear degrees of freedom and a large amount of close-lying electronic excited states.

Toward Simulation of Fe(II) Low-Spin → High-Spin Photoswitching by Synergistic Spin-Vibronic Dynamics

A new theoretical approach is presented and applied for the simulation of Fe(II) low-spin (LS, singlet, t_2g⁶e_g⁰) → high-spin (HS, quintet, t_2g⁴e_g²) photoswitching dynamics of the octahedral model complex [Fe(NCH)₆]²⁺. The utilized synergistic methodology heavily exploits the strengths of complementary electronic structure and spin-vibronic dynamics methods. Specifically, we perform 3D quantum dynamics (QD) and full-dimensional trajectory surface hopping (TSH, in conjunction with a linear vibronic coupling model), with the modes for QD selected by TSH. We follow a hybrid approach which is based on the application of time-dependent density functional theory (TD-DFT) excited-state potential energy surfaces (PESs) and multiconfigurational second-order perturbation theory (CASPT2) spin–orbit couplings (SOCs). Our method delivers accurate singlet–triplet–quintet intersystem crossing (ISC) dynamics, as assessed by comparison to our recent high-level ab initio simulations and related time-resolved experimental data. Furthermore, we investigate the capability of our simulations to identify the location of ISCs. Finally, we assess the approximation of constant SOCs (calculated at the Franck–Condon geometry), whose validity has central importance for the combination of TD-DFT PESs and CASPT2 SOCs. This efficient methodology will have a key role in simulating LS → HS dynamics for more complicated cases, involving higher density of states and varying electronic character, as well as the analysis of ultrafast experiments.

Vibronic Photoexcitation Dynamics of Perylene Diimide: Computational Insights

Perylene diimide (PDI) represents a prototype material for organic optoelectronic devices because of its strong optical absorbance, chemical stability, efficient energy transfer, and optical and chemical tunability. Herein, we analyze in detail the vibronic relaxation of its photoexcitation using nonadiabatic excited-state molecular dynamics simulations. We find that after the absorption of a photon, which excites the electron to the second excited state, S₂, induced vibronic dynamics features persistent modulations in the spatial localization of electronic and vibrational excitations. These energy exchanges are dictated by strong vibronic couplings that overcome structural disorders and thermal fluctuations. Specifically, the electronic wavefunction periodically swaps between localizations on the right and left sides of the molecule. Within 1 ps of such dynamics, a nonradiative transition to the lowest electronic state, S₁, takes place, resulting in a complete delocalization of the wavefunction. The observed vibronic dynamics emerges following the electronic energy deposition in the direction that excites a combination of two dominant vibrational normal modes. This behavior is maintained even with a chemical substitution that breaks the symmetry of the molecule. We believe that our findings elucidate the nature of the complex dynamics of the optically excited states and, therefore, contribute to the development of tunable functionalities of PDIs and their derivatives.

Micro-Solvated DMABN: Excited State Quantum Dynamics and Dual Fluorescence Spectra

In this work, we report a complete analysis by theoretical and spectroscopic methods of the short-time behaviour of 4-(dimethylamino)benzonitrile (DMABN) in the gas phase as well as in cyclohexane, tetrahydrofuran, acetonitrile, and water solution, after excitation to the La state. The spectroscopic properties of DMABN were investigated experimentally using UV absorption and fluorescence emission spectroscopy. The computational study was developed at different electronic structure levels and using the Polarisable Continuum Model (PCM) and explicit solvent molecules to reproduce the solvent environment. Additionally, excited state quantum dynamics simulations in the diabatic picture using the direct dynamics variational multiconfigurational Gaussian (DD-vMCG) method were performed, the largest quantum dynamics "on-the-fly" simulations performed with this method until now. The comparison with fully converged multilayer multiconfigurational time-dependent Hartree (ML-MCTDH) dynamics on parametrised linear vibronic coupling (LVC) potentials show very similar population decays and evolution of the nuclear wavepacket. The ring C=C stretching and three methyl tilting modes are identified as the responsible motions for the internal conversion from the La to the Lb states. No major differences are observed in the ultrafast initial decay in different solvents, but we show that this effect depends strongly on the level of electronic structure used.

Time-resolved relaxation and fragmentation of polycyclic aromatic hydrocarbons investigated in the ultrafast XUV-IR regime

Polycyclic aromatic hydrocarbons (PAHs) play an important role in interstellar chemistry and are subject to high energy photons that can induce excitation, ionization, and fragmentation. Previous studies have demonstrated electronic relaxation of parent PAH monocations over 10-100 femtoseconds as a result of beyond-Born-Oppenheimer coupling between the electronic and nuclear dynamics. Here, we investigate three PAH molecules: fluorene, phenanthrene, and pyrene, using ultrafast XUV and IR laser pulses. Simultaneous measurements of the ion yields, ion momenta, and electron momenta as a function of laser pulse delay allow a detailed insight into the various molecular processes. We report relaxation times for the electronically excited PAH^*, PAH^+* and PAH^2+* states, and show the time-dependent conversion between fragmentation pathways. Additionally, using recoil-frame covariance analysis between ion images, we demonstrate that the dissociation of the PAH²⁺ ions favors reaction pathways involving two-body breakup and/or loss of neutral fragments totaling an even number of carbon atoms.

Surface Hopping Dynamics on Vibronic Coupling Models

The simulation of photoinduced non-adiabatic dynamics is of great relevance in many scientific disciplines, ranging from physics and materials science to chemistry and biology. Upon light irradiation, different relaxation processes take place in which electronic and nuclear motion are intimately coupled. These are best described by the time-dependent molecular Schrödinger equation, but its solution poses fundamental practical challenges to contemporary theoretical chemistry. Two widely used and complementary approaches to this problem are multiconfigurational time-dependent Hartree (MCTDH) and trajectory surface hopping (SH). MCTDH is an accurate fully quantum-mechanical technique but often is feasible only in reduced dimensionality, in combination with approximate vibronic coupling (VC) Hamiltonians, or both (i.e., reduced-dimensional VC potentials). In contrast, SH is a quantum–classical technique that neglects most nuclear quantum effects but allows nuclear dynamics in full dimensionality by calculating potential energy surfaces on the fly. If nuclear quantum effects do not play a central role and a linear VC (LVC) Hamiltonian is appropriate—e.g., for stiff molecules that generally keep their conformation in the excited state—then it seems advantageous to combine the efficient LVC and SH techniques. In this Account, we describe how surface hopping based on an LVC Hamiltonian (SH/LVC)—as recently implemented in the SHARC surface hopping package—can provide an economical and automated approach to simulate non-adiabatic dynamics. First, we illustrate the potential of SH/LVC in a number of showcases, including intersystem crossing in SO₂, intra-Rydberg dynamics in acetone, and several photophysical studies on large transition-metal complexes, which would be much more demanding or impossible to perform with other methods. While all of the applications provide very useful insights into light-induced phenomena, they also hint at difficulties faced by the SH/LVC methodology that need to be addressed in the future. Second, we contend that the SH/LVC approach can be useful to benchmark SH itself. By the use of the same (LVC) potentials as MCTDH calculations have employed for decades and by relying on the efficiency of SH/LVC, it is possible to directly compare multiple SH test calculations with a MCTDH reference and ponder the accuracy of various correction algorithms behind the SH methodology, such as decoherence corrections or momentum rescaling schemes. Third, we demonstrate how the efficiency of SH/LVC can also be exploited to identify essential nuclear and electronic degrees of freedom to be employed in more accurate MCTDH calculations. Lastly, we show that SH/LVC is able to advance the development of SH protocols that can describe nuclear dynamics including explicit laser fields—a very challenging endeavor for trajectory-based schemes. To end, this Account compiles the typical costs of contemporary SH simulations, evidencing the great advantages of using parametrized potentials. The LVC model is a sleeping beauty that, kissed by SH, is fueling the field of excited-state molecular dynamics. We hope that this Account will stimulate future research in this direction, leveraging the advantages of the SH/VC schemes to larger extents and extending their applicability to uncharted territories.

Efficient quantum dynamics simulations of complex molecular systems: A unified treatment of dynamic and static disorder

We present a unified and highly numerically efficient formalism for the simulation of quantum dynamics of complex molecular systems, which takes into account both temperature effects and static disorder. The methodology is based on the thermo-field dynamics formalism, and Gaussian static disorder is included into simulations via auxiliary bosonic operators. This approach, combined with the tensor-train/matrix-product state representation of the thermalized stochastic wave function, is applied to study the effect of dynamic and static disorders in charge-transfer processes in model organic semiconductor chains employing the Su-Schrieffer-Heeger (Holstein-Peierls) model Hamiltonian.

Machine Learning for Electronically Excited States of Molecules

Electronically excited states of molecules are at the heart of photochemistry, photophysics, as well as photobiology and also play a role in material science. Their theoretical description requires highly accurate quantum chemical calculations, which are computationally expensive. In this review, we focus on not only how machine learning is employed to speed up such excited-state simulations but also how this branch of artificial intelligence can be used to advance this exciting research field in all its aspects. Discussed applications of machine learning for excited states include excited-state dynamics simulations, static calculations of absorption spectra, as well as many others. In order to put these studies into context, we discuss the promises and pitfalls of the involved machine learning techniques. Since the latter are mostly based on quantum chemistry calculations, we also provide a short introduction into excited-state electronic structure methods and approaches for nonadiabatic dynamics simulations and describe tricks and problems when using them in machine learning for excited states of molecules.

Machine Learning for Electronically Excited States of Molecules

Electronically excited states of molecules are at the heart of photochemistry, photophysics, as well as photobiology and also play a role in material science. Their theoretical description requires highly accurate quantum chemical calculations, which are computationally expensive. In this review, we focus on not only how machine learning is employed to speed up such excited-state simulations but also how this branch of artificial intelligence can be used to advance this exciting research field in all its aspects. Discussed applications of machine learning for excited states include excited-state dynamics simulations, static calculations of absorption spectra, as well as many others. In order to put these studies into context, we discuss the promises and pitfalls of the involved machine learning techniques. Since the latter are mostly based on quantum chemistry calculations, we also provide a short introduction into excited-state electronic structure methods and approaches for nonadiabatic dynamics simulations and describe tricks and problems when using them in machine learning for excited states of molecules.

The Quest to Simulate Excited-State Dynamics of Transition Metal Complexes

This Perspective describes current computational efforts in the field of simulating photodynamics of transition metal complexes. We present the typical workflows and feature the strengths and limitations of the different contemporary approaches. From electronic structure methods suitable to describe transition metal complexes to approaches able to simulate their nuclear dynamics under the effect of light, we give particular attention to build a bridge between theory and experiment by critically discussing the different models commonly adopted in the interpretation of spectroscopic experiments and the simulation of particular observables. Thereby, we review all the studies of excited-state dynamics on transition metal complexes, both in gas phase and in solution from reduced to full dimensionality.

Ultrafast and long-time excited state kinetics of an NIR-emissive vanadium(iii) complex II. Elucidating triplet-to-singlet excited-state dynamics

We report the non-adiabatic dynamics of VIIICl3(ddpd), a complex based on the Earth-abundant first-row transition metal vanadium with a d2 electronic configuration which is able to emit phosphorescence in solution in the near-infrared spectral region. Trajectory surface-hopping dynamics based on linear vibronic coupling potentials obtained with CASSCF provide molecular-level insights into the intersystem crossing from triplet to singlet metal-centered states. While the majority of the singlet population undergoes back-intersystem crossing to the triplet manifold, 1-2% remains stable during the 10 ps simulation time, enabling the phosphorescence described in Dorn et al. Chem. Sci., 2021, DOI: 10.1039/D1SC02137K. Competing with intersystem crossing, two different relaxation channels via internal conversion through the triplet manifold occur. The nuclear motion that drives the dynamics through the different electronic states corresponds mainly to the increase of all metal-ligand bond distances as well as the decrease of the angles of trans-coordinated ligand atoms. Both motions lead to a decrease in the ligand-field splitting, which stabilizes the interconfigurational excited states populated during the dynamics. Analysis of the electronic character of the states reveals that increasing and stabilizing the singlet population, which in turn can result in enhanced phosphorescence, could be accomplished by further increasing the ligand-field strength.

QM/MM Nonadiabatic Dynamics: the SHARC/COBRAMM Approach

We present the SHARC/COBRAMM approach to enable easy and efficient excited-state dynamics simulations at different levels of electronic structure theory in the presence of complex environments using a quantum mechanics/molecular mechanics (QM/MM) setup. SHARC is a trajectory surface-hoping method that can incorporate the simultaneous effects of nonadiabatic and spin-orbit couplings in the excited-state dynamics of molecular systems. COBRAMM allows ground- and excited-state QM/MM calculations using a subtractive scheme, with electrostatic embedding and a hydrogen link-atom approach. The combination of both free and open-source program packages provides a modular and extensive framework to model nonadiabatic processes after light irradiation from the atomistic scale to the nano-scale. As an example, the relaxation of acrolein from S₁ to T₁ in solution is provided.

Fragment Diabatization Linear Vibronic Coupling Model for Quantum Dynamics of Multichromophoric Systems: Population of the Charge-Transfer State in the Photoexcited Guanine-Cytosine Pair

We introduce a method (FrD-LVC) based on a fragment diabatization (FrD) for the parametrization of a linear vibronic coupling (LVC) model suitable for studying the photophysics of multichromophore systems. In combination with effective quantum dynamics (QD) propagations with multilayer multiconfigurational time-dependent Hartree (ML-MCTDH), the FrD-LVC approach gives access to the study of the competition between intrachromophore decays, like those at conical intersections, and interchromophore processes, like exciton localization/delocalization and the involvement of charge-transfer (CT) states. We used FrD-LVC parametrized with time-dependent density functional theory (TD-DFT) calculations, adopting either CAM-B3LYP or ωB97X-D functionals, to study the ultrafast photoexcited QD of a guanine-cytosine (GC) hydrogen-bonded pair, within a Watson-Crick arrangement, considering up to 12 coupled diabatic electronic states and the effect of all of the 99 vibrational coordinates. The bright excited states localized on C and, especially, on G are predicted to be strongly coupled to the G → C CT state, which is efficiently and quickly populated after an excitation to any of the four lowest energy bright local excited states. Our QD simulations show that more than 80% of the excited population on G and ∼50% of that on C decay to this CT state in less than 50 fs. We investigate the role of vibronic effects in the population of the CT state and show that it depends mainly on its large reorganization energy so that it can occur even when it is significantly less stable than the bright states in the Franck-Condon region. At the same time, we document that the formation of the GC pair almost suppresses the involvement of dark nπ* excited states in the photoactivated dynamics.

Modeling radiative and non-radiative pathways at both the Franck-Condon and Herzberg-Teller approximation level

Here, we present a concise model that can predict the photoluminescent properties of a given compound from first principles, both within and beyond the Franck-Condon approximation. The formalism required to compute fluorescence, Internal Conversion (IC), and Inter-System Crossing (ISC) is discussed. The IC mechanism, in particular, is a difficult pathway to compute due to difficulties associated with the computation of required bosonic configurations and non-adiabatic coupling elements. Here, we offer a discussion and breakdown on how to model these pathways at the Density Functional Theory (DFT) level with respect to its computational implementation, strengths, and current limitations. The model is then used to compute the photoluminescent quantum yield (PLQY) of a number of small but important compounds: anthracene, tetracene, pentacene, diketo-pyrrolo-pyrrole (DPP), and Perylene Diimide (PDI) within a polarizable continuum model. Rate constants for fluorescence, IC, and ISC compare well for the most part with respect to experiment, despite triplet energies being overestimated to a degree. The resulting PLQYs are promising with respect to the level of theory being DFT. While we obtained a positive result for PDI within the Franck-Condon limit, the other systems require a second order correction. Recomputing quantum yields with Herzberg-Teller terms yields PLQYs of 0.19, 0.08, 0.04, 0.70, and 0.99 for anthracene, tetracene, pentacene, DPP, and PDI, respectively. Based on these results, we are confident that the presented methodology is sound with respect to the level of quantum chemistry and presents an important stepping stone in the search for a tool to predict the properties of larger coupled systems.

Simulation of Nonlinear Femtosecond Signals at Finite Temperature via a Thermo Field Dynamics-Tensor Train Method: General Theory and Application to Time- and Frequency-Resolved Fluorescence of the Fenna-Matthews-Olson Complex

Addressing needs of contemporary nonlinear femtosecond optical spectroscopy, we have developed a fully quantum, numerically accurate wave function-based approach for the calculation of third-order spectroscopic signals of polyatomic molecules and molecular aggregates at finite temperature. The systems are described by multimode nonadiabatic vibronic-coupling Hamiltonians, in which diagonal terms are treated in harmonic approximation, while off-diagonal interstate couplings are assumed to be coordinate independent. The approach is based on the Thermo Field Dynamics (TFD) representation of quantum mechanics and tensor-train (TT) machinery for efficient numerical simulation of quantum evolution of systems with many degrees of freedom. The developed TFD-TT approach is applied to the calculation of time- and frequency-resolved fluorescence spectra of the Fenna-Matthews-Olson (FMO) antenna complex at room temperature taking into account finite time-frequency resolution in fluorescence detection, orientational averaging, and static disorder.

Molecular excited states through a machine learning lens

Theoretical simulations of electronic excitations and associated processes in molecules are indispensable for fundamental research and technological innovations. However, such simulations are notoriously challenging to perform with quantum mechanical methods. Advances in machine learning open many new avenues for assisting molecular excited-state simulations. In this Review, we track such progress, assess the current state of the art and highlight the critical issues to solve in the future. We overview a broad range of machine learning applications in excited-state research, which include the prediction of molecular properties, improvements of quantum mechanical methods for the calculations of excited-state properties and the search for new materials. Machine learning approaches can help us understand hidden factors that influence photo-processes, leading to a better control of such processes and new rules for the design of materials for optoelectronic applications.

Velocity Adjustment in Surface Hopping: Ethylene as a Case Study of the Maximum Error Caused by Direction Choice

The most common surface hopping dynamics algorithms require velocity adjustment after hopping to ensure total-energy conservation. Based on the semiclassical analysis, this adjustment must be made parallel to the nonadiabatic coupling vector's direction. Nevertheless, this direction is not always known, and the common practice has been to adjust the velocity in either the linear momentum or velocity directions. This paper benchmarks surface hopping dynamics of photoexcited ethylene with velocity adjustment in several directions, including those of the nonadiabatic coupling vector, the momentum, and the energy gradient difference. It is shown that differences in time constants and structural evolution fall within the statistical uncertainty of the method considering up to 500 trajectories in each dynamics set, rendering the three approaches statistically equivalent. For larger ensembles beyond 1000 trajectories, significant differences between the results arise, limiting the validity of adjustment in alternative directions. Other possible adjustment directions (velocity, single-state gradients, angular momentum) are evaluated as well. Given the small size of ethylene, the results reported in this paper should be considered an upper limit for the error caused by the choice of the velocity-adjustment direction on surface hopping dynamics.

Modeling Spin-Crossover Dynamics

In this article, we review nonadiabatic molecular dynamics (NAMD) methods for modeling spin-crossover transitions. First, we discuss different representations of electronic states employed in the grid-based and direct NAMD simulations. The nature of interstate couplings in different representations is highlighted, with the main focus on nonadiabatic and spin-orbit couplings. Second, we describe three NAMD methods that have been used to simulate spin-crossover dynamics, including trajectory surface hopping, ab initio multiple spawning, and multiconfiguration time-dependent Hartree. Some aspects of employing different electronic structure methods to obtain information about potential energy surfaces and interstate couplings for NAMD simulations are also discussed. Third, representative applications of NAMD to spin crossovers in molecular systems of different sizes and complexities are highlighted. Finally, we pose several fundamental questions related to spin-dependent processes. These questions should be possible to address with future methodological developments in NAMD.

Validating fewest-switches surface hopping in the presence of laser fields

The capability of fewest-switches surface hopping (FSSH) to describe non-adiabatic dynamics under explicit excitation with external fields is evaluated. Different FSSH parameters are benchmarked against multi-configurational time dependent Hartree (MCTDH) reference calculations using SO₂ and 2-thiocytosine as model, yet realistic, molecular systems. Qualitatively, FSSH is able to reproduce the trends in the MCTDH dynamics with (also without) an explicit external field; however, no set of FSSH parameters is ideal. The adequate treatment of the overcoherence in FSSH is revealed as the driving factor to improve the description of the excitation process with respect to the MCTDH reference. Here, two corrections were tested: the augmented-FSSH (AFSSH) correction and the energy-based decoherence correction. A dependence on the employed basis is detected in AFSSH, performing better when spin-orbit and external laser field couplings are treated as off-diagonal elements instead of projecting them onto the diagonal of the Hamilton operator. In the presence of an electric field, the excited state dynamics was found to depend strongly on the vector used to rescale the kinetic energy along after a transition between surfaces. For SO₂, recurrence of the excited wave packet throughout the duration of the applied laser pulse is observed for laser pulses (>100 fs), resulting in additional interferences missed by FSSH and only visible in variational multi-configurational Gaussian when utilizing a large number of Gaussian basis functions. This feature vanishes when going toward larger molecules, such as 2-thiocytosine, where this effect is barely visible in a laser pulse 200 fs long.

Nonadiabatic Absorption Spectra and Ultrafast Dynamics of DNA and RNA Photoexcited Nucleobases

We have recently proposed a protocol for Quantum Dynamics (QD) calculations, which is based on a parameterisation of Linear Vibronic Coupling (LVC) Hamiltonians with Time Dependent (TD) Density Functional Theory (TD-DFT), and exploits the latest developments in multiconfigurational TD-Hartree methods for an effective wave packet propagation. In this contribution we explore the potentialities of this approach to compute nonadiabatic vibronic spectra and ultrafast dynamics, by applying it to the five nucleobases present in DNA and RNA. For all of them we computed the absorption spectra and the dynamics of ultrafast internal conversion (100 fs timescale), fully coupling the first 2-3 bright states and all the close by dark states, for a total of 6-9 states, and including all the normal coordinates. We adopted two different functionals, CAM-B3LYP and PBE0, and tested the effect of the basis set. Computed spectra are in good agreement with the available experimental data, remarkably improving over pure electronic computations, but also with respect to vibronic spectra obtained neglecting inter-state couplings. Our QD simulations indicate an effective population transfer from the lowest energy bright excited states to the close-lying dark excited states for uracil, thymine and adenine. Dynamics from higher-energy states show an ultrafast depopulation toward the more stable ones. The proposed protocol is sufficiently general and automatic to promise to become useful for widespread applications.

Parameterization of a linear vibronic coupling model with multiconfigurational electronic structure methods to study the quantum dynamics of photoexcited pyrene

With this work, we present a protocol for the parameterization of a Linear Vibronic Coupling (LVC) Hamiltonian for quantum dynamics using highly accurate multiconfigurational electronic structure methods such as RASPT2/RASSCF, combined with a maximum-overlap diabatization technique. Our approach is fully portable and can be applied to many medium-size rigid molecules whose excited state dynamics requires a quantum description. We present our model and discuss the details of the electronic structure calculations needed for the parameterization, analyzing critical situations that could arise in the case of strongly interacting excited states. The protocol was applied to the simulation of the excited state dynamics of the pyrene molecule, starting from either the first or the second bright state (S₂ or S₅). The LVC model was benchmarked against state-of-the-art quantum mechanical calculations with optimizations and energy scans and turned out to be very accurate. The dynamics simulations, performed including all active normal coordinates with the multilayer multiconfigurational time-dependent Hartree method, show good agreement with the available experimental data, endorsing prediction of the excited state mechanism, especially for S₅, whose ultrafast deactivation mechanism was not yet clearly understood.