Assessing the performance of trajectory surface hopping methods: Ultrafast internal conversion in pyrazine

Tracking the Electron Density Changes in Excited States: A Computational Study of Pyrazine

The development of X-ray free-electron lasers has enabled ultrafast X-ray diffraction (XRD) experiments, which are capable of resolving electronic and vibrational transitions and structural changes in molecules or capturing molecular movies. While time-resolved XRD has attracted more attention, the extraction of information from signals is challenging and requires theoretical support. In this work, we combined X-ray scattering theory and a trajectory surface hopping approach to resolve dynamical changes in the electronic structure of photoexcited molecules by studying the time evolution of electron density changes between electronic excited states and ground state. Using the pyrazine molecule as an example, we show that key features of reaction pathways can be identified, enabling the capture of structural changes associated with electronic transitions for a photoexcited molecule.

Interconnection between Polarization-Detected and Population-Detected Signals: Theoretical Results and Ab Initio Simulations

Most of spectroscopic signals are specified by the nonlinear laser-induced polarization. In recent years, population-detection of signals becomes a trend in femtosecond spectroscopy. Polarization-detected (PD) and population-detected signals are fundamentally different, because they are determined by photoinduced processes acting on disparate time scales. In this work, we consider the fluorescence-detected (FD) N-wave-mixing (NWM) signal as a representative example of population-detected signals, derive a rigorous expression for this signal, and discuss its approximate variants suitable for numerical simulations. This leads us to the definition of the phenomenological FD (PFD) signal, which contains as a special case all definitions of FD signals available in the literature. Then we formulate and prove the population-polarization equivalence (PPE) theorem, which states that PFD NWM signals produced by (possibly strong) laser pulses can be evaluated as conventional PD signals in which the effective polarization is determined by the PFD transition dipole moment operator. We use the PPE theorem for the construction of the ab initio protocol for the simulation of PFD 4WM signals. As an example, we calculate electronic two-dimensional (2D) PFD spectra of the gas-phase pyrazine and compare them with the corresponding PD 2D spectra.

Quantum Dynamics from Classical Trajectories

Nonadiabatic molecular dynamics plays an essential role in exploring the time evolution of molecular systems. Various methods have been developed for this study, with varying accuracy and computational cost. One very successful among them is trajectory surface hopping, which propagates nuclei as classical trajectories using forces from a quantum description of the electrons and incorporates nonadiabatic effects through stochastic state changes during each trajectory propagation. A statistical analysis of an ensemble of the independent trajectories recovers the simulated system's behavior. This approach can give good results, but it is known to overlook nuclear quantum effects, leading to inaccurate predictions. Here, we present quantum dynamics from classical trajectories (QDCT), a new protocol to recover the quantum wavepacket from the classical trajectories generated by surface hopping. In this first QDCT implementation, we apply it to recover results at the multiple spawning level from postprocessing surface hopping precomputed trajectories. With a series of examples, we demonstrate QDCT's potential to improve the accuracy of the dynamics, correct decoherence effects, and diagnose problems or increase confidence in surface hopping results. All that comes at virtually no computational cost since no new electronic calculation is required.

On-the-Fly Simulation of Two-Dimensional Fluorescence-Excitation Spectra

Two-dimensional (2D) fluorescence-excitation (2D-FLEX) spectroscopy is a recently proposed nonlinear femtosecond technique for the detection of photoinduced dynamics. The method records a time-resolved fluorescence signal in its excitation- and detection-frequency dependence and hence combines the exclusive detection of excited state dynamics (fluorescence) with signals resolved in both excitation and emission frequencies (2D electronic spectroscopy). In this work, we develop an on-the-fly protocol for the simulation of 2D-FLEX spectra of molecular systems, which is based on interfacing the classical doorway-window representation of spectroscopic responses with trajectory surface hopping simulations. Applying this methodology to gas-phase pyrazine, we show that femtosecond 2D-FLEX spectra can deliver detailed information that is otherwise obtainable via attosecond spectroscopy.

Surface hopping molecular dynamics simulation of ultrafast methyl iodide photodissociation mapped by Coulomb explosion imaging

We present a highly efficient approach to directly and reliably simulate photodissociation followed by Coulomb explosion of methyl iodide. In order to achieve statistical reliability, more than 40 000 trajectories are calculated on accurate potential energy surfaces of both the neutral molecule and the doubly charged cation. Non-adiabatic effects during photodissociation are treated using a Landau-Zener surface hopping algorithm. The simulation is performed analogous to a recent pump-probe experiment using coincident ion momentum imaging [Ziaee et al., Phys. Chem. Chem. Phys., 2023, 25, 9999-10010]. At large pump-probe delays, the simulated delay-dependent kinetic energy release signals show overall good agreement with the experiment, with two major dissociation channels leading to I(2P3/2) and I*(2P1/2) products. At short pump-probe delays, the simulated kinetic energy release differs significantly from the values obtained by a purely Coulombic approximation or a one-dimensional description of the dicationic potential energy surfaces, and shows a clear bifurcation near 12 fs, owing to non-adiabatic transitions through a conical intersection. The proposed approach is particularly suitable and efficient in simulating processes that highly rely on statistics or for identifying rare reaction channels.

Ultrafast Photoinduced Dynamics in 1,3-Cyclohexadiene: A Comparison of Trajectory Surface Hopping Schemes†

Photoinduced nonadiabatic processes play a crucial role in a wide range of disciplines, from fundamental steps in biology to modern applications in advanced materials science. A theoretical understanding of these processes is highly desirable, and trajectory surface hopping (TSH) has proven to be a well-suited framework for a wide range of systems. In this work, we present a comprehensive comparison between two TSH algorithms, the conventional Tully's fewest switches surface hopping (FSSH) scheme and the Landau-Zener surface hopping (LZSH), to study the photoinduced ring-opening of 1,3-cyclohexadiene (CHD) to 1,3,5-hexatriene at the spin-flip time-dependent density functional theory (SF-TDDFT) level of theory. Additionally, we compare our results with a literature study at the extended multistate complete active space second-order perturbation theory method (XMS-CASPT2) level of theory. Our results show that the average population and lifetimes estimated with LZSH using SF-TDDFT are closer to the literature (using multireference methods) than those estimated with FSSH using SF-TDDFT. The latter speaks in favor of applying LZSH in combination with the SF-TDDFT method to study larger and more complex systems such as molecular photoswitches where the CHD molecule acts as a backbone. In addition, we present an implementation of Tully's FSSH algorithm as an extension to the PySurf software package.

A size-consistent multi-state mapping approach to surface hopping

We develop a multi-state generalization of the recently proposed mapping approach to surface hopping (MASH) for the simulation of electronically nonadiabatic dynamics. This new approach extends the original MASH method to be able to treat systems with more than two electronic states. It differs from previous approaches in that it is size consistent and rigorously recovers the original two-state MASH in the appropriate limits. We demonstrate the accuracy of the method by applying it to a series of model systems for which exact benchmark results are available, and we find that the method is well suited to the simulation of photochemical relaxation processes.

Understanding the Photoprocesses in Biological Systems: Need for Accurate Multireference Treatment

Light-matter interaction is crucial to life itself and revolves around many of the central processes in biology. The need for understanding these photochemical and photophysical processes cannot be overemphasized. Interaction of light with biological systems starts with the absorption of light and subsequent phenomena that occur in the excited states of the system. However, excited states are typically difficult to understand within the mean field approximation of quantum chemical methods. Therefore, suitable multireference methods and methodologies have been developed to understand these phenomena. In this Perspective, we will describe a few methods and methodologies suitable for these descriptions and discuss some persisting difficulties.

MLatom Software Ecosystem for Surface Hopping Dynamics in Python with Quantum Mechanical and Machine Learning Methods

We present an open-source MLatom@XACS software ecosystem for on-the-fly surface hopping nonadiabatic dynamics based on the Landau-Zener-Belyaev-Lebedev algorithm. The dynamics can be performed via Python API with a wide range of quantum mechanical (QM) and machine learning (ML) methods, including ab initio QM (CASSCF and ADC(2)), semiempirical QM methods (e.g., AM1, PM3, OMx, and ODMx), and many types of ML potentials (e.g., KREG, ANI, and MACE). Combinations of QM and ML methods can also be used. While the user can build their own combinations, we provide AIQM1, which is based on Δ-learning and can be used out-of-the-box. We showcase how AIQM1 reproduces the isomerization quantum yield of trans-azobenzene at a low cost. We provide example scripts that, in dozens of lines, enable the user to obtain the final population plots by simply providing the initial geometry of a molecule. Thus, those scripts perform geometry optimization, normal mode calculations, initial condition sampling, parallel trajectories propagation, population analysis, and final result plotting. Given the capabilities of MLatom to be used for training different ML models, this ecosystem can be seamlessly integrated into the protocols building ML models for nonadiabatic dynamics. In the future, a deeper and more efficient integration of MLatom with Newton-X will enable a vast range of functionalities for surface hopping dynamics, such as fewest-switches surface hopping, to facilitate similar workflows via the Python API.

Exploring the Mechanisms behind Non-aromatic Fluorescence with the Density Functional Tight Binding Method

Recent experimental findings reveal nonconventional fluorescence emission in biological systems devoid of conjugated bonds or aromatic compounds, termed non-aromatic fluorescence (NAF). This phenomenon is exclusive to aggregated or solid states and remains absent in monomeric solutions. Previous studies focused on small model systems in vacuum show that the carbonyl stretching mode along with strong interaction of short hydrogen bonds (SHBs) remains the primary vibrational mode explaining NAF in these systems. In order to simulate larger model systems taking into account the effects of the surrounding environment, in this work we propose using the density functional tight-binding (DFTB) method in combination with non-adiabatic molecular dynamics (NAMD) and the mixed quantum/molecular mechanics (QM/MM) approach. We investigate the mechanism behind NAF in the crystal structure of l-pyroglutamine-ammonium, comparing it with the related nonfluorescent amino acid l-glutamine. Our results extend our previous findings to more realistic systems, demonstrating the efficiency and robustness of the proposed DFTB method in the context of NAMD in biological systems. Furthermore, due to its inherent low computational cost, this method allows for a better sampling of the nonradiative events at the conical intersection which is crucial for a complete understanding of this phenomenon. Beyond contributing to the ongoing exploration of NAF, this work paves the way for future application of this method in more complex biological systems such as amyloid aggregates, biomaterials, and non-aromatic proteins.

The Role of Aqueous Solvation on the Intersystem Crossing of Nitrophenols

The photochemistry of nitrophenols is a source of smog as nitrous acid is formed from their photolysis. Nevertheless, computational studies of the photochemistry of these widespread toxic molecules are scarce. In this work, the initial photodeactivation of ortho-nitrophenol and para-nitrophenol is modeled, both in gas phase and in aqueous solution to simulate atmospheric and aerosol environments. A large number of excited states, six for ortho-nitrophenol and 11 for para-nitrophenol, have been included and were all populated during the decay. Moreover, periodic time-dependent density functional theory (TDDFT) is used for both the explicitly included solvent and the solute. A comparison to periodic QM/MM (TDDFT/MM), with electrostatic embedding, is made, showing notable differences between the decays of solvated nitrophenols simulated with QM/MM and full (TD)DFT. A reduced intersystem crossing in aqueous solution could be observed thanks to the surface hopping approach using explicit, periodic TDDFT solvation including spin-orbit couplings.

Artificial-Intelligence-Enhanced On-the-Fly Simulation of Nonlinear Time-Resolved Spectra

Time-resolved spectroscopy is an important tool for unraveling the minute details of structural changes in molecules of biological and technological significance. The nonlinear femtosecond signals detected for such systems must be interpreted, but it is a challenging task for which theoretical simulations are often indispensable. Accurate simulations of transient absorption or two-dimensional electronic spectra are, however, computationally very expensive, prohibiting the wider adoption of existing first-principles methods. Here, we report an artificial-intelligence-enhanced protocol to drastically reduce the computational cost of simulating nonlinear time-resolved electronic spectra, which makes such simulations affordable for polyatomic molecules of increasing size. The protocol is based on the doorway-window approach for the on-the-fly surface-hopping simulations. We show its applicability for the prototypical molecule of pyrazine for which it produces spectra with high precision with respect to ab initio reference while cutting the computational cost by at least 95% compared to pure first-principles simulations.

Which Electronic Structure Method to Choose in Trajectory Surface Hopping Dynamics Simulations? Azomethane as a Case Study

Nonadiabatic dynamics simulations have become a standard approach to explore photochemical reactions. Such simulations require underlying potential energy surfaces and couplings between them, calculated at a chosen level of theory, yet this aspect is rarely assessed. Here, in combination with the popular trajectory surface hopping dynamics method, we use a high-accuracy XMS-CASPT2 electronic structure level as a benchmark for assessing the performances of various post-Hartree-Fock methods (namely, CIS, ADC(2), CC2, and CASSCF) and exchange-correlation functionals (PBE, PBE0, and CAM-B3LYP) in a TD-DFT/TDA context, using the isomerization around a double bond as test case. Different relaxation pathways are identified, and the ability of the different methods to reproduce their relative importance and time scale is discussed. The results show that multireference electronic structure methods should be preferred, when studying nonadiabatic decay between excited and ground states. If not affordable, TD-DFT with TDA and hybrid functionals and ADC(2) are efficient alternatives but overestimate the nonradiative decay yield and thus may miss deexcitation pathways.

Benchmarking non-adiabatic quantum dynamics using the molecular Tully models

On-the-fly non-adiabatic dynamics methods are becoming more important as tools to characterise the time evolution of a system after absorbing light. These methods, which calculate quantities such as state energies, gradients and interstate couplings at every time step, circumvent the requirement for pre-computed potential energy surfaces. There are a number of different algorithms used, the most common being Tully Surface Hopping (TSH), but all are approximate solutions to the time-dependent Schrödinger equation and benchmarking is required to understand their accuracy and performance. For this, a common set of systems and observables are required to compare them. In this work, we validate the on-the-fly direct dynamics variational multi-configuration Gaussian (DD-vMCG) method using three molecular systems recently suggested by Ibele and Curchod as molecular versions of the Tully model systems used to test one-dimensional non-adiabatic behaviour [Ibele et al., Phys. Chem. Chem. Phys. 2020, 22, 15183-15196]. Parametrised linear vibronic potential energy surfaces for each of the systems were also tested and compared to on-the-fly results. The molecules, which we term the Ibele-Curchod models, are ethene, DMABN and fulvene and the authors used them to test and compare several versions of the Ab Initio Multiple Spawning (AIMS) method alongside TSH. The three systems present different deactivation pathways after excitation to their ππ* bright states. When comparing DD-vMCG to AIMS and TSH, we obtain crucial differences in some cases, for which an explanation is provided by the classical nature and the chosen initial conditions of the TSH simulations.

Prediction of fluorescence quantum yields using the extended thawed Gaussian approximation

Spontaneous emission and internal conversion rates are calculated within harmonic approximations and compared to the results obtained within the semi-classical extended thawed Gaussian approximation (ETGA). This is the first application of the ETGA in the calculation of internal conversion and emission rates for real molecular systems, namely, formaldehyde, fluorobenzene, azulene, and a dicyano-squaraine dye. The viability of the models as black-box tools for prediction of spontaneous emission and internal conversion rates is assessed. All calculations were done using a consistent protocol in order to investigate how different methods perform without previous experimental knowledge using density functional theory (DFT) and time-dependent DFT (TD-DFT) with B3LYP, PBE0, ωB97XD, and CAM-B3LYP functionals. Contrasting the results with experimental data shows that there are further improvements required before theoretical predictions of emission and internal conversion rates can be used as reliable indicators for the photo-luminescence properties of molecules. We find that the ETGA performs rather similar to the vertical harmonical model. Including anharmonicities in the calculation of internal conversion rates has a moderate effect on the quantitative results in the studied systems. The emission rates are fairly stable with respect to computational parameters, but the internal conversion rate reveals itself to be highly dependent on the choice of the spectral line shape function, particularly the width of the Lorentzian function, associated with homogeneous broadening.

Tuning UV Pump X-ray Probe Spectroscopy on the Nitrogen K Edge Reveals the Radiationless Relaxation of Pyrazine: Ab Initio Simulations Using the Quasiclassical Doorway-Window Approximation

Transient absorption UV pump X-ray probe spectroscopy has been established as a versatile technique for the exploration of ultrafast photoinduced dynamics in valence-excited states. In this work, an ab initio theoretical framework for the simulation of time-resolved UV pump X-ray probe spectra is presented. The method is based on the description of the radiation-matter interaction in the classical doorway-window approximation and a surface-hopping algorithm for the nonadiabatic nuclear excited-state dynamics. Using the second-order algebraic-diagrammatic construction scheme for excited states, UV pump X-ray probe signals were simulated for the carbon and nitrogen K edges of pyrazine, assuming a duration of 5 fs of the UV pump and X-ray probe pulses. It is predicted that spectra measured at the nitrogen K edge carry much richer information about the ultrafast nonadiabatic dynamics in the valence-excited states of pyrazine than those measured at the carbon K edge.

Ab initio simulation of peak evolutions and beating maps for electronic two-dimensional signals of a polyatomic chromophore

By employing the doorway-window (DW) on-the-fly simulation protocol, we performed ab initio simulations of peak evolutions and beating maps of electronic two-dimensional (2D) spectra of a polyatomic molecule in the gas phase. As the system under study, we chose pyrazine, which is a paradigmatic example of photodynamics dominated by conical intersections (CIs). From the technical perspective, we demonstrate that the DW protocol is a numerically efficient methodology suitable for simulations of 2D spectra for a wide range of excitation/detection frequencies and population times. From the information content perspective, we show that peak evolutions and beating maps not only reveal timescales of transitions through CIs but also pinpoint the most relevant coupling and tuning modes active at these CIs.

Internal conversion rates from the extended thawed Gaussian approximation: Theory and validation

The theoretical prediction of the rates of nonradiative processes in molecules is fundamental in assessing their emissive properties. In this context, global harmonic models have been widely used to simulate vibronic spectra as well as internal conversion rates and to predict photoluminescence quantum yields. However, these simplified models suffer from the limitations that are inherent to the harmonic approximation and can have a severe effect on the calculated internal conversion rates. Therefore, the development of more accurate semiclassical methods is highly desirable. Here, we introduce a procedure for the calculation of nonradiative rates in the framework of the time-dependent semi-classical Extended Thawed Gaussian Approximation (ETGA). We systematically investigate the performance of the ETGA method by comparing it to the adiabatic and vertical harmonic methods, which belong to the class of widely used global harmonic models. Its performance is tested in potentials that cannot be treated adequately by global harmonic models, beginning with Morse potentials of varying anharmonicity followed by a double well potential. The calculated radiative and nonradiative internal conversion rates are compared to reference values based on exact quantum dynamics. We find that the ETGA has the capability to predict internal conversion rates in anharmonic systems with an appreciable energy gap, whereas the global harmonic models prove to be insufficient.

Machine learning of double-valued nonadiabatic coupling vectors around conical intersections

In recent years, machine learning has had an enormous success in fitting ab initio potential-energy surfaces to enable efficient simulations of molecules in their ground electronic state. In order to extend this approach to excited-state dynamics, one must not only learn the potentials but also nonadiabatic coupling vectors (NACs). There is a particular difficulty in learning NACs in systems that exhibit conical intersections, as due to the geometric-phase effect, the NACs may be double-valued and are, thus, not suitable as training data for standard machine-learning techniques. In this work, we introduce a set of auxiliary single-valued functions from which the NACs can be reconstructed, thus enabling a reliable machine-learning approach.

Equation-of-Motion Methods for the Calculation of Femtosecond Time-Resolved 4-Wave-Mixing and N-Wave-Mixing Signals

Femtosecond nonlinear spectroscopy is the main tool for the time-resolved detection of photophysical and photochemical processes. Since most systems of chemical interest are rather complex, theoretical support is indispensable for the extraction of the intrinsic system dynamics from the detected spectroscopic responses. There exist two alternative theoretical formalisms for the calculation of spectroscopic signals, the nonlinear response-function (NRF) approach and the spectroscopic equation-of-motion (EOM) approach. In the NRF formalism, the system-field interaction is assumed to be sufficiently weak and is treated in lowest-order perturbation theory for each laser pulse interacting with the sample. The conceptual alternative to the NRF method is the extraction of the spectroscopic signals from the solutions of quantum mechanical, semiclassical, or quasiclassical EOMs which govern the time evolution of the material system interacting with the radiation field of the laser pulses. The NRF formalism and its applications to a broad range of material systems and spectroscopic signals have been comprehensively reviewed in the literature. This article provides a detailed review of the suite of EOM methods, including applications to 4-wave-mixing and N-wave-mixing signals detected with weak or strong fields. Under certain circumstances, the spectroscopic EOM methods may be more efficient than the NRF method for the computation of various nonlinear spectroscopic signals.

Trajectory Surface Hopping for a Polarizable Embedding QM/MM Formulation

We present the implementation of trajectory surface-hopping nonadiabatic dynamics for a polarizable embedding QM/MM formulation. Time-dependent density functional theory was used at the quantum mechanical level of theory, whereas the molecular mechanics description involved the polarizable AMOEBA force field. This implementation has been obtained by integrating the surface-hopping program Newton-X NS with an interface between the Gaussian 16 and the Tinker suites of codes to calculate QM/AMOEBA energies and forces. The implementation has been tested on a photoinduced electron-driven proton-transfer reaction involving pyrimidine and a hydrogen-bonded water surrounded by a small cluster of water molecules and within a large water droplet.

Ab initio trajectory surface-hopping dynamics studies of excited-state proton-coupled electron transfer reactions in trianisoleheptazine-phenol complexes

The excited-state proton-coupled electron-transfer (PCET) reaction in hydrogen-bonded complexes of trianisoleheptazine (TAHz), a chromophore related to polymeric carbon nitrides widely used in hydrogen-evolution photocatalysis, with several phenol derivatives were recently studied by Schlenker and coworkers with time-resolved photoluminescence quenching and pump-probe experiments. A pronounced dependence of the PCET reactivity on the electron-donating/electron-withdrawing character of the substituents on phenol was found, with indications of a barrierless or nearly barrierless PCET reaction for the most strongly electron-donating substituent, methoxy. In the present work, the excited-state PCET dynamics was explored with first-principles nonadiabatic dynamics simulations using the TDDFT/ωB97X-D electronic-structure model for two selected complexes, TAHz-phenol and TAHz-methoxyphenol. The qualitative reliability of the TDDFT/ωB97X-D electronic-structure model was assessed by extensive benchmarking of excitation energies and potential-energy profiles against a wave-function-based ab initio method, the algebraic-diagrammatic construction of second order (ADC(2)). The nonadiabatic dynamics simulations provide temporally and structurally resolved insights into paradigmatic PCET reactions in TAHz-phenol complexes. The radiationless relaxation of the photoexcited bright ¹ππ* state to the long-lived dark S₁ state of TAHz occurs in less than 100 fs. The ensuing PCET reaction on the adiabatic S₁ surface is faster in TAHz-methoxyphenol complexes than in TAHz-phenol complexes due to a lower H-atom-transfer barrier, as observed in the experiments. The relaxation of the complexes to the electronic ground state is found to occur exclusively via PCET within the 250 fs time window covered by the present simulations, confirming the essential role of the hydrogen bond for the fluorescence quenching process. The absolute values of the computed PCET time constants are significantly shorter than those extracted from time-resolved photoluminescence measurements for mixtures of TAHz with phenolic substrates in toluene. The possible origins of this discrepancy are discussed.

Mixed-quantum-classical or fully-quantized dynamics? A unified code to compare methods

Three methods for non-adiabatic dynamics are compared to highlight their capabilities. Multi-configurational time-dependent Hartree is a full grid-based solution to the time-dependent Schrödinger equation, variational multi-configurational Gaussian (vMCG) uses a less flexible but unrestricted Gaussian wavepacket basis, and trajectory surface hopping (TSH) replaces the nuclear wavepacket with a swarm of classical trajectories. Calculations with all methods using a model Hamiltonian were performed. The vMCG and TSH were also then run in a direct dynamics mode, with the potential energy surfaces calculated on-the-fly using quantum chemistry calculations. All dynamics calculations used the Quantics package, with the TSH calculations using a new interface to a surface hopping code. A novel approach to calculate adiabatic populations from grid-based quantum dynamics using a time-dependent discrete variable representation is presented, allowing a proper comparison of methods. This article is part of the theme issue 'Chemistry without the Born-Oppenheimer approximation'.

Water Oxidation and Hydrogen Evolution with Organic Photooxidants: A Theoretical Perspective

In this Perspective, we discuss a novel water-splitting scenario, namely the direct oxidation of water molecules by organic photooxidants in hydrogen-bonded chromophore-water complexes. In comparison with the established scenario of semiconductor-based water splitting, the distance of electron transfer processes is thereby reduced from mesoscopic scales to the Ångström scale, and the time scale is reduced from milliseconds to femtoseconds, which suppresses competing loss processes. The concept is illustrated by computational studies for the heptazine-H₂O complex. The excited-state landscape of this complex has been characterized with ab initio electronic-structure methods and the proton-coupled electron-transfer dynamics has been explored with nonadiabatic dynamics simulations. A unique feature of the heptazine chromophore is the existence of a low-lying and exceptionally long-lived ¹ππ* state in which a substantial part of the photon energy can be stored for hundreds of nanoseconds and is available for the oxidation of water molecules. The calculations reveal that the absorption spectra and the photochemical functionalities of heptazine chromophores can be systematically tailored by chemical substitution. The options of harvesting hydrogen and the problems posed by the high reactivity of OH radicals are discussed.

Fewest switches surface hopping with Baeck-An couplings

In the Baeck-An (BA) approximation, first-order nonadiabatic coupling vectors are given in terms of adiabatic energy gaps and the second derivative of the gaps with respect to the coupling coordinate. In this paper, a time-dependent (TD) BA approximation is derived, where the couplings are computed from the energy gaps and their second time-derivatives. TD-BA couplings can be directly used in fewest switches surface hopping, enabling nonadiabatic dynamics with any electronic structure methods able to provide excitation energies and energy gradients. Test results of surface hopping with TD-BA couplings for ethylene and fulvene show that the TD-BA approximation delivers a qualitatively correct picture of the dynamics and a semiquantitative agreement with reference data computed with exact couplings. Nevertheless, TD-BA does not perform well in situations conjugating strong couplings and small velocities. Considered the uncertainties in the method, TD-BA couplings could be a competitive approach for inexpensive, exploratory dynamics with a small trajectories ensemble. We also assessed the potential use of TD-BA couplings for surface hopping dynamics with time-dependent density functional theory (TDDFT), but the results are not encouraging due to singlet instabilities near the crossing seam with the ground state.

Fewest switches surface hopping with Baeck-An couplings

In the Baeck-An (BA) approximation, first-order nonadiabatic coupling vectors are given in terms of adiabatic energy gaps and the second derivative of the gaps with respect to the coupling coordinate. In this paper, a time-dependent (TD) BA approximation is derived, where the couplings are computed from the energy gaps and their second time-derivatives. TD-BA couplings can be directly used in fewest switches surface hopping, enabling nonadiabatic dynamics with any electronic structure methods able to provide excitation energies and energy gradients. Test results of surface hopping with TD-BA couplings for ethylene and fulvene show that the TD-BA approximation delivers a qualitatively correct picture of the dynamics and a semiquantitative agreement with reference data computed with exact couplings. Nevertheless, TD-BA does not perform well in situations conjugating strong couplings and small velocities. Considered the uncertainties in the method, TD-BA couplings could be a competitive approach for inexpensive, exploratory dynamics with a small trajectories ensemble. We also assessed the potential use of TD-BA couplings for surface hopping dynamics with time-dependent density functional theory (TDDFT), but the results are not encouraging due to singlet instabilities near the crossing seam with the ground state.

Diabatic States of Molecules

Quantitative simulations of electronically nonadiabatic molecular processes require both accurate dynamics algorithms and accurate electronic structure information. Direct semiclassical nonadiabatic dynamics is expensive due to the high cost of electronic structure calculations, and hence it is limited to small systems, limited ensemble averaging, ultrafast processes, and/or electronic structure methods that are only semiquantitatively accurate. The cost of dynamics calculations can be made manageable if analytic fits are made to the electronic structure data, and such fits are most conveniently carried out in a diabatic representation because the surfaces are smooth and the couplings between states are smooth scalar functions. Diabatic representations, unlike the adiabatic ones produced by most electronic structure methods, are not unique, and finding suitable diabatic representations often involves time-consuming nonsystematic diabatization steps. The biggest drawback of using diabatic bases is that it can require large amounts of effort to perform a globally consistent diabatization, and one of our goals has been to develop methods to do this efficiently and automatically. In this Feature Article, we introduce the mathematical framework of diabatic representations, and we discuss diabatization methods, including adiabatic-to-diabatic transformations and recent progress toward the goal of automatization.

Ab Initio Quasiclassical Simulation of Femtosecond Time-Resolved Two-Dimensional Electronic Spectra of Pyrazine

Two-dimensional (2D) electronic spectroscopy is a powerful nonlinear technique which provides spectroscopic information on two frequency axes as well as dynamical information as a function of the so-called waiting time. Herein, an ab initio theoretical framework for the simulation of electronic 2D spectra has been developed. The method is based on the classical approximation to the doorway-window representation of three-pulse photon-echo signals and the description of nuclear motion by classical trajectories. Nonadiabatic effects are taken into account by a trajectory surface-hopping algorithm. 2D electronic spectra were simulated with ab initio on-the-fly trajectory calculations using the ADC(2) electronic-structure method for the pyrazine molecule, which is a benchmark system for ultrafast radiationless decay through conical intersections. It is demonstrated that 2D spectroscopy with subfemtosecond UV pulses can provide unprecedented detailed information on the ultrafast photodynamics of polyatomic molecules.

Ab Initio Nonadiabatic Surface-Hopping Trajectory Simulations of Photocatalytic Water Oxidation and Hydrogen Evolution with the Heptazine Chromophore

In the past decade, polymeric carbon nitrides consisting of heptazine (Hz) building blocks emerged as highly promising materials for photocatalytic hydrogen evolution from water or sacrificial electron donors with near-ultraviolet light. However, the complexity of these materials and their poor characterization at the atomic level are detrimental to the unraveling of the detailed mechanisms of the reactions leading to hydrogen evolution. Recently, it has been shown that a derivative of the Hz molecule, trianisole-heptazine (TAHz), is a potent photobase, which readily oxidizes various derivatives of phenol and even water by an excited-state proton-coupled electron-transfer (PCET) reaction. Energy profiles along minimum-energy reaction paths and relaxed PCET potential-energy surfaces, which previously were computed with ab initio electronic-structure methods for complexes of Hz and TAHz with protic substrates, led to qualitative insights. To obtain more quantitative insight, reaction dynamics simulations are required. In the present work, the time scales of the electron and proton transfer processes and the branching ratios of competing channels were explored with ab initio on-the-fly quasiclassical surface-hopping trajectory simulations for the hydrogen-bonded Hz-H₂O complex. By the analysis of representative trajectories, detailed insight into the interplay of various nonadiabatic electronic transitions, electron transfer, proton transfer, and vibrational energy relaxation is obtained. The HzH radicals which are formed by the photoreduction of Hz can disproportionate to Hz and HzH₂ in an exothermic reaction with a low reaction barrier. The time scales and branching ratios of competing channels in H-atom photodetachment from the HzH₂ molecule also were explored with ab initio surface-hopping simulations. These results delineate for the first time a quantitatively supported scenario of water oxidation and hydrogen evolution with a molecular carbon nitride photocatalyst.

X-ray transient absorption reveals the 1Au (nπ*) state of pyrazine in electronic relaxation

Electronic relaxation in organic chromophores often proceeds via states not directly accessible by photoexcitation. We report on the photoinduced dynamics of pyrazine that involves such states, excited by a 267 nm laser and probed with X-ray transient absorption spectroscopy in a table-top setup. In addition to the previously characterized ¹B_2u (ππ*) (S₂) and ¹B_3u (nπ*) (S₁) states, the participation of the optically dark ¹A_u (nπ*) state is assigned by a combination of experimental X-ray core-to-valence spectroscopy, electronic structure calculations, nonadiabatic dynamics simulations, and X-ray spectral computations. Despite ¹A_u (nπ*) and ¹B_3u (nπ*) states having similar energies at relaxed geometry, their X-ray absorption spectra differ largely in transition energy and oscillator strength. The ¹A_u (nπ*) state is populated in 200 ± 50 femtoseconds after electronic excitation and plays a key role in the relaxation of pyrazine to the ground state.

Combined Surface-Hopping, Dyson Orbital, and B-Spline Approach for the Computation of Time-Resolved Photoelectron Spectroscopy Signals: The Internal Conversion in Pyrazine

A computational protocol for simulating time-resolved photoelectron signals of medium-sized molecules is presented. The procedure is based on a trajectory surface-hopping description of the excited-state dynamics and a combined Dyson orbital and multicenter B-spline approach for the computation of cross sections and asymmetry parameters. The accuracy of the procedure has been illustrated for the case of ultrafast internal conversion of gas-phase pyrazine excited to the ¹B_2u(ππ*) state. The simulated spectra and the asymmetry map are compared to the experimental data, and a very good agreement was obtained without applying any energy-dependent rescaling or broadening. An interesting side result of this work is the finding that the signature of the ¹A_u(nπ*) state is indistinguishable from that of the ¹B_3u(nπ*) state in the time-resolved photoelectron spectrum. By locating four symmetrically equivalent minima on the lowest-excited (S₁) adiabatic potential energy surface of pyrazine, we revealed the strong vibronic coupling of the ¹A_u(nπ*) and ¹B_3u(nπ*) states near the S₁ ← S₀ band origin.

Identification of an ultrafast internal conversion pathway of pyrazine by time-resolved vacuum ultraviolet photoelectron spectrum simulations

The internal conversion from the optically bright S₂ (¹B_2u, ππ*) state to the dark S₁ (¹B_3u, nπ*) state in pyrazine is a standard benchmark for experimental and theoretical studies on ultrafast radiationless decay. Since 2008, a few theoretical groups have suggested significant contributions of other dark states S₃ (¹A_u, nπ*) and S₄ (¹B_2g, nπ*) to the decay of S₂. We have previously reported the results of nuclear wave packet simulations [Kanno et al., Phys. Chem. Chem. Phys. 17, 2012 (2015)] and photoelectron spectrum calculations [Mignolet et al., Chem. Phys. 515, 704 (2018)] that support the conventional two-state picture. In this article, the two different approaches, i.e., wave packet simulation and photoelectron spectrum calculation, are combined: We computed the time-resolved vacuum ultraviolet photoelectron spectrum and photoelectron angular distribution for the ionization of the wave packet transferred from S₂ to S₁. The present results reproduce almost all the characteristic features of the corresponding experimental time-resolved spectrum [Horio et al., J. Chem. Phys. 145, 044306 (2016)], such as a rapid change from a three-band to two-band structure. This further supports the existence and character of the widely accepted pathway (S₂ → S₁) of ultrafast internal conversion in pyrazine.

Spiers Memorial Lecture: Introduction to ultrafast spectroscopy and imaging of photochemical reactions

A brief overview is presented on ultrafast spectroscopy and imaging of photochemical reactions by highlighting several experimental studies reported in the last five years. A particular focus is placed on new experiments performed using high-order harmonic generation, X-ray free electron lasers, and relativistic electron beams. Exploration of fundamental chemical reaction dynamics using these advanced experimental methodologies is in an early stage, and exciting new research opportunities await in this rapidly expanding and advancing research field. At the same time, there is no experimental methodology that provides all aspects of the electronic and structural dynamics in a single experiment, and investigations using different methodologies with various perspectives need to be considered in a comprehensive manner.

Trajectory surface-hopping photoinduced dynamics from Rydberg states of trimethylamine

We present a computational study on nonadiabatic excited-state dynamics initiated from the 3p Rydberg states of trimethylamine (TMA). We utilise a methodology based on full-dimensional (39 D) trajectory surface-hopping (TSH) simulations, in which propagation is carried out on on-the-fly density functional theory (DFT)/time-dependent DFT (TD-DFT) potentials. Both our electronic structure benchmarks to high-level ab initio methods (EOM-CCSD, CASPT2) and TSH simulations demonstrate high-accuracy of the applied CAM-B3LYP functional for the description of Rydberg excited states. Based on our excited-state simulations, we construct the following mechanistic picture: when pumped resonantly to the 3p Rydberg manifold, TMA coherently vibrates along the planarisation mode with a period of 104 fs and an exponential coherence decay time constant of 240 fs. Nonadiabatic dynamics occur on a faster (∼1 ps) and a slower (∼3 ps) timescale, along the N-C stretching mode by mixing with a dissociative σN-C* state. As a minor relaxation channel, 3p → 3s internal conversion occurs via branching at the σN-C*/3s intersection. We find that photodissociaton is hardly observable within 3 ps (1%), which is a failure of the range-separated hybrid CAM-B3LYP functional, as a consequence of its static electron correlation deficiency at long range. In contrast, pure DFT (GGA-BLYP) provides an accurate long-range description (19% dissociation yield), also supported by comparison to recent ultrafast experiments, even if the Rydberg state energies are significantly underestimated (>1 eV). Finally, we reveal the crucial role of vibrational coherence and energy transfer from the planarisation mode for N-C bond activation and resulting nonadiabatic dynamics. The present work illustrates the importance of nuclear-electronic coupling for excited-state dynamics and branching at conical intersections.

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.

A fast and robust trajectory surface hopping method: Application to the intermolecular photodissociation of a carbon dioxide dimer cation (CO2)2. J Chem Phys 2021;154:164108. [PMID: 33940846 DOI: 10.1063/5.0045402]

Our recently developed trajectory surface hopping method uses numerical time derivatives of adiabatic potential gradients to estimate the nonadiabatic transition probability and the hopping direction. To demonstrate the practicality of the novel method, we applied it to the intermolecular photodissociation of a carbon dioxide dimer cation (CO₂)₂ ⁺. Our simulations reproduced the measured velocity distribution of CO₂ ⁺ fragments consisting of two (fast and slow) components and revealed that nonadiabatic transitions occur promptly toward the electronic ground state regardless of the fragment velocity. The structure of (CO₂)₂ ⁺ at optical excitation governs the fate of subsequent nonadiabatic dynamics leading to a fast or slow dissociation. Our method gave similar results to the fewest switches algorithm at lower computational expense. Our fast and robust surface hopping method is promising for the investigation of nonadiabatic dynamics in large and complex systems.

Simulation of UV absorption spectra and relaxation dynamics of uracil and uracil-water clusters

Despite many studies, the mechanisms of nonradiative relaxation of uracil in the gas phase and in aqueous solution are still not fully resolved. Here we combine theoretical UV absorption spectroscopy with nonadiabatic dynamics simulations to identify the photophysical mechanisms that can give rise to experimentally observed decay time constants. We first compute and theoretically assign the electronic spectra of uracil using the second-order algebraic-diagrammatic-construction (ADC(2)) method. The obtained electronic states, their energy differences and state-specific solvation effects are the prerequisites for understanding the photodynamics. We then use nonadiabatic trajectory-surface-hopping dynamics simulations to investigate the photoinduced dynamics of uracil and uracil-water clusters. In contrast to previous studies, we found that a single mechanism - the ethylenic twist around the C[double bond, length as m-dash]C bond - is responsible for the ultrafast component of the nonradiative decay, both in the gas phase and in solution. Very good agreement with the experimentally determined ultrashort decay time constants is obtained.

Exploring the effects of quantum decoherence on the excited-state dynamics of molecular systems

Fewest-switches surface hopping (FSSH) is employed in order to investigate the nonadiabatic excited-state dynamics of thiophene and related compounds and hence to establish a connection between the electronic system, the critical points in configuration space and the deactivation dynamics. The potential-energy surfaces of the studied molecules were calculated with complete active space self-consistent field and time-dependent density-functional theory. They are analyzed thoroughly to locate and optimize minimum-energy conical intersections, which are essential to the dynamics of the system. The influence of decoherence on the dynamics is examined by employing different decoherence schemes. We find that irrespective of the employed decoherence algorithm, the population dynamics of thiophene give results which are sound with the expectations grounded on the analysis of the potential-energy surface. A more detailed look at single trajectories as well as on the excited-state lifetimes, however, reveals a substantial dependence on how decoherence is accounted for. In order to connect these findings, we describe how ensemble averaging cures some of the overcoherence problems of uncorrected FSSH. Eventually, we identify carbon–sulfur bond cleavage as a common feature accompanying electronic transitions between different states in the simulations of all thiophene-related compounds studied in this work, which is of interest due to their relevance in organic photovoltaics.

Ab Initio Surface-Hopping Simulation of Femtosecond Transient-Absorption Pump-Probe Signals of Nonadiabatic Excited-State Dynamics Using the Doorway-Window Representation

An ab initio theoretical framework for the simulation of femtosecond time-resolved transient absorption (TA) pump-probe (PP) spectra with quasi-classical trajectories is presented. The simulations are based on the classical approximation to the doorway-window (DW) representation of third-order four-wave-mixing signals. The DW formula accounts for the finite duration and spectral shape of the pump and probe pulses. In the classical DW formalism, classical trajectories are stochastically sampled from a positive definite doorway distribution, and the signals are evaluated by averaging over a positive definite window distribution. Nonadiabatic excited-state dynamics is described by a stochastic surface-hopping algorithm. The method has been implemented for the pyrazine molecule with the second-order algebraic-diagrammatic construction (ADC(2)) ab initio electronic-structure method. The methodology is illustrated by ab initio simulations of the ground-state bleach, stimulated emission, and excited-state absorption contributions to the TA PP spectrum of gas-phase pyrazine.

PySurf: A Framework for Database Accelerated Direct Dynamics

The greatest restriction to the theoretical study of the dynamics of photoinduced processes is computationally expensive electronic structure calculations. Machine learning algorithms have the potential to reduce the number of these computations significantly. Here, PySurf is introduced as an innovative code framework, which is specifically designed for rapid prototyping and development tasks for data science applications in computational chemistry. It comes with powerful Plugin and Workflow engines, which allows intuitive customization for individual tasks. Data is automatically stored through the database framework, which enables additional interpolation of properties in previously evaluated regions of the conformational space. To illustrate the potential of the framework, a code for nonadiabatic surface hopping simulations based on the Landau-Zener algorithm is presented here. Deriving gradients from the interpolated potential energy surfaces allows for full-dimensional nonadiabatic surface hopping simulations using only adiabatic energies (energy only). Simulations of a pyrazine model and ab initio-based calculations of the SO2 molecule show that energy-only calculations with PySurf are able to correctly predict the nonadiabatic dynamics of these prototype systems. The results reveal the degree of sophistication, which can be achieved by the database accelerated energy-only surface hopping simulations being competitive to commonly used semiclassical approaches.

Photoinduced water oxidation in pyrimidine-water clusters: a combined experimental and theoretical study

The photocatalytic oxidation of water with molecular or polymeric N-heterocyclic chromophores is a topic of high current interest in the context of artificial photosynthesis, that is, the conversion of solar energy to clean fuels. Hydrogen-bonded clusters of N-heterocycles with water molecules in a molecular beam are simple model systems for which the basic mechanisms of photochemical water oxidation can be studied under well-defined conditions. In this work, we explored the photoinduced H-atom transfer reaction in pyrimidine-water clusters yielding pyrimidinyl and hydroxyl radicals with laser spectroscopy, mass spectrometry and trajectory-based ab initio molecular dynamics simulations. The oxidation of water by photoexcited pyrimidine is unequivocally confirmed by the detection of the pyrimidinyl radical. The dynamics simulations provide information on the time scales and branching ratios of the reaction. While relaxation to local minima of the S₁ potential-energy surface is the dominant reaction channel, the H-atom transfer reaction occurs on ultrafast time scales (faster than about 100 fs) with a branching ratio of a few percent.

Photochemistry of Amylene Double Bond in Clusters on Free Argon Nanoparticles

We have investigated reactivity of double bond in 2-methyl-2-butene (also trimethylethylene or amylene) in the excited and ionized states. In a combined experimental and theoretical study, we focused on both the intermolecular and intramolecular reactions. In a molecular beam experiment, we have sequentially picked up several amylene molecules on the surface of argon nanoparticles Ar_M, M̅ ≈ 90, acting as a cold support. Ionization with 70 eV electrons yields mass spectra strongly dominated by amylene cluster ions Am(Am)_n⁺. Interestingly, upon multiphoton ionization with 193 nm (6.4 eV) photons, a new strong fragment series appears in the spectra, nominally corresponding to an addition of two carbon atoms, i.e., (Am)_nC₂⁺. This difference between electron and photoionization suggests a reaction in an excited state of amylene with a neighboring amylene molecule. We used techniques of nonadiabatic molecular dynamics to study the reactivity of amylene molecules both in the excited and in ionized states. Possible reaction pathways are proposed, substantiating the observed differences between the electron ionization and photoionization mass spectra.

Performance of Mixed Quantum-Classical Approaches on Modeling the Crossover from Hopping to Bandlike Charge Transport in Organic Semiconductors

In the present study, several mixed quantum-classical (MQC) methods are applied to on-the-fly nonadiabatic molecular dynamics simulations of hole transport in molecular organic semiconductors (OSCs). The tested MQC methods contain the mean-field Ehrenfest (MFE), trajectory surface hopping (TSH) approaches based on Tully's fewest switches surface hopping (FSSH) and the global flux surface hopping (GFSH), the latter in the diabatic/adiabatic representation, and a Landau-Zener type trajectory surface hopping (LZSH). We also tested several correction schemes which were proposed to identify trivial crossings and to remove unphysical long-range charge transfers due to decoherence corrections. In addition, several cost-effective approaches for the nuclear velocity adjustment after an energy-allowed/energy-forbidden hop are investigated with respect to detailed balance and internal consistency conditions. To model a broad spectrum of OSCs with different charge transport characteristics, we derived from the anthracene structural model the construction of two additional models by uniformly scaling down the electronic couplings by the factors of 0.1 and 0.5. Anthracene shows a bandlike charge transport mechanism, characterized by slightly delocalized charge carriers 'diffusing' through the crystal. For smaller couplings, the mechanism changes to a hopping type, characteristically differing in the charge delocalization and temperature dependence. The MFE and corrected adiabatic TSH approaches are able to quantitatively reproduce the expected behavior, while the diabatic LZSH method fails for large couplings, as do approaches which are based on the hopping of localized charge between neighboring sites. Moreover, we find that while the hole mobility of the anthracene crystal simulated using the celebrated Marcus theory is in good agreement with the experimental value, its agreement has to be regarded as an accident due to the overestimation of the prefactor in the Marcus rate equation.

Modeling nonadiabatic dynamics in condensed matter materials: some recent advances and applications

This review focuses on recent developments in the field of nonadiabatic molecular dynamics (NA-MD), with particular attention given to condensed-matter systems. NA-MD simulations for small molecular systems can be performed using high-level electronic structure (ES) calculations, methods accounting for the quantization of nuclear motion, and using fewer approximations in the dynamical methodology itself. Modeling condensed-matter systems imposes many limitations on various aspects of NA-MD computations, requiring approximations at various levels of theory-from the ES, to the ways in which the coupling of electrons and nuclei are accounted for. Nonetheless, the approximate treatment of NA-MD in condensed-phase materials has gained a spin lately in many applied studies. A number of advancements of the methodology and computational tools have been undertaken, including general-purpose methods, as well as those tailored to nanoscale and condensed matter systems. This review summarizes such methodological and software developments, puts them into the broader context of existing approaches, and highlights some of the challenges that remain to be solved.