Recommendations for Velocity Adjustment in Surface Hopping

On-the-fly simulations of transient absorption pump-probe spectra: Combining mapping dynamics with doorway-window protocol

We have constructed an ab initio protocol for the simulation of transient-absorption (TA) pump-probe (PP) signals of realistic polyatomic systems. The protocol is based on interfacing the doorway-window representation of spectroscopic signals with the on-the-fly mapping Hamiltonian dynamics approach at the symmetrical quasi-classical/Meyer-Miller level. The methodology is applied to the simulation of TA PP signals of two molecular systems, azobenzene and cis-hepta-3,5,7-trieniminium cation. For both molecules, the TA PP spectra were demonstrated to give a direct fingerprint of the excited state wavepacket dynamics and internal conversion, which permits the monitoring of the isomerization pathways en route to the final photoproducts.

Charting electronic-state manifolds across molecules with multi-state learning and gap-driven dynamics via efficient and robust active learning

We present a robust protocol for affordable learning of electronic states to accelerate photophysical and photochemical molecular simulations. The protocol solves several issues precluding the widespread use of machine learning (ML) in excited-state simulations. We introduce a novel physics-informed multi-state ML model that can learn an arbitrary number of excited states across molecules, with accuracy better or similar to the accuracy of learning ground-state energies, where information on excited-state energies improves the quality of ground-state predictions. We also present gap-driven dynamics for accelerated sampling of the small-gap regions, which proves crucial for stable surface-hopping dynamics. Together, multi-state learning and gap-driven dynamics enable efficient active learning, furnishing robust models for surface-hopping simulations and helping to uncover long-time-scale oscillations in cis-azobenzene photoisomerization. Our active-learning protocol includes sampling based on physics-informed uncertainty quantification, ensuring the quality of each adiabatic surface, low error in energy gaps, and precise calculation of the hopping probability.

SPADE 1.0: A Simulation Package for Non-Adiabatic Dynamics in Extended Systems

Nonadiabatic molecular dynamics (NAMD) simulations are crucial for revealing the underlying mechanisms of photochemical and photophysical processes. Typical NAMD simulation software packages rely on on-the-fly ab initio electronic structure and nonadiabatic coupling calculations, and thus become challenging when dealing with large complex systems. We here introduce a new Simulation Package for non-Adiabatic Dynamics in Extended systems (SPADE), which is designed to address the limitations of traditional surface hopping methods in dealing with these problems. By design, SPADE enables the users to define arbitrary quasi-diabatic Hamiltonians through parametrized functions and incorporates a variety of algorithms (e.g., global flux hopping probabilities, complex crossing and decoherence corrections), which can realize efficient and reliable NAMD simulations without using nonadiabatic couplings at all. All the employed methods and expressions for diabatic Hamiltonian matrix elements can be flexibly set through the input files. SPADE is mainly written in Fortran based on a modular design and has a great capacity for further implementation of new methods. SPADE can be used to simulate both model and atomistic systems as long as proper Hamiltonians are provided. As demonstrations, a series of representative models are studied to show the main features and capabilities.

Generalization of Quantum-Trajectory Surface Hopping to Multiple Quantum States

In this work, we present a generalization of the quantum trajectory surface hopping (QTSH) to multiple states and its implementation in the Libra package for nonadiabatic dynamics. In lieu of the ad hoc velocity rescaling used in many trajectory-based surface hopping approaches, QTSH utilizes quantum forces to evolve nuclear degrees of freedom continuously. It also lifts the unphysical constraint of enforcing the total energy conservation at the individual trajectory level and rather conserves the total energy at the trajectory ensemble level. Leveraging our new implementation of the multistate QTSH, we perform a comparative analysis of this method with the conventional fewest switches surface hopping approach. We combine the QTSH and decoherence corrections based on the simplified decay of mixing (SDM) and exact factorization (XF), leading to the QTSH-SDM and QTSH-XF schemes. Using the Holstein, superexchange, and phenol model Hamiltonians, we assess the relative accuracy of the resulting combined schemes in reproducing branching ratios, population, and coherence dynamics for a broad range of initial conditions. We observe that the decoherence correction in QTSH is crucial to improve energy conservation as well as the internal consistency between the population from the quantum probability and active state.

Transient-Absorption Pump-Probe Spectra as Information-Rich Observables: Case Study of Fulvene

Conical intersections (CIs) are the most efficient channels of photodeactivation and energy transfer, while femtosecond spectroscopy is the main experimental tool delivering information on molecular CI-driven photoinduced processes. In this work, we undertake a comprehensive ab initio investigation of the CI-mediated internal conversion in fulvene by simulating evolutions of electronic populations, bond lengths and angles, and time-resolved transient absorption (TA) pump-probe (PP) spectra. TA PP spectra are evaluated on the fly by combining the symmetrical quasiclassical/Meyer-Miller-Stock-Thoss (SQC/MMST) dynamics and the doorway-window representation of spectroscopic signals. We show that the simulated time-resolved TA PP spectra reveal not only the population dynamics but also the key nuclear motions as well as mode-mode couplings. We also demonstrate that TA PP signals are not only experimental observables: They can also be considered as information-rich purely theoretical observables, which deliver more information on the CI-driven dynamics than conventional electronic populations. This information can be extracted by the appropriate theoretical analyses of time-resolved TA PP signals.

Assessing the performance of coupled-trajectory schemes on full-dimensional two-state linear vibronic coupling models

We investigate the performance of coupled-trajectory methods for nonadiabatic molecular dynamics in simulating the photodynamics of 4-(dimethylamino)benzonitrile (DMABN) and fulvene, with electronic structure provided by linear vibrational coupling models. We focus on the coupled-trajectory mixed quantum-classical (CTMQC) algorithm and on the (combined) coupled-trajectory Tully surface hopping [(C)CTTSH] in comparison to independent-trajectory approaches, such as multi-trajectory Ehrenfest and Tully surface hopping. Our analysis includes not only electronic populations but also additional electronic and nuclear properties in position and momentum space. For both DMABN and fulvene, the recently developed CCTTSH algorithm successfully resolves the internal inconsistencies of coupled-trajectory Tully surface hopping. Instead, we find that DMABN highlights a significant weakness of CTMQC, which arises when the trajectories remain for a long time in the vicinity of a region of strong nonadiabaticity.

ULaMDyn: enhancing excited-state dynamics analysis through streamlined unsupervised learning

The analysis of nonadiabatic molecular dynamics (NAMD) data presents significant challenges due to its high dimensionality and complexity. To address these issues, we introduce ULaMDyn, a Python-based, open-source package designed to automate the unsupervised analysis of large datasets generated by NAMD simulations. ULaMDyn integrates seamlessly with the Newton-X platform and employs advanced dimensionality reduction and clustering techniques to uncover hidden patterns in molecular trajectories, enabling a more intuitive understanding of excited-state processes. Using the photochemical dynamics of fulvene as a test case, we demonstrate how ULaMDyn efficiently identifies critical molecular geometries and critical nonadiabatic transitions. The package offers a streamlined, scalable solution for interpreting large NAMD datasets. It is poised to facilitate advances in the study of excited-state dynamics across a wide range of molecular systems.

Legion: A Platform for Gaussian Wavepacket Nonadiabatic Dynamics

Nonadiabatic molecular dynamics is crucial in investigating the time evolution of excited states in molecular systems. Among the various methods for performing such dynamics, those employing frozen Gaussian wavepacket propagation, particularly the multiple spawning approach, offer a favorable balance between computational cost and reliability. It propagates on-the-fly trajectories used to build and propagate the nuclear wavepacket. Despite its potential, efficient, flexible, and easily accessible software for Gaussian wavepacket propagation is less common compared to other methods, such as surface hopping. To address this, we present Legion, a software that facilitates the development and application of classical-trajectory-guided quantum wavepacket methods. The version presented here already contains a highly flexible and fully functional ab initio multiple spawning implementation, with different strategies to improve efficiency. Legion is written in Python for data management and NumPy/Fortran for numerical operations. It is created under the umbrella of the Newton-X platform and inherits all of its electronic structure interfaces beyond other direct interfaces. It also contains new approximations that allow it to circumvent the computation of the nonadiabatic coupling, extending the electronic structure methods that can be used for multiple spawning dynamics. We test, validate, and demonstrate Legion's functionalities through multiple spawning dynamics of fulvene (CASSCF and CASPT2) and DMABN (TDDFT).

Assessing Nonadiabatic Dynamics Methods in Long Timescales

Nonadiabatic dynamics simulations complement time-resolved experiments by revealing ultrafast excited-state mechanistic information in photochemical reactions. Understanding the relaxation mechanisms of photoexcited molecules finds application in energy, material, and medicinal research. However, with substantial computational costs, the nonadiabatic dynamics simulations have been restricted to ultrafast timescales, typically less than a few picoseconds, thus neglecting a wide range of photoactivated processes occurring in much longer timescales. Before developing new methodologies, we must ask: How well do the popular nonadiabatic dynamics methods perform in a long timescale simulation? In this study, we employ the multiconfiguration time-dependent Hartree (MCTDH) and its multilayer variants (ML-MCTDH), ab initio multiple spawning (AIMS), and fewest-switches surface hopping (FSSH) methodologies to simulate the excited-states dynamics of a weakly coupled multidimensional Spin-Boson model Hamiltonian designed for a long timescale decay behavior. Our study assures that despite having very different theoretical backgrounds, all the above methods deliver qualitatively similar results. While quantum dynamics would be very costly for long timescale simulations, the trajectory-based approaches are paving the way for future advancements.

Ultrafast Dynamics of Diketopyrrolopyrrole Dimers

Diketopyrrolopyrroles (DPPs) have attracted attention for their potential applications in organic photovoltaics due to their tunable optical properties and charge-carrier mobilities. In this study, we investigate the excited-state dynamics of a DPP dimer using time-dependent density functional theory (TDDFT) and nonadiabatic molecular dynamics simulations. Our results reveal a near-barrierless hydrogen migration state intersection that facilitates ultrafast internal conversion with a lifetime of about 400 fs, leading to fluorescence quenching. Electronic density analysis along the relaxation pathway confirms a hydrogen atom transfer mechanism. These findings highlight the critical role of state intersections in the photophysical properties of DPP dimers, providing new insights for the design of functionalized DPP systems aimed at suppressing nonradiative decay for enhanced performance in photovoltaic applications.

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" Nonadiabatic Dynamics Simulation on the Ultrafast Photoisomerization of a Molecular Photoswitch Iminothioindoxyl: An RMS-CASPT2 Investigation

Iminothioindoxyl (ITI) is a new class of photoswitch that exhibits many excellent properties including well-separated absorption bands in the visible region for both conformers, ultrafast Z to E photoisomerization as well as the millisecond reisomerization at room temperature for the E isomer, and switchable ability in both solids and various solvents. However, the underlying ultrafast photoisomerization mechanism at the atomic level remains unclear. In this work, we have employed a combination of high-level RMS-CASPT2-based static electronic structure calculations and nonadiabatic dynamics simulations to investigate the ultrafast photoisomerization dynamics of ITI. Based on the minimum-energy structures, minimum-energy conical intersections, linear interpolation internal coordinate paths, and nonadiabatic dynamics simulations, the overall photoisomerization scenario of ITI upon excitation is established. Upon excitation around 416 nm, the molecule will be excited to the S2 state considering its close energy to the experimentally measured absorption maximum and larger oscillator strength, from which ultrafast decay of S2 to S1 state can take place efficiently with a time constant of 62 fs. However, the photoisomerization is not likely to complete in the S2 state since the dihedral associated with the Z to E isomerization changes little during the relaxation. Upon relaxing to the S1 state, the molecule will decay to the S0 state ultrafast with a time constant of 232 fs. In contrast, the decay of the S1 state is important for the isomerization considering that the dihedral related to the isomerization of the hopping structures is close to 90°. Therefore, the S1/S0 intersection region should be important for the isomerization of ITI. Arriving at the S0 state, the molecule can either go back to the original Z reactant or isomerize to the E products. At the end of the 500 fs simulation time, the E configuration accounts for nearly 37% of the final structures. Moreover, the photoisomerization mechanism is different from the isomerization mechanism in the ground state; i.e., instead of the inversion mechanism in the ground state, the photoisomerization prefers the rotation mechanism. Our results not only agree well with previous experimental studies but also provide some novel insights that could be helpful for future improvements in the performance of the ITI photoswitches.

Exciton energy transfer inside cavity-A benchmark study of polaritonic dynamics using the surface hopping method

Strong coupling between the molecular system and photon inside the cavity generates polaritons, which can alter reaction rates by orders of magnitude. In this work, we benchmark the surface hopping method to simulate non-adiabatic dynamics in a cavity. The comparison is made against a numerically exact method (the hierarchical equations of motion) for a model system investigating excitonic energy transfer for a broad range of parameters. Surface hopping captures the effects of the radiation mode well, both at resonance and off-resonance. We have further investigated parameters that can increase or decrease the rate of population transfer, and we find that surface hopping in general can capture both effects well. Finally, we show that the dipole self-energy term within our parameter regime does not significantly affect the system's dynamics.

Energy-Conserving Surface Hopping for Auger Processes

Auger-type processes are ubiquitous in nanoscale materials because quantum confinement enhances Coulomb interactions, and there exist large densities of states. Modeling Auger processes requires the modification of nonadiabatic (NA) molecular dynamics algorithms to include transitions caused by both NA and Coulomb couplings. The system is split into quantum and classical subsystems, e.g., electrons and vibrations, and as a result, energy conservation becomes nontrivial. In surface hopping, an electronic transition induced by NA coupling is accompanied by a classical velocity readjustment to ensure conservation of the total quantum-classical energy. A different treatment is needed for Auger transitions driven by Coulomb interactions. We develop a nonadiabatic molecular dynamics methodology that meticulously differentiates the energy redistribution accompanying hops induced by the NA coupling and the Coulomb interaction and correctly conserves the total energy at each transition. If the transition is driven by a Coulomb interaction, the hop energy is redistributed within the quantum electronic subsystem only. If the transition is NA, the energy is redistributed between the quantum and classical subsystems. Properly maintaining energy conservation for both types of transitions is crucial to generate a correct order of events, obtain accurate transition times, maintain a proper statistical distribution of state populations, and reach thermodynamic equilibrium. We test the method with biexciton annihilation and Auger-assisted hot electron relaxation in a CdSe quantum dot. The sequence of Auger and phonon-driven processes and the calculated time scales are in excellent agreement with the experimental results. The developed approach can be coupled with any surface-hopping method and provides a crucial practical advance to study charge-carrier dynamics in the nanoscale and condensed matter systems.

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.

Generalized Semiclassical Ehrenfest Method: A Route to Wave Function-Free Photochemistry and Nonadiabatic Dynamics with Only Potential Energies and Gradients

We reconsider recent methods by which direct dynamics calculations of electronically nonadiabatic processes can be carried out while requiring only adiabatic potential energies and their gradients. We show that these methods can be understood in terms of a new generalization of the well-known semiclassical Ehrenfest method. This is convenient because it eliminates the need to evaluate electronic wave functions and their matrix elements along the mixed quantum-classical trajectories. The new approximations and procedures enabling this advance are the curvature-driven approximation to the time-derivative coupling, the generalized semiclassical Ehrenfest method, and a new gradient correction scheme called the time-derivative matrix (TDM) scheme. When spin-orbit coupling is present, one can carry out dynamics calculations in the fully adiabatic basis using potential energies and gradients calculated without spin-orbit coupling plus the spin-orbit coupling matrix elements. Even when spin-orbit coupling is neglected, the method is useful because it allows calculations by electronic structure methods for which nonadiabatic coupling vectors are unavailable. In order to place the new considerations in context, the article starts out with a review of background material on trajectory surface hopping, the semiclassical Ehrenfest scheme, and methods for incorporating decoherence. We consider both internal conversion and intersystem crossing. We also review several examples from our group of successful applications of the curvature-driven approximation.

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.

Nuclear Quantum Effects on Nonadiabatic Dynamics of a Green Fluorescent Protein Chromophore Analogue: Ring-Polymer Surface-Hopping Simulation

Herein, we have used the "on-the-fly" ring-polymer surface-hopping simulation method with the centroid approximation (RPSH-CA), in combination with the multireference OM2/MRCI electronic structure calculations to study the photoinduced dynamics of a green fluorescent protein (GFP) chromophore analogue in the gas phase, i.e., o-HBI, at 50, 100, and 300 K with 1, 5, 10, and 15 beads (3600 1 ps trajectories). The electronic structure calculations identified five new minimum-energy conical intersection (MECI) structures, which, together with the previous one, play crucial roles in the excited-state decay dynamics of o-HBI. It is also found that the excited-state intramolecular proton transfer (ESIPT) occurs in an ultrafast manner and is completed within 20 fs in all the simulation conditions because there is no barrier associated with this ESIPT process in the S1 state. However, the other excited-state dynamical results are strongly related to the number of beads. At 50 and 100 K, the nuclear quantum effects (NQEs) are very important; therefore, the excited-state dynamical results change significantly with the bead number. For example, the S1 decay time deduced from time-dependent state populations becomes longer as the bead number increases. Nevertheless, an essentially convergent trend is observed when the bead number is close to 10. In contrast, at 300 K, the NQEs become weaker and the above dynamical results converge very quickly even with 1 bead. Most importantly, the NQEs seriously affect the excited-state decay mechanism of o-HBI. At 50 and 100 K, most trajectories decay to the S0 state via perpendicular keto MECIs, whereas, at 300 K, only twisted keto MECIs are responsible for the excited-state decay. The present work not only comprehensively explores the temperature-dependent photoinduced dynamics of o-HBI, but also demonstrates the importance and necessity of NQEs in nonadiabatic dynamics simulations, especially at relatively low temperatures.

Prediction Challenge: Simulating Rydberg photoexcited cyclobutanone with surface hopping dynamics based on different electronic structure methods

This research examines the nonadiabatic dynamics of cyclobutanone after excitation into the n → 3s Rydberg S2 state. It stems from our contribution to the Special Topic of the Journal of Chemical Physics to test the predictive capability of computational chemistry against unseen experimental data. Decoherence-corrected fewest-switches surface hopping was used to simulate nonadiabatic dynamics with full and approximated nonadiabatic couplings. Several simulation sets were computed with different electronic structure methods, including a multiconfigurational wavefunction [multiconfigurational self-consistent field (MCSCF)] specially built to describe dissociative channels, multireference semiempirical approach, time-dependent density functional theory, algebraic diagrammatic construction, and coupled cluster. MCSCF dynamics predicts a slow deactivation of the S2 state (10 ps), followed by an ultrafast population transfer from S1 to S0 (<100 fs). CO elimination (C3 channel) dominates over C2H4 formation (C2 channel). These findings radically differ from the other methods, which predicted S2 lifetimes 10-250 times shorter and C2 channel predominance. These results suggest that routine electronic structure methods may hold low predictive power for the outcome of nonadiabatic dynamics.