Quantum dynamics simulations using Gaussian wavepackets: the vMCG method

Accelerating wavepacket propagation with machine learning

In this work, we discuss the use of a recently introduced machine learning (ML) technique known as Fourier neural operators (FNO) as an efficient alternative to the traditional solution of the time-dependent Schrödinger equation (TDSE). FNOs are ML models which are employed in the approximated solution of partial differential equations. For a wavepacket propagating in an anharmonic potential and for a tunneling system, we show that the FNO approach can accurately and faithfully model wavepacket propagation via the density. Additionally, we demonstrate that FNOs can be a suitable replacement for traditional TDSE solvers in cases where the results of the quantum dynamical simulation are required repeatedly such as in the case of parameter optimization problems (e.g., control). The speed-up from the FNO method allows for its combination with the Markov-chain Monte Carlo approach in applications that involve solving inverse problems such as optimal and coherent laser control of the outcome of dynamical processes.

Modeling Photodissociation: Quantum Dynamics Simulations of Methanol

A comprehensive computational study of the gas-phase photodissociation dynamics of methanol is presented. Using a multiconfigurational active space based method (RASSCF) to obtain multidimensional potential energy surfaces (PESs) on-the-fly, direct quantum dynamics simulations were run using the variational multi-configurational Gaussian method (DD-vMCG). Different initial excitation energies were simulated to investigate the dependence of the branching ratios on the electronic state being populated. A detailed mechanistic explanation is provided for the observed differences with respect to the excitation energy. Population of the lowest lying excited state of methanol leads to rapid hydroxyl hydrogen loss as the main dissociation channel. This is rationalized by the strongly dissociative nature of the PES cut along the O-H stretching coordinate, confirmed by the broad feature in the absorption spectrum. In contrast, more energetic excitations lead mainly to C-O bond breaking. Again, analysis of the diabatic surfaces offers a clear explanation in terms of the nature of the electronic states involved and the coupling between them. The type of calculations presented, as well as the subsequent analysis of the results, should be seen as a general workflow for the modeling of photochemical reactions.

Perspective on Theoretical and Experimental Advances in Atmospheric Photochemistry

Research that explores the chemistry of Earth's atmosphere is central to the current understanding of global challenges such as climate change, stratospheric ozone depletion, and poor air quality in urban areas. This research is a synergistic combination of three established domains: earth observation, for example, using satellites, and in situ field measurements; computer modeling of the atmosphere and its chemistry; and laboratory measurements of the properties and reactivity of gas-phase molecules and aerosol particles. The complexity of the interconnected chemical and photochemical reactions which determine the composition of the atmosphere challenges the capacity of laboratory studies to provide the spectroscopic, photochemical, and kinetic data required for computer models. Here, we consider whether predictions from computational chemistry using modern electronic structure theory and nonadiabatic dynamics simulations are becoming sufficiently accurate to supplement quantitative laboratory data for wavelength-dependent absorption cross-sections, photochemical quantum yields, and reaction rate coefficients. Drawing on presentations and discussions from the CECAM workshop on Theoretical and Experimental Advances in Atmospheric Photochemistry held in March 2024, we describe key concepts in the theory of photochemistry, survey the state-of-the-art in computational photochemistry methods, and compare their capabilities with modern experimental laboratory techniques. From such considerations, we offer a perspective on the scope of computational (photo)chemistry methods based on rigorous electronic structure theory to become a fourth core domain of research in atmospheric chemistry.

Time evolution as an optimization problem: The hydrogen atom in strong laser fields in a basis of time-dependent Gaussian wave packets

Recent advances in attosecond science have made it increasingly important to develop stable, reliable, and accurate algorithms and methods to model the time evolution of atoms and molecules in intense laser fields. A key process in attosecond science is high-harmonic generation, which is challenging to model with fixed Gaussian basis sets, as it produces high-energy electrons, with a resulting rapidly varying and highly oscillatory wave function that extends over dozens of ångström. Recently, Rothe's method, where time evolution is rephrased as an optimization problem, has been applied to the one-dimensional Schrödinger equation. Here, we apply Rothe's method to the hydrogen wave function and demonstrate that thawed, complex-valued Gaussian wave packets with time-dependent width, center, and momentum parameters are able to reproduce spectra obtained from essentially exact grid calculations for high-harmonic generation with only 50-181 Gaussians for field strengths up to 5 × 1014 W/cm2. This paves the way for the inclusion of continuum contributions into real-time, time-dependent electronic-structure theory with Gaussian basis sets for strong fields and eventually accurate simulations of the time evolution of molecules without the Born-Oppenheimer approximation.

An Exhaustive Quantum-Classical Study of C6F6+ Using the Newly Formulated Parallel TDDVR Method

We recently implemented our parallelized quantum-classical dynamical approach, known as the Time-Dependent Discrete Variable Representation (TDDVR) method, which is applied to the spectroscopically important hexafluorobenzene (HFBz) radical cation, where several conical intersections exist in their seven lowest excited electronic states (S11B2u, S21E1g, S31B1u, S41E1u, and S51A2u) considering degeneracy among potential energy surfaces (PESs), to demonstrate their various dynamical aspects. This new parallel version shows almost linear scalability with increasing number of computing processors. To get photoelectron (PE) spectra, Mass-Analyzed Threshold Ionization (MATI) spectra, population dynamics, and many other dynamical observables, the first-principles dynamics is applied at the state-of-the-art level to the corresponding Hamiltonian, where the Jahn-Teller (JT) and pseudo-Jahn-Teller (PJT) type interactions are involved in those coupled seven electronic states. The quantum-classical method is used to thoroughly analyze the effects of these couplings on the nuclear dynamics of the involved electronic states, and the findings are compared with those observables obtained from experiments. Intrinsic dynamical properties are explained using the reduced densities of the wave packet (WP) in a coupled electronic manifold. The PE and MATI spectra of HFBz computed using TDDVR are found to be in good agreement with earlier experimental data and other theoretically simulated spectra.

Controlling Electronic Coherences and the Curvature Induced by the Derivative Coupling at a Conical Intersection: A Quantum Ehrenfest (QuEh) Protocol for Reaction Path Following Application to "Channel 3" Benzene Photochemistry

We report a protocol for the implementation of "reaction path following" from a transition state through a conical intersection, including both the path curvature induced by the derivative coupling and the corresponding induced electronic coherences. This protocol focuses on the "central" Gaussian wavepacket (initially unexcited) in the quantum Ehrenfest (QuEh) method. Like the reaction path following, the normal mode corresponding to the imaginary frequency at the transition state is given an initial momentum. The protocol is applied to the "channel 3" radiationless decay of benzene. We also demonstrate that one can enhance the effect of the derivative coupling and the electronic coherence with an IR pulse.

Optimizing Conical Intersections without Explicit Use of Non-Adiabatic Couplings

We present two alternative methods for optimizing minimum energy conical intersection (MECI) molecular geometries without knowledge of the derivative coupling (DC). These methods are based on the utilization of Lagrange multipliers: (i) one method uses an approximate calculation of the DC, while the other (ii) do not require the DC. Both methods use the fact that information on the DC is contained in the Hessian of the squared energy difference. Tests done on a set of small molecular systems, in comparison with other methods, show the ability of the proposed methods to optimize MECIs. Finally, we apply the methods to the furimamide molecule, to optimize and characterize its S1/S2 MECI, and to optimizing the S0/S1 MECI of the silver trimer.

Quantum Quality with Classical Cost: Ab Initio Nonadiabatic Dynamics Simulations Using the Mapping Approach to Surface Hopping

Nonadiabatic dynamics methods are an essential tool for investigating photochemical processes. In the context of employing first-principles electronic structure techniques, such simulations can be carried out in a practical manner using semiclassical trajectory-based methods or wave packet approaches. While all approaches applicable to first-principles simulations are necessarily approximate, it is commonly thought that wave packet approaches offer inherent advantages over their semiclassical counterparts in terms of accuracy and that this trait simply comes at a higher computational cost. Here we demonstrate that the mapping approach to surface hopping (MASH), a recently introduced trajectory-based nonadiabatic dynamics method, can be efficiently applied in tandem with ab initio electronic structure. Our results even suggest that MASH may provide more accurate results than on-the-fly wave packet techniques, all at a much lower computational cost.

Using a multistate mapping approach to surface hopping to predict the ultrafast electron diffraction signal of gas-phase cyclobutanone

Using the recently developed multistate mapping approach to surface hopping (multistate MASH) method combined with SA(3)-CASSCF(12,12)/aug-cc-pVDZ electronic structure calculations, the gas-phase isotropic ultrafast electron diffraction (UED) of cyclobutanone is predicted and analyzed. After excitation into the n-3s Rydberg state (S2), cyclobutanone can relax through two S2/S1 conical intersections, one characterized by compression of the CO bond and the other by dissociation of the α-CC bond. Subsequent transfer into the ground state (S0) is then achieved via two additional S1/S0 conical intersections that lead to three reaction pathways: α ring-opening, ethene/ketene production, and CO liberation. The isotropic gas-phase UED signal is predicted from the multistate MASH simulations, allowing for a direct comparison to the experimental data. This work, which is a contribution to the cyclobutanone prediction challenge, facilitates the identification of the main photoproducts in the UED signal and thereby emphasizes the importance of dynamics simulations for the interpretation of ultrafast experiments.

Finite-Temperature Correlation Functions Obtained from Combined Real- and Imaginary-Time Propagation of Variational Thawed Gaussian Wavepackets

We apply the so-called variational Gaussian wavepacket approximation (VGA) for conducting both real- and imaginary-time dynamics to calculate thermal correlation functions. By considering strongly anharmonic systems, such as a quartic potential and a double-well potential at high and low temperatures, it is shown that this method is partially able to account for tunneling. This is contrary to other popular many-body methods, such as ring polymer molecular dynamics and the classical Wigner method, which fail in this respect. It is a historical peculiarity that no one has considered the VGA method for representing both the Boltzmann operator and the real-time propagation. This method should be well suited for molecular systems containing many atoms.

Simulating Attochemistry: Which Dynamics Method to Use?

Attochemistry aims to exploit the properties of coherent electronic wavepackets excited via attosecond pulses to control the formation of photoproducts. Such molecular processes can, in principle, be simulated with various nonadiabatic dynamics methods, yet the impact of the approximations underlying the methods is rarely assessed. The performances of widely used mixed quantum-classical approaches, Tully surface hopping, and classical Ehrenfest methods are evaluated against the high-accuracy DD-vMCG quantum dynamics. This comparison is conducted for the valence ionization of fluorobenzene. Analyzing the nuclear motion induced in the branching space of the nearby conical intersection, the results show that the mixed quantum-classical methods reproduce quantitatively the average motion of a quantum wavepacket when initiated on a single electronic state. However, they fail to properly capture the nuclear motion induced by an electronic wavepacket along the derivative coupling, the latter originating from the quantum electronic coherence property, key to attochemistry.

Time dependent vibrational electronic coupled cluster (VECC) theory for non-adiabatic nuclear dynamics

A time-dependent vibrational electronic coupled-cluster (VECC) approach is proposed to simulate photo-electron/UV-VIS absorption spectra as well as time-dependent properties for non-adiabatic vibronic models, going beyond the Born-Oppenheimer approximation. A detailed derivation of the equations of motion and a motivation for the ansatz are presented. The VECC method employs second-quantized bosonic construction operators and a mixed linear and exponential ansatz to form a compact representation of the time-dependent wave-function. Importantly, the method does not require a basis set, has only a few user-defined inputs, and has a classical (polynomial) scaling with respect to the number of degrees of freedom (of the vibronic model), resulting in a favorable computational cost. In benchmark applications to small models and molecules, the VECC method provides accurate results compared to multi-configurational time-dependent Hartree calculations when predicting short-time dynamical properties (i.e., photo-electron/UV-VIS absorption spectra) for non-adiabatic vibronic models. To illustrate the capabilities, the VECC method is also successfully applied to a large vibronic model for hexahelicene with 14 electronic states and 63 normal modes, developed in the group by Aranda and Santoro [J. Chem. Theory Comput. 17, 1691, (2021)].

Quantum dynamics of excited state proton transfer in green fluorescent protein

Photoexcitation of green fluorescent protein (GFP) triggers long-range proton transfer along a "wire" of neighboring protein residues, which, in turn, activates its characteristic green fluorescence. The GFP proton wire is one of the simplest, most well-characterized models of biological proton transfer but remains challenging to simulate due to the sensitivity of its energetics to the surrounding protein conformation and the possibility of non-classical behavior associated with the movement of lightweight protons. Using a direct dynamics variational multiconfigurational Gaussian wavepacket method to provide a fully quantum description of both electrons and nuclei, we explore the mechanism of excited state proton transfer in a high-dimensional model of the GFP chromophore cluster over the first two picoseconds following excitation. During our simulation, we observe the sequential starts of two of the three proton transfers along the wire, confirming the predictions of previous studies that the overall process starts from the end of the wire furthest from the fluorescent chromophore and proceeds in a concerted but asynchronous manner. Furthermore, by comparing the full quantum dynamics to a set of classical trajectories, we provide unambiguous evidence that tunneling plays a critical role in facilitating the leading proton transfer.

Quasi-direct Quantum Molecular Dynamics: The Time-Dependent Adaptive Density-Guided Approach for Potential Energy Surface Construction

We present a new quasi-direct quantum molecular dynamics computational method which offers a compromise between quantum dynamics using a precomputed potential energy surface (PES) and fully direct quantum dynamics. This method is termed the time-dependent adaptive density-guided approach (TD-ADGA) and is a method for constructing a PES on the fly during a dynamics simulation. This is achieved by acquisition of new single-point (SP) calculations and refitting of the PES, depending on the need of the dynamics. The TD-ADGA is a further development of the adaptive density-guided approach (ADGA) for PES construction where the placement of SPs is guided by the density of the nuclear wave function. In TD-ADGA, the ADGA framework has been integrated into the time propagation of the time-dependent nuclear wave function and we use the reduced one-mode density of this wave function to guide when and where new SPs are placed. The PES is thus extended or updated if the wave function moves into new areas or if a certain area becomes more important. Here, we derive equations for the reduced one-mode density for the time-dependent Hartree (TDH) method and for multiconfiguration time-dependent Hartree (MCTDH) methods, but the TD-ADGA can be used with any time-dependent wave function method as long as a density is available. The TD-ADGA method has been investigated on molecular systems containing single- and double-minimum potentials and on single-mode and multi-mode systems. We explore different approaches to handle the fact that the TD-ADGA involves a PES that changes during the computation and show how results can be obtained that are in very good agreement with results obtained by using an accurate reference PES. Dynamics with TD-ADGA is essentially a black box procedure, where only the initialization of the system and how to compute SPs must be provided. The TD-ADGA thus makes it easier to carry out quantum molecular dynamics and the quasi-direct framework opens up the possibility to compute quantum dynamics accurately for larger molecular systems.

Resolving the Experimental Photoelectron Spectra of CAl3Si. J Phys Chem A 2024;128:355-369. [PMID: 38189257 DOI: 10.1021/acs.jpca.3c06295] [Citation(s) in RCA: 0] [Impact Index Per Article: 0]

The experimental photoelectron spectra concerning the six electronic states of CAl3Si- are resolved through electronic structure calculations and quantum nuclear dynamics in this study. It incorporates a model diabatic Hamiltonian to evaluate the coupling parameters and fit the potential energy curves (PECs). The analysis of these PECs showed us that there are sufficient nonadiabatic effects in the photoelectron spectra through the presence of various conical intersections. Poisson intensity distributions (PIDs) and the wave packet density plots are utilized for assigning the fundamental and first overtone excitations. The nuclear dynamics study is accomplished by employing time-dependent (TD) and time-independent (TI) quantum chemistry methods. Ultimately, our theoretical results concurred well with the experimental findings exhibiting vibronic coupling amidst the nearly positioned electronic states.

Photoelectron spectra of benzene: Can path dependent diabatic surfaces provide unique observables?

While carrying out Beyond Born-Oppenheimer theory based diabatization, the solutions of adiabatic-to-diabatic transformation equations depend on the paths of integration over two-dimensional cross-sections of multi-dimensional space of nuclear degrees of freedom. It is shown that such path-dependent solutions leading to diabatic potential energy surface matrices computed along any two different paths are related through an orthogonal matrix, and thereby, those surface matrices should provide unique observables. While exploring the numerical validity of the theoretical framework, we construct diabatic Hamiltonians for the five low-lying electronic states (X̃2E1g, B̃2E2g, and C̃2A2u) of benzene radical cation (C6H6+) along three different approaches of contour integration over two dimensional nuclear planes constituted by seven non-adiabatically active normal modes. Three different diabatic surface matrices are further employed to generate the photoelectron spectra of the benzene molecule (C6H6). It is interesting to note that the spectral peak positions and intensity patterns for all three cases are almost close to each other and also exhibit very good agreement with the experimental results.

Quantum Dynamics of Photodissociation: Recent Advances and Challenges

Recent advances in constructing accurate potential energy surfaces and nonadiabatic couplings from high-level ab initio data have revealed detailed potential landscapes in not only the ground electronic state but also excited ones. They enabled quantitatively accurate characterization of photoexcited reactive systems using quantum mechanical methods. In this Perspective, we survey the recent progress in quantum mechanical studies of adiabatic and nonadiabatic photodissociation dynamics, focusing on initial state control and product energy disposal. These new insights helped to understand quantum effects in small prototypical systems, and the results serve as benchmarks for developing more approximate theoretical methods.

What Controls the Quality of Photodynamical Simulations? Electronic Structure Versus Nonadiabatic Algorithm

The field of nonadiabatic dynamics has matured over the last decade with a range of algorithms and electronic structure methods available at the moment. While the community currently focuses more on developing and benchmarking new nonadiabatic dynamics algorithms, the underlying electronic structure controls the outcome of nonadiabatic simulations. Yet, the electronic-structure sensitivity analysis is typically neglected. In this work, we present a sensitivity analysis of the nonadiabatic dynamics of cyclopropanone to electronic structure methods and nonadiabatic dynamics algorithms. In particular, we compare wave function-based CASSCF, FOMO-CASCI, MS- and XMS-CASPT2, density-functional REKS, and semiempirical MRCI-OM3 electronic structure methods with the Landau-Zener surface hopping, fewest switches surface hopping, and ab initio multiple spawning with informed stochastic selection algorithms. The results clearly demonstrate that the electronic structure choice significantly influences the accuracy of nonadiabatic dynamics for cyclopropanone even when the potential energy surfaces exhibit qualitative and quantitative similarities. Thus, selecting the electronic structure solely on the basis of the mapping of potential energy surfaces can be misleading. Conversely, we observe no discernible differences in the performance of the nonadiabatic dynamics algorithms across the various methods. Based on the above results, we discuss the present-day practice in computational photodynamics.

Efficient time-dependent vibrational coupled cluster computations with time-dependent basis sets at the two-mode coupling level: Full and hybrid TDMVCC[2]

The computation of the nuclear quantum dynamics of molecules is challenging, requiring both accuracy and efficiency to be applicable to systems of interest. Recently, theories have been developed for employing time-dependent basis functions (denoted modals) with vibrational coupled cluster theory (TDMVCC). The TDMVCC method was introduced along with a pilot implementation, which illustrated good accuracy in benchmark computations. In this paper, we report an efficient implementation of TDMVCC, covering the case where the wave function and Hamiltonian contain up to two-mode couplings. After a careful regrouping of terms, the wave function can be propagated with a cubic computational scaling with respect to the number of degrees of freedom. We discuss the use of a restricted set of active one-mode basis functions for each mode, as well as two interesting limits: (i) the use of a full active basis where the variational modal determination amounts essentially to the variational determination of a time-dependent reference state for the cluster expansion; and (ii) the use of a single function as an active basis for some degrees of freedom. The latter case defines a hybrid TDMVCC/TDH (time-dependent Hartree) approach that can obtain even lower computational scaling. The resulting computational scaling for hybrid and full TDMVCC[2] is illustrated for polyaromatic hydrocarbons with up to 264 modes. Finally, computations on the internal vibrational redistribution of benzoic acid (39 modes) are used to show the faster convergence of TDMVCC/TDH hybrid computations towards TDMVCC compared to simple neglect of some degrees of freedom.

HORTENSIA, a program package for the simulation of nonadiabatic autoionization dynamics in molecules

We present a program package for the simulation of ultrafast vibration-induced autoionization dynamics in molecular anions in the manifold of the adiabatic anionic states and the discretized ionization continuum. This program, called HORTENSIA (Hopping Real-time Trajectories for Electron-ejection by Nonadiabatic Self-Ionization in Anions), is based on the nonadiabatic surface-hopping methodology, wherein nuclei are propagated as an ensemble along classical trajectories in the quantum-mechanical potential created by the electronic density of the molecular system. The electronic Schrödinger equation is numerically integrated along the trajectory, providing the time evolution of electronic state coefficients, from which switching probabilities into discrete electronic states are determined. In the case of a discretized continuum state, this hopping event is interpreted as the ejection on an electron. The derived diabatic and nonadiabatic couplings in the time-dependent electronic Schrödinger equation are calculated from anionic and neutral wavefunctions obtained from quantum-chemical calculations with commercially available program packages interfaced with our program. Based on this methodology, we demonstrate the simulation of autoionization electron kinetic energy spectra that are both time- and angle-resolved. In addition, the program yields data that can be interpreted easily with respect to geometric characteristics, such as bonding distances and angles, which facilitate the detection of molecular configurations important for the autoionization process. Furthermore, several useful extensions are included, namely, tools for the generation of initial conditions and input files as well as for the evaluation of output files, all of this both through console commands and a graphical user interface.

Probing the sensitivity of ab initio multiple spawning to its parameters

Full multiple spawning (FMS) offers a strategy to simulate the nonadiabatic dynamics of molecular systems by describing their nuclear wavefunctions by a linear combination of coupled trajectory basis functions (TBFs). Applying a series of controlled approximations to the full multiple spawning (FMS) equations leads to the ab initio multiple spawning (AIMS), which is compatible with an on-the-fly propagation of the TBFs and an accurate description of nonadiabatic processes. The AIMS strategy and its numerical implementations, however, rely on a series of user-defined parameters. Herein, we investigate the influence of these parameters on the electronic-state population of two molecular systems- trans-azomethane and a two-dimensional model of the butatriene cation. This work highlights the stability of AIMS with respect to most of its parameters, underlines the specific parameters that require particular attention from the user of the method, and offers prescriptions for an informed selection of their value.

Supplementary Information

The online version contains supplementary material available at 10.1007/s00214-023-03004-w.

Simulating optical linear absorption for mesoscale molecular aggregates: An adaptive hierarchy of pure states approach

In this paper, we present dyadic adaptive HOPS (DadHOPS), a new method for calculating linear absorption spectra for large molecular aggregates. This method combines the adaptive HOPS (adHOPS) framework, which uses locality to improve computational scaling, with the dyadic HOPS method previously developed to calculate linear and nonlinear spectroscopic signals. To construct a local representation of dyadic HOPS, we introduce an initial state decomposition that reconstructs the linear absorption spectra from a sum over locally excited initial conditions. We demonstrate the sum over initial conditions can be efficiently Monte Carlo sampled and that the corresponding calculations achieve size-invariant [i.e., O(1)] scaling for sufficiently large aggregates while trivially incorporating static disorder in the Hamiltonian. We present calculations on the photosystem I core complex to explore the behavior of the initial state decomposition in complex molecular aggregates as well as proof-of-concept DadHOPS calculations on an artificial molecular aggregate inspired by perylene bis-imide to demonstrate the size-invariance of the method.

An on-the-fly deep neural network for simulating time-resolved spectroscopy: predicting the ultrafast ring opening dynamics of 1,2-dithiane

Revolutionary developments in ultrafast light source technology are enabling experimental spectroscopists to probe the structural dynamics of molecules and materials on the femtosecond timescale. The capacity to investigate ultrafast processes afforded by these resources accordingly inspires theoreticians to carry out high-level simulations which facilitate the interpretation of the underlying dynamics probed during these ultrafast experiments. In this Article, we implement a deep neural network (DNN) to convert excited-state molecular dynamics simulations into time-resolved spectroscopic signals. Our DNN is trained on-the-fly from first-principles theoretical data obtained from a set of time-evolving molecular dynamics. The train-test process iterates for each time-step of the dynamics data until the network can predict spectra with sufficient accuracy to replace the computationally intensive quantum chemistry calculations required to produce them, at which point it simulates the time-resolved spectra for longer timescales. The potential of this approach is demonstrated by probing dynamics of the ring opening of 1,2-dithiane using sulphur K-edge X-ray absorption spectroscopy. The benefits of this strategy will be more markedly apparent for simulations of larger systems which will exhibit a more notable computational burden, making this approach applicable to the study of a diverse range of complex chemical dynamics.

A Structure-Based Gaussian Expansion for Quantum Reaction Dynamics in Molecules: Application to Hydrogen Tunneling in Malonaldehyde

We developed an approximate method for quantum reaction dynamics simulations, namely, a structure-based Gaussian (SBG) expansion approach, where SBG bases for the expansion of the wave function Ψ, expressed by a product of single-atom Cartesian Gaussians centered at the positions of respective nuclei, are mainly placed around critical structures on reaction pathways such as on the intrinsic reaction coordinate (IRC) through a transition state. In the present approach, the "pseudo-lattice points" at which SBGs are deployed are selected in a perturbative manner so as to make moderate the expansion length. We first applied the SBG idea to a two-dimensional quadruple-well model and obtained accurate tunneling splitting values between the lowest four states. We then applied it to hydrogen tunneling in malonaldehyde and achieved a tunneling splitting of 27.1 cm^-1 with only 875 SBGs at the MP2/6-31G(d,p) level of theory, in good agreement with 25 cm^-1 by the more elaborate multiconfiguration time-dependent Hartree method. Reasonable results were also obtained for singly and doubly deuterated malonaldehyde. We analyzed the tunneling states by utilizing expansion coefficients of individual SBGs and found that 40-45% of the SBGs in Ψ are nonplanar structures and SBGs away from the IRC contribute a little to hydrogen transfer.

Ring Polymer Molecular Dynamics with Electronic Transitions

Full quantum dynamics of molecules and materials is of fundamental importance, which requires a faithful description of simultaneous quantum motions of the electron and nuclei. A new scheme is developed for nonadiabatic simulations of coupled electron-nuclear quantum dynamics with electronic transitions based on the Ehrenfest theorem and ring polymer molecular dynamics. Built upon the isomorphic ring polymer Hamiltonian, time-dependent multistate electronic Schrödinger equations are solved self-consistently with approximate equation of motions for nuclei. Each bead bears a distinct electronic configuration and thus moves on a specific effective potential. This independent-bead approach provides an accurate description of the real-time electronic population and quantum nuclear trajectory, maintaining a good agreement with the exact quantum solution. Implementation of first-principles calculations enables us to simulate photoinduced proton transfer in H_{2}O-H_{2}O^{+} where we find a good agreement with experiment.

Factorized Electron-Nuclear Dynamics with an Effective Complex Potential

We present a quantum dynamics approach for molecular systems based on wave function factorization into components describing the light and heavy particles, such as electrons and nuclei. The dynamics of the nuclear subsystem can be viewed as motion of the trajectories defined in the nuclear subspace, evolving according to the average nuclear momentum of the full wave function. The probability density flow between the nuclear and electronic subsystems is facilitated by the imaginary potential, derived to ensure a physically meaningful normalization of the electronic wave function for each configuration of the nuclei, and conservation of the probability density associated with each trajectory in the Lagrangian frame of reference. The imaginary potential, defined in the nuclear subspace, depends on the momentum variance in the nuclear coordinates averaged over the electronic component of the wave function. An effective real potential, driving the dynamics of the nuclear subsystem, is defined to minimize motion of the electronic wave function in the nuclear degrees of freedom. Illustration and the analysis of the formalism are given for a two-dimensional model system of vibrationally nonadiabatic dynamics.

Mind the GAP: quantifying the breakdown of the linear vibronic coupling Hamiltonian

Excited state dynamics play a critical role across a broad range of scientific fields. Importantly, the highly non-equilibrium nature of the states generated by photoexcitation means that excited state simulations should usually include an accurate description of the coupled electronic-nuclear motion, which often requires solving the time-dependent Schrödinger equation (TDSE). One of the biggest challenges for these simulations is the requirement to calculate the PES over which the nuclei evolve. An effective approach for addressing this challenge is to use the approximate linear vibronic coupling (LVC) Hamiltonian, which enables a model potential to be parameterised using relatively few quantum chemistry calculations. However, this approach is only valid provided there are no large amplitude motions in the excited state dynamics. In this paper we introduce and deploy a metric, the global anharmonicity parameter (GAP), which can be used to assess the accuracy of an LVC potential. Following its derivation, we illustrate its utility by applying it to three molecules exhibiting different rigidity in their excited states.

The hierarchy of Davydov's Ansätze: From guesswork to numerically "exact" many-body wave functions

This Perspective presents an overview of the development of the hierarchy of Davydov's Ansätze and a few of their applications in many-body problems in computational chemical physics. Davydov's solitons originated in the investigation of vibrational energy transport in proteins in the 1970s. Momentum-space projection of these solitary waves turned up to be accurate variational ground-state wave functions for the extended Holstein molecular crystal model, lending unambiguous evidence to the absence of formal quantum phase transitions in Holstein systems. The multiple Davydov Ansätze have been proposed, with increasing Ansatz multiplicity, as incremental improvements of their single-Ansatz parents. For a given Hamiltonian, the time-dependent variational formalism is utilized to extract accurate dynamic and spectroscopic properties using Davydov's Ansätze as its trial states. A quantity proven to disappear for large multiplicities, the Ansatz relative deviation is introduced to quantify how closely the Schrödinger equation is obeyed. Three finite-temperature extensions to the time-dependent variation scheme are elaborated, i.e., the Monte Carlo importance sampling, the method of thermofield dynamics, and the method of displaced number states. To demonstrate the versatility of the methodology, this Perspective provides applications of Davydov's Ansätze to the generalized Holstein Hamiltonian, variants of the spin-boson model, and systems of cavity-assisted singlet fission, where accurate dynamic and spectroscopic properties of the many-body systems are given by the Davydov trial states.

Quantum-classical dynamics of vibration-induced autoionization in molecules

We present a novel method for the simulation of the vibration-induced autoionization dynamics in molecular anions in the framework of the quantum-classical surface hopping approach. Classical trajectories starting from quantum initial conditions are propagated on a quantum-mechanical potential energy surface while allowing for autoionization through transitions into discretized continuum states. These transitions are induced by the couplings between the electronic states of the bound anionic system and the electron-detached system composed of the neutral molecule and the free electron. A discretization scheme for the detached system is introduced, and a set of formulas is derived that enable the approximate calculation of couplings between the bound and free-electron states. We demonstrate our method on the example of the anion of vinylidene, a high-energy isomer of acetylene, for which detailed experimental data are available. Our results provide information on the time scale of the autoionization process and give insight into the energetic and angular distribution of the ejected electrons, as well as the associated changes in the molecular geometry. We identify the formation of structures with reduced C-C bond lengths and T-like conformations through bending of the CH₂ group with respect to the C-C axis and point out the role of autoionization as a driving process for the isomerization to acetylene.

Extending the Applicability of the Multiple-Spawning Framework for Nonadiabatic Molecular Dynamics

Ab initio multiple-spawning (AIMS) describes the nonadiabatic dynamics of molecules by expanding nuclear wave functions in a basis of traveling multidimensional Gaussians called trajectory basis functions (TBFs). New TBFs can be spawned whenever nuclear amplitude is transferred between electronic states due to nonadiabatic transitions. While the adaptive size of the TBF basis grants AIMS its characteristic accuracy in describing nonadiabatic processes, it also leads to a fast and uncontrolled growth of the number of TBFs, penalizing computational efficiency. A different flavor of AIMS, called AIMS with informed stochastic selections (AIMSWISS), has recently been proposed to reduce the number of TBFs dramatically. Herein, we test the performance of AIMSWISS for a series of challenging nonadiabatic processes─photodynamics of two-dimensional model systems, 1,2-dithiane and chromium (0) hexacarbonyl─and show that this method is robust and extends the range of molecular systems that can be simulated within the multiple-spawning framework.

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.

Bivariational time-dependent wave functions with biorthogonal adaptive basis sets: General formulation and regularization of equations of motion through polar decomposition

We derive general bivariational equations of motion (EOMs) for time-dependent wave functions with biorthogonal time-dependent basis sets. The time-dependent basis functions are linearly parameterized and their fully variational time evolution is ensured by solving a set of so-called constraint equations, which we derive for arbitrary wave function expansions. The formalism allows division of the basis set into an active basis and a secondary basis, ensuring a flexible and compact wave function. We show how the EOMs specialize to a few common wave function forms, including coupled cluster and linearly expanded wave functions. It is demonstrated, for the first time, that the propagation of such wave functions is not unconditionally stable when a secondary basis is employed. The main signature of the instability is a strong increase in non-orthogonality, which eventually causes the calculation to fail; specifically, the biorthogonal active bra and ket bases tend toward spanning different spaces. Although formally allowed, this causes severe numerical issues. We identify the source of this problem by reparametrizing the time-dependent basis set through polar decomposition. Subsequent analysis allows us to remove the instability by setting appropriate matrix elements to zero. Although this solution is not fully variational, we find essentially no deviation in terms of autocorrelation functions relative to the variational formulation. We expect that the results presented here will be useful for the formal analysis of bivariational time-dependent wave functions for electronic and nuclear dynamics in general and for the practical implementation of time-dependent CC wave functions in particular.

Variational Approach for Linearly Dependent Moving Bases in Quantum Dynamics: Application to Gaussian Functions

In this paper, we present a variational treatment of the linear dependence for a non-orthogonal time-dependent basis set in solving the Schrödinger equation. The method is based on (i) the definition of a linearly independent working space and (ii) a variational construction of the propagator over finite time steps. The second point allows the method to properly account for changes in the dimensionality of the working space along the time evolution. In particular, the time evolution is represented by a semi-unitary transformation. Tests are carried out on a quartic double-well potential with Gaussian basis functions whose centers evolve according to classical equations of motion. We show that the resulting dynamics converges to the exact one and is unitary by construction.

Long-Range Nonequilibrium Coherent Tunneling Induced by Fractional Vibronic Resonances

We study the influence of a linear energy bias on a nonequilibrium excitation on a chain of molecules coupled to local vibrations (a tilted Holstein model) using both a random-walk rate kernel theory and a nonperturbative, massively parallelized adaptive-basis algorithm. We uncover structured and discrete vibronic resonance behavior fundamentally different from both linear response theory and homogeneous polaron dynamics. Remarkably, resonance between the phonon energy ℏω and the bias δ_ϵ occurs not only at integer but also fractional ratios δ_ϵ/(ℏω) = m/n, which effect long-range n-bond m-phonon tunneling. These observations are reproduced in a model calculation of a recently demonstrated Cy3 system, and the effect of dipole-dipole-type non-nearest-neighbor coupling and vibrationally relaxed initial states is also considered. Potential applications range from molecular electronics to optical lattices and artificial light harvesting via vibronic engineering of coherent quantum transport.

How electronic superpositions drive nuclear motion following the creation of a localized hole in the glycine radical cation

In this work, we have studied the nuclear and electron dynamics in the glycine cation starting from localized hole states using the quantum Ehrenfest method. The nuclear dynamics is controlled both by the initial gradient and by the instantaneous gradient that results from the oscillatory electron dynamics (charge migration). We have used the Fourier transform (FT) of the spin densities to identify the “normal modes” of the electron dynamics. We observe an isomorphic relationship between the electron dynamics normal modes and the nuclear dynamics, seen in the vibrational normal modes. The FT spectra obtained this way show bands that are characteristic of the energy differences between the adiabatic hole states. These bands contain individual peaks that are in one-to-one correspondence with atom pair (+·) ↔ (·+) resonances, which, in turn, stimulate nuclear motion involving the atom pair. With such understanding, we anticipate “designer” coherent superpositions that can drive nuclear motion in a particular direction.

Real-time non-adiabatic dynamics in the one-dimensional Holstein model: Trajectory-based vs exact methods

We benchmark a set of quantum-chemistry methods, including multitrajectory Ehrenfest, fewest-switches surface-hopping, and multiconfigurational-Ehrenfest dynamics, against exact quantum-many-body techniques by studying real-time dynamics in the Holstein model. This is a paradigmatic model in condensed matter theory incorporating a local coupling of electrons to Einstein phonons. For the two-site and three-site Holstein model, we discuss the exact and quantum-chemistry methods in terms of the Born-Huang formalism, covering different initial states, which either start on a single Born-Oppenheimer surface, or with the electron localized to a single site. For extended systems with up to 51 sites, we address both the physics of single Holstein polarons and the dynamics of charge-density waves at finite electron densities. For these extended systems, we compare the quantum-chemistry methods to exact dynamics obtained from time-dependent density matrix renormalization group calculations with local basis optimization (DMRG-LBO). We observe that the multitrajectory Ehrenfest method, in general, only captures the ultrashort time dynamics accurately. In contrast, the surface-hopping method with suitable corrections provides a much better description of the long-time behavior but struggles with the short-time description of coherences between different Born-Oppenheimer states. We show that the multiconfigurational Ehrenfest method yields a significant improvement over the multitrajectory Ehrenfest method and can be converged to the exact results in small systems with moderate computational efforts. We further observe that for extended systems, this convergence is slower with respect to the number of configurations. Our benchmark study demonstrates that DMRG-LBO is a useful tool for assessing the quality of the quantum-chemistry methods.

Excited state non-adiabatic dynamics of large photoswitchable molecules using a chemically transferable machine learning potential

Light-induced chemical processes are ubiquitous in nature and have widespread technological applications. For example, photoisomerization can allow a drug with a photo-switchable scaffold such as azobenzene to be activated with light. In principle, photoswitches with desired photophysical properties like high isomerization quantum yields can be identified through virtual screening with reactive simulations. In practice, these simulations are rarely used for screening, since they require hundreds of trajectories and expensive quantum chemical methods to account for non-adiabatic excited state effects. Here we introduce a diabatic artificial neural network (DANN), based on diabatic states, to accelerate such simulations for azobenzene derivatives. The network is six orders of magnitude faster than the quantum chemistry method used for training. DANN is transferable to azobenzene molecules outside the training set, predicting quantum yields for unseen species that are correlated with experiment. We use the model to virtually screen 3100 hypothetical molecules, and identify novel species with high predicted quantum yields. The model predictions are confirmed using high-accuracy non-adiabatic dynamics. Our results pave the way for fast and accurate virtual screening of photoactive compounds.

Controlling energy conservation in quantum dynamics with independently moving basis functions: Application to multi-configuration Ehrenfest

Application of the time-dependent variational principle to a linear combination of frozen-width Gaussians describing the nuclear wavefunction provides a formalism where the total energy is conserved. The computational downside of this formalism is that trajectories of individual Gaussians are solutions of a coupled system of differential equations, limiting implementation to serial propagation algorithms. To allow for parallelization and acceleration of the computation, independent trajectories based on simplified equations of motion were suggested. Unfortunately, within practical realizations involving finite Gaussian bases, this simplification leads to breaking the energy conservation. We offer a solution for this problem by using Lagrange multipliers to ensure the energy and norm conservation regardless of basis function trajectories or basis completeness. We illustrate our approach within the multi-configurational Ehrenfest method considering a linear vibronic coupling model.

Smolyak Algorithm Adapted to a System-Bath Separation: Application to an Encapsulated Molecule with Large-Amplitude Motions

A Smolyak algorithm adapted to system-bath separation is proposed for rigorous quantum simulations. This technique combines a sparse grid method with the system-bath concept in a specific configuration without limitations on the form of the Hamiltonian, thus achieving a highly efficient convergence of the excitation transitions for the "system" part. Our approach provides a general way to overcome the perennial convergence problem for the standard Smolyak algorithm and enables the simulation of floppy molecules with more than a hundred degrees of freedom. The efficiency of the present method is illustrated on the simulation of H₂ caged in an sII clathrate hydrate including two kinds of cage modes. The transition energies are converged by increasing the number of normal modes of water molecules. Our results confirm the triplet splittings of both translational and rotational (j = 1) transitions of the H₂ molecule. Furthermore, they show a slight increase of the translational transitions with respect to the ones in a rigid cage.

The moving crude adiabatic alternative to the adiabatic representation in excited state dynamics

The choice of the electronic representation in on-the-fly quantum dynamics is crucial. The adiabatic representation is appealing since adiabatic states are readily available from quantum chemistry packages. The nuclear wavepackets are then expanded in a basis of Gaussian functions, which follow trajectories to explore the potential energy surfaces and approximate the potential using a local expansion of the adiabatic quantities. Nevertheless, the adiabatic representation is plagued with severe limitations when conical intersections are involved: the diagonal Born-Oppenheimer corrections (DBOCs) are non-integrable, and the geometric phase effect on the nuclear wavepackets cannot be accounted for unless a model is available. To circumvent these difficulties, the moving crude adiabatic (MCA) representation was proposed and successfully tested in low energy dynamics where the wavepacket skirts the conical intersection. We assess the MCA representation in the case of non-adiabatic transitions through conical intersections. First, we show that using a Gaussian basis in the adiabatic representation indeed exhibits the aforementioned difficulties with a special emphasis on the possibility to regularize the DBOC terms. Then, we show that MCA is indeed able to properly model non-adiabatic transitions. Tests are done on linear vibronic coupling models for the bis(methylene) adamantyl cation and the butatriene cation. This article is part of the theme issue 'Chemistry without the Born-Oppenheimer approximation'.

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'.

Quantum dynamics of the photoinduced charge separation in a symmetric donor–acceptor–donor triad: The role of vibronic couplings, symmetry and temperature

The photoinduced charge separation in a symmetric donor–acceptor–donor (D–A–D) triad is studied quantum mechanically using a realistic diabatic vibronic coupling model. The model includes a locally excited DA*D state and two charge-transfer states D+A−D and DA−D+ and is constructed according to a procedure generally applicable to semirigid D–A–D structures and based on energies, forces, and force constants obtained by quantum chemical calculations. In this case, the electronic structure is described by time-dependent density functional theory, and the corrected linear response is used in conjunction with the polarizable continuum model to account for state-specific solvent effects. The multimode dynamics following the photoexcitation to the locally excited state are simulated by the hybrid Gaussian-multiconfigurational time-dependent Hartree method, and temperature effects are included using thermo field theory. The dynamics are connected to the transient absorption spectrum obtained in recent experiments, which is simulated and fully assigned from first principles. It is found that the charge separation is mediated by symmetry-breaking vibrations of relatively low frequency, which implies that temperature should be accounted for to obtain reliable estimates of the charge transfer rate.

Non-Markovian Vibrational Relaxation Dynamics at Surfaces

Vibrational dynamics of adsorbates near surfaces plays both an important role for applied surface science and as model lab for studying fundamental problems of open quantum systems. We employ a previously developed model for the relaxation of a D-Si-Si bending mode at a D:Si(100)-(2$\times$1) surface, induced by a ``bath' of more than $2000$ phonon modes [U. Lorenz, P. Saalfrank, Chem. Phys. {\bf 482}, 69 (2017)], to extend previous work along various directions. First, we use a Hierarchical Effective Mode (HEM) model [E.W. Fischer, F. Bouakline, M. Werther, P. Saalfrank, J. Chem. Phys. {\bf 153}, 064704 (2020)] to study relaxation of higher excited vibrational states than hitherto done, by solving a high-dimensional system-bath time-dependent Schr\"odinger equation (TDSE). In the HEM approach, (many) real bath modes are replaced by (much less) effective bath modes. Accordingly, we are able to examine scaling laws for vibrational relaxation lifetimes for a realistic surface science problem. Second, we compare the performance of the multilayer multiconfigurational time-dependent Hartree (ML-MCTDH) approach with the recently developed coherent-state based multi-Davydov D2 {\it ansatz} [N. Zhou, Z. Huang, J. Zhu, V. Chernyak, Y. Zhao, {J. Chem. Phys.} {\bf 143}, 014113 (2015)]. Both approaches work well, with some computational advantages for the latter in the presented context. Third, we apply open-system density matrix theory in comparison with basically ``exact' solutions of the multi-mode TDSEs. Specifically, we use an open-system Liouville-von Neumann (LvN) equation treating vibration-phonon coupling as Markovian dissipation in Lindblad form to quantify effects beyond the Born-Markov approximation.

Applicability of the Thawed Gaussian Wavepacket Dynamics to the Calculation of Vibronic Spectra of Molecules with Double-Well Potential Energy Surfaces

Simulating vibrationally resolved electronic spectra of anharmonic systems, especially those involving double-well potential energy surfaces, often requires expensive quantum dynamics methods. Here, we explore the applicability and limitations of the recently proposed single-Hessian thawed Gaussian approximation for the simulation of spectra of systems with double-well potentials, including 1,2,4,5-tetrafluorobenzene, ammonia, phosphine, and arsine. This semiclassical wavepacket approach is shown to be more robust and to provide more accurate spectra than the conventional harmonic approximation. Specifically, we identify two cases in which the Gaussian wavepacket method is especially useful due to the breakdown of the harmonic approximation: (i) when the nuclear wavepacket is initially at the top of the potential barrier but delocalized over both wells, e.g., along a low-frequency mode, and (ii) when the wavepacket has enough energy to classically go over the low potential energy barrier connecting the two wells. The method is efficient and requires only a single classical ab initio molecular dynamics trajectory, in addition to the data required to compute the harmonic spectra. We also present an improved algorithm for computing the wavepacket autocorrelation function, which guarantees that the evaluated correlation function is continuous for arbitrary size of the time step.

NQCDynamics.jl: A Julia package for nonadiabatic quantum classical molecular dynamics in the condensed phase

Accurate and efficient methods to simulate nonadiabatic and quantum nuclear effects in high-dimensional and dissipative systems are crucial for the prediction of chemical dynamics in the condensed phase. To facilitate effective development, code sharing, and uptake of newly developed dynamics methods, it is important that software implementations can be easily accessed and built upon. Using the Julia programming language, we have developed the NQCDynamics.jl package, which provides a framework for established and emerging methods for performing semiclassical and mixed quantum-classical dynamics in the condensed phase. The code provides several interfaces to existing atomistic simulation frameworks, electronic structure codes, and machine learning representations. In addition to the existing methods, the package provides infrastructure for developing and deploying new dynamics methods, which we hope will benefit reproducibility and code sharing in the field of condensed phase quantum dynamics. Herein, we present our code design choices and the specific Julia programming features from which they benefit. We further demonstrate the capabilities of the package on two examples of chemical dynamics in the condensed phase: the population dynamics of the spin-boson model as described by a wide variety of semiclassical and mixed quantum-classical nonadiabatic methods and the reactive scattering of H₂ on Ag(111) using the molecular dynamics with electronic friction method. Together, they exemplify the broad scope of the package to study effective model Hamiltonians and realistic atomistic systems.

Describing the photo-isomerization of a retinal chromophore model with coupled and quantum trajectories

The exact factorization of the electron-nuclear wavefunction is applied to the study of the photo- isomerization of a retinal chromophore model. We describe such an ultrafast nonadiabatic process by analyzing the time-dependent potentials of the theory and by mimicking nuclear dynamics with quantum and coupled trajectories. The time-dependent vector and scalar potentials are the signature of the exact factorization, as they guide nuclear dynamics by encoding the complete electronic dynamics and including excited-state effects. Analysis of the potentials is, thus, essential - when possible - to predict the time-dependent behavior of the system of interest. In this work, we employ the exact time-dependent potentials, available for the numerically-exactly solvable model used here, to propagate quantum nuclear trajectories representing the isomerization reaction of the retinal chromophore. The quantum trajectories are the best possible trajectory-based description of the reaction when using the exact-factorization formalism, and thus allow us to assess the performance of the coupled-trajectory, fully approximate, schemes derived from the exact-factorization equations.

Simulation of absorption spectra of molecular aggregates: A hierarchy of stochastic pure state approach

Simulation of spectroscopic observables for molecular aggregates with strong and structured coupling of electronic excitation to vibrational degrees of freedom is an important but challenging task. The Hierarchy of Pure States (HOPS) provides a formally exact solution based on local, stochastic trajectories. Exploiting the localization of HOPS for the simulation of absorption spectra in large aggregates requires a formulation in terms of normalized trajectories. Here, we provide a normalized dyadic equation where the ket- and bra-states are propagated in different electronic Hilbert spaces. This work opens the door to applying adaptive HOPS methods for the simulation of absorption spectra.

Time-resolved photoelectron spectroscopy: the continuing evolution of a mature technique

Time-resolved photoelectron spectroscopy (TRPES) has become one of the most widespread techniques for probing nonadiabatic dynamics in the excited electronic states of molecules. Furthermore, the complementary development of ab initio approaches for the simulation of TRPES signals has enabled the interpretation of these transient spectra in terms of underlying coupled electronic-nuclear dynamics. In this perspective, we discuss the current state-of-the-art approaches, including efforts to push femtosecond pulses into vacuum ultraviolet and soft X-ray regimes as well as the utilization of novel polarizations to use time-resolved optical activity as a probe of nonadiabatic dynamics. We close this perspective with a forward-looking prospectus on the new areas of application for this technique.

Beyond Born-Oppenheimer based diabatic surfaces of 1,3,5-C6H3F3+ to generate the photoelectron spectra using time-dependent discrete variable representation approach

In this article, Beyond Born-Oppenheimer (BBO) treatment is implemented to construct diabatic potential energy surfaces (PESs) of 1,3,5-C₆H₃F₃⁺ over a series [eighteen (18)] of two-dimensional (2D) nuclear planes constituted with eleven normal modes (Q₂, Q_9x, Q_9y, Q_13x, Q_13y, Q_18x, Q_18y, Q_10x, Q_10y, Q_12x and Q_12y) to include all possible nonadiabatic interactions among six coupled electronic states (X̃²E'', , B̃²E' and ). We had formulated explicit expressions of adiabatic to diabatic transformation (ADT) equations [S. Mukherjee, J. Dutta, B. Mukherjee, S. Sardar and S. Adhikari, J. Chem. Phys., 2019, 150, 064308] for the same system forming six state sub-Hilbert space and at present, these ADT equations are solved by incorporating MRCI level ab initio adiabatic PESs and CP-MCSCF calculated nonadiabatic coupling terms (NACTs) to derive diabatic PESs and couplings. Such single-valued, smooth, symmetric and continuous diabatic surface matrices are utilized to carry out multi-state multi-mode nuclear dynamics with the help of time-dependent discrete variable representation (TDDVR) methodology to compute the photoelectron (PE) spectra of 1,3,5-C₆H₃F₃. Our theoretically calculated spectra for X̃²E'', and states using BBO treatment and TDDVR dynamics show peak by peak correspondence with the experimental results as well as better than the findings of the multi-configuration time-dependent Hartree (MCTDH) method.