Trajectory Surface Hopping Research Articles

Machine Learning Nonadiabatic Dynamics: Eliminating Phase Freedom of Nonadiabatic Couplings with the State-Interaction State-Averaged Spin-Restricted Ensemble-Referenced Kohn-Sham Approach.

Excited-state molecular dynamics (ESMD) simulations near conical intersections (CIs) pose significant challenges when using machine learning potentials (MLPs). Although MLPs have gained recognition for their integration into mixed quantum-classical (MQC) methods, such as trajectory surface hopping (TSH), and their capacity to model correlated electron-nuclear dynamics efficiently, difficulties persist in managing nonadiabatic dynamics. Specifically, singularities at CIs and double-valued coupling elements result in discontinuities that disrupt the smoothness of predictive functions. Partial solutions have been provided by learning diabatic Hamiltonians with phaseless loss functions to these challenges. However, a definitive method for addressing the discontinuities caused by CIs and double-valued coupling elements has yet to be developed. Here, we introduce the phaseless coupling term, Δ2, derived from the square of the off-diagonal elements of the diabatic Hamiltonian in the state-interaction state-averaged spin-restricted ensemble-referenced Kohn-Sham (SI-SA-REKS, briefly SSR)(2,2) formalism. This approach improves the stability and accuracy of the MLP model by addressing the issues arising from CI singularities and double-valued coupling functions. We apply this method to the penta-2,4-dieniminium cation (PSB3), demonstrating its effectiveness in improving MLP training for ML-based nonadiabatic dynamics. Our results show that the Δ2-based ML-ESMD method can reproduce ab initio ESMD simulations, underscoring its potential and efficiency for broader applications, particularly in large-scale and long-time scale ESMD simulations.

Simulation of Ultrafast Excited-State Dynamics in Fe(II) Complexes: Assessment of Electronic Structure Descriptions.

The assessment of electronic structure descriptions utilized in the simulation of the ultrafast excited-state dynamics of Fe(II) complexes is presented. Herein, we evaluate the performance of the RPBE, OPBE, BLYP, B3LYP, B3LYP*, PBE0, TPSSh, CAM-B3LYP, and LC-BLYP (time-dependent) density functional theory (DFT/TD-DFT) methods in full-dimensional trajectory surface hopping (TSH) simulations carried out on linear vibronic coupling (LVC) potentials. We exploit the existence of time-resolved X-ray emission spectroscopy (XES) data for the [Fe(bmip)2]2+ and [Fe(terpy)2]2+ prototypes for dynamics between metal-to-ligand charge-transfer (MLCT) and metal-centered (MC) states, which serve as a reference to benchmark the calculations (bmip = 2,6-bis(3-methyl-imidazole-1-ylidine)-pyridine, terpy = 2,2':6',2″-terpyridine). The results show that the simulated ultrafast population dynamics between MLCT and MC states with various spin multiplicities (singlet, triplet, and quintet) highly depend on the utilized DFT/TD-DFT method, with the percentage of exact (Hartree-Fock) exchange being the governing factor. Importantly, B3LYP* and TPSSh are the only DFT/TD-DFT methods with satisfactory performance, best reproducing the experimentally resolved dynamics for both complexes, signaling an optimal balance in the description of MLCT-MC energetics. This work demonstrates the power of combining TSH/LVC dynamics simulations with time-resolved experimental reference data to benchmark full-dimensional potential energy surfaces.

Spin-Orbit Coupling and Admixture Coefficients in SA-CASSCF and MS-CASPT2, and Triplet Excitation Yield Simulated via Trajectory Surface Hopping and Calibrated SA-CASSCF in 1,2-Dioxetane Derivatives.

The energy gaps, spin-orbit coupling (SOC), and admixture coefficients over a series of the configurations are evaluated by the SA-CASSCF/6-31G, SA-CASSCF/6-31G*, SA-CASSCF/ANO-RCC-VDZP, and MS-CASPT2/ANO-RCC-VDZP to reveal the extent of the inaccuracy of the SA-CASSCF. By comparing the mean absolute errors for the energy gaps and the admixture coefficient magnitudes (ACMs) measured between the SA-CASSCF/6-31G, SA-CASSCF/6-31G*, or SA-CASSCF/ANO-RCC-VDZP and the MS-CASPT2/ANO-RCC-VDZP, the SA-CASSCF/6-31G is selected as the electronic structure method in the nonadiabatic molecular dynamics simulation. The major components of the ACMs of the SA-CASSCF/6-31G and MS-CASPT2/ANO-RCC-VDZP are identified and compared; we find that the ACMs are underestimated by the SA-CASSCF/6-31G, which is verified by the reasonable triplet quantum yield simulated by the trajectory surface hopping and the calibrated SA-CASSCF/6-31G. The magnitude of the singlet-triplet mixing positively correlates to the hopping probability between the mixed singlet and triplet states, which is confirmed by the computed S-T transition probability.

What Two-Dimensional Electronic Spectroscopy Can Tell Us about Energy Transfer in Dendrimers: Ab Initio Simulations.

Two-dimensional (2D) electronic spectra of the phenylene ethynylene dendrimer with 2-ring and 3-ring branches were evaluated by combining the on-the-fly trajectory surface hopping nonadiabatic dynamics and the doorway-window simulation protocol. The ground state bleach (GSB), stimulated emission (SE), and excited-state absorption (ESA) contributions to the 2D signal were obtained and carefully analyzed. The results demonstrate that the ultrafast intramolecular nonadiabatic excited-state energy transfer (EET) from the 2-ring to the 3-ring units is comprehensively characterized by the SE and ESA signals. It is proven that the monitoring of the 2D ESA signal is especially convenient, because it is spectrally well separated from the SE and GSB signals. Hence, 2D electronic spectroscopy provides a detailed and multifaceted view of the intramolecular EET process in dendrimers.

Nonadiabatic ab initio chemical reaction dynamics for the photoisomerization reaction of 3,5-dimethylisoxazole via the S1 electronic state.

Nonadiabatic ab initio molecular dynamics simulations were performed to explore the photoisomerization pathway from isoxazole (iso-OXA) to oxazole (OXA), considering four electronic states. The XMS-CASPT2 and SA4-CASSCF theories were employed to describe these electronic structures, which were caused by 12 electrons in 11 orbitals with the cc-pVDZ + sp diffuse basis set; the Gaussian s- and p-type diffuse functions were extracted from Dunning's aug-cc-pVDZ function. The potential energy and its gradient at each time step were computed on-the-fly at these levels in the time evolution of the classical trajectory. When the two electronic states were close to each other, the trajectory surface hopping (TSH) judgment between the two adjacent states was carried out by the anteater procedure based on the Zhu-Nakamura formula (ZN-TSH). The two different excited state lifetimes were found to exist in the first electronic state (S1), estimated at 10.77 and 119.81 fs. Upon photoexcitation, the N-O bond breaks and energetically relaxes to the ground state (S0). In the pathway leading to the main product, azirine formation, the 5-membered ring retains a planar structure while undergoing a non-adiabatic transition with an increasing N-O bond distance. Furthermore, it was verified that a 1,2-shift takes place in the pathway that results in the production of ketenimine, causing a nonadiabatic transition.

Ultrafast structural dynamics of UV photoexcited cis,cis-1,3-cyclooctadiene observed with time-resolved electron diffraction.

Conjugated diene molecules are highly reactive upon photoexcitation and can relax through multiple reaction channels that depend on the position of the double bonds and the degree of molecular rigidity. Understanding the photoinduced dynamics of these molecules is crucial for establishing general rules governing the relaxation and product formation. Here, we investigate the femtosecond time-resolved photoinduced excited-state structural dynamics of cis,cis-1,3-cyclooctadiene, a large-flexible cyclic conjugated diene molecule, upon excitation with 200 nm using mega-electron-volt ultrafast electron diffraction and trajectory surface hopping dynamics simulations. We tracked the photoinduced structural changes from the Franck-Condon region through the conical intersection seam to the ground state. Our findings revealed a novel primary reaction coordinate involving ring distortion, where the ring stretches along one axis and compresses along the perpendicular axis. The nuclear wavepacket remains compact along this reaction coordinate until it reaches the conical intersection seam, and it rapidly spreads as it approaches the ground state, where multiple products are formed.

Photoisomerization Dynamics of Azo-Escitalopram Using Surface Hopping and a Semiempirical Method.

The photoisomerization dynamics of azo-escitalopram, a synthetic photoswitchable inhibitor of the human serotonin transporter, is investigated in both gas-phase and water. We use the trajectory surface hopping method─as implemented in SHARC─interfaced with the floating occupation molecular orbital-configuration interaction semiempirical method to calculate on-the-fly energies, forces, and couplings. The inclusion of explicit water molecules is enabled using an electrostatic quantum mechanics/molecular mechanics framework. We find that the photoisomerization quantum yield of trans-azo-escitalopram is wavelength- and environment-dependent, with n → π* excitation yielding higher quantum yields than π → π* excitation. Additionally, we observe the formation of two distinct cis-isomers in the photoisomerization from the most thermodynamically stable trans-isomer, with formation rates influenced by both the excitation window and the surrounding environment. We predict longer excited-state lifetimes than those reported for azobenzene, suggesting that the escitalopram moiety contributes to prolonged lifetimes and slower torsional motions.

Sensitivity Analysis in Photodynamics: How Does the Electronic Structure Control cis-Stilbene Photodynamics?

The techniques of computational photodynamics are increasingly employed to unravel reaction mechanisms and interpret experiments. However, misinterpretations in nonadiabatic dynamics caused by inaccurate underlying potentials are often difficult to foresee. This work focuses on revealing the systematic errors in the nonadiabatic simulations due to the underlying potentials and suggests a thrifty approach to evaluate the sensitivity of the simulations to the potential. This issue is exemplified in the photochemistry of cis-stilbene, where similar experimental outcomes have been differently interpreted based on the electronic structure methods supporting nonadiabatic dynamics. We examine the predictions of cis-stilbene photochemistry using trajectory surface hopping methods coupled with various electronic structure methods (OM3-MRCISD, SA2-CASSCF, XMS-SA2-CASPT2, and XMS-SA3-CASPT2) and assess their ability to interpret experimental observations. While the excited-state lifetimes and calculated photoelectron spectra show consistency with experiments, the reaction quantum yields vary significantly: either completely suppressing cyclization or isomerization. Intriguingly, analyzing stationary points on the potential energy surface does not hint at any major discrepancy, making the electronic structure methods seemingly reliable when treated separately. We show that performing an ensemble of simulations with different potentials provides an estimate of the electronic structure sensitivity. However, this ensemble approach is costly. Thus, we propose running nonadiabatic simulations with an external bias at a resource-efficient underlying potential (semiempirical or machine-learned) for the sensitivity analysis. We demonstrate this approach using a semiempirical OM3-MRCISD method with a harmonic bias toward cyclization.

Non-adiabatic Couplings in Surface Hopping with Tight Binding Density Functional Theory: The Case of Molecular Motors.

Nonadiabatic molecular dynamics (NAMD) has become an essential computational technique for studying the photophysical relaxation of molecular systems after light absorption. These phenomena require approximations that go beyond the Born-Oppenheimer approximation, and the accuracy of the results heavily depends on the electronic structure theory employed. Sophisticated electronic methods, however, make these techniques computationally expensive, even for medium size systems. Consequently, simulations are often performed on simplified models to interpret the experimental results. In this context, a variety of techniques have been developed to perform NAMD using approximate methods, particularly density functional tight binding (DFTB). Despite the use of these techniques on large systems, where ab initio methods are computationally prohibitive, a comprehensive validation has been lacking. In this work, we present a new implementation of trajectory surface hopping combined with DFTB, utilizing nonadiabatic coupling vectors. We selected the methaniminium cation and furan systems for validation, providing an exhaustive comparison with the higher-level electronic structure methods. As a case study, we simulated a system from the class of molecular motors, which has been extensively studied experimentally but remains challenging to simulate with ab initio methods due to its inherent complexity. Our approach effectively captures the key photophysical mechanism of dihedral rotation after the absorption of light. Additionally, we successfully reproduced the transition from the bright to dark states observed in the time-dependent fluorescence experiments, providing valuable insights into this critical part of the photophysical behavior in molecular motors.

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

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

Frustrated charge transfer in vibrationally inelastic Ar++N2 collisions via hard collision glory scattering

Vibrational energy transfer in collisions between ions and neutrals is a fundamental process in interstellar media, planetary atmospheres, and plasmas. The conventional wisdom is that glancing collisions with large impact parameters are forward-scattered with low vibrational excitation, while hard collisions with small impact parameters are sideway- or backward-scattered with relatively high vibrational excitation. Here, we report experimental observations with a three-dimensional velocity-map imaging crossed-beam apparatus in the inelastic scattering process Ar++N2(v′′ = 0, J′′)→Ar++N2(v′, J′), where all the vibrationally excited N2 products are dominated by forward scattering, contradicting the textbook model. Trajectory surface hopping calculations not only reproduced the experimental observation qualitatively, but also revealed that the vibrational excitation mainly occurs through a transient charge-transfer process. The hard collision glory mechanism, which has so far only been observed in inelastic rotational energy transfer between neutrals, is shown to play a major role for vibrational excitation in the inelastic Ar++N2 collision, via the frustrated charge transfer process.

Multiconfigurational Surface Hopping: a Time-Dependent Variational Approach with Momentum-Jump Trajectories.

The Ehrenfest mean field dynamics and trajectory surface hopping have been widely used in nonadiabatic dynamics simulations. Based on the time-dependent variational principle (TDVP), the multiconfigurational Ehrenfest (MCE) method has also been developed and can be regarded as a multiconfigurational extension of the traditional Ehrenfest dynamics. However, it is not straightforward to apply the TDVP to surface hopping trajectories because there exists momentum jump during surface hops. To solve this problem, we here propose a multiconfigurational surface hopping (MCSH) method, where continuous momenta are obtained by linear interpolation and the interpolated trajectories are used to construct the basis functions for TDVP in a postprocessing manner. As demonstrated in a series of representative spin-boson models, MCSH achieves high accuracy with only several hundred trajectory bases and can uniformly improve the performance of surface hopping. In principle, MCSH can be combined with all kinds of mixed quantum-classical trajectories and thus has the potential to properly describe general nonadiabatic dynamics.

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

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

Influence of Singlet Oxygen in H + O2 Collision Reaction.

The collision reaction of H + O2 = OH + O is a pivotal step in combustion. To investigate the influence of singlet oxygen on this reaction, we computed potential energy surfaces (PESs) for all six lowest states using high-level ab initio methods and coupled them with embedded atom neural network (EANN) fitting. By integrating quasi-classical trajectory (QCT) with trajectory surface hopping (TSH) based on the fitted PESs, we simulated the dynamics of both ground- and excited-states to derive the reaction rate constants for the forward and reverse processes. The results reveal that the forward reaction facilitates radical generation, promoting combustion reactions. Furthermore, calculations of reverse reaction rate constants indicate that all electronic states ultimately yield ground-state oxygen, leading to radical deactivation and exerting an inhibitory effect on combustion processes.

The fewest switches surface hopping as an optimisation problem

Several ways of defining the probabilities of quantum state transitions in nonadiabatic trajectory surface hopping simulations have been proposed and successfully applied to date. Despite their success, there is a question about the uniqueness of the ways to define such probabilities – some formulations require mathematically-motivated but still ad hoc assumptions to resolve the otherwise under-determined problem. In this work, I present a new approach (termed FSSH-3) to define the hop proposal probabilities. Unlike other approaches, it does not require any ad hoc assumptions and is based on the least squares optimisation of certain functionals combined with particular choices of initial conditions for the target variables. A comprehensive comparative study of several established surface hopping methodologies is conducted using several multiple-state model Hamiltonians. The recently reported formulation of the FSSH-2 method is extended to the case of multiple states and validated. It is demonstrated that the performance of all studied methods, including the new FSSH-3 (in two variants) is robust with respect to integration time step used but only if local diabatisation approach is used. Under such conditions, all methods yield nearly identical results, in excellent agreement with fully quantum simulations, even with sufficiently large integration time steps.

Non-adiabatic dynamics of photoexcited cyclobutanone: Predicting structural measurements from trajectory surface hopping with XMS-CASPT2 simulations.

Over the years, theoretical calculations and scalable computer simulations have complemented ultrafast experiments, as they offer the advantage of overcoming experimental restrictions and having access to the whole dynamics. This synergy between theory and experiment promises to yield a deeper understanding of photochemical processes, offering valuable insights into the behavior of complex systems at the molecular level. However, the ability of theoretical models to predict ultrafast experimental outcomes has remained largely unexplored. In this work, we aim to predict the electron diffraction signals of an upcoming ultrafast photochemical experiment using high-level electronic structure calculations and non-adiabatic dynamics simulations. In particular, we perform trajectory surface hopping with extended multi-state complete active space with second order perturbation simulations for understanding the photodissociation of cyclobutanone (CB) upon excitation at 200nm. Spin-orbit couplings are considered for investigating the role of triplet states. Our simulations capture the bond cleavage after ultrafast relaxation from the 3s Rydberg state, leading to the formation of the previously observed primary photoproducts: CO + cyclopropane/propene (C3 products), ketene, and ethene (C2 products). The ratio of the C3:C2 products is found to be about 1:1. Within 700fs, the majority of trajectories transition to their electronic ground state, with a small fraction conserving the initial cyclobutanone ring structure. We found a minimal influence of triplet states during the early stages of the dynamics, with their significance increasing at later times. We simulate MeV-ultrafast electron diffraction (UED) patterns from our trajectory results, linking the observed features with specific photoproducts and the underlying structural dynamics. Our analysis reveals highly intense features in the UED signals corresponding to the photochemical processes of CB. These features offer valuable insights into the experimental monitoring of ring opening dynamics and the formation of C3 and C2 photoproducts.

How Do the Position and Number of Methyl Substituents Affect the Photochemical Process of Criegee Intermediate? Trajectory Surface-Hopping Dynamics of Four-Carbon CIs.

Electronic-structure calculations combined with nonadiabatic trajectory surface-hopping (TSH) dynamic simulations were carried out on two alkenyl-substituted Criegee intermediates (CIs), i.e., propenyl-substituted CI (PCI) and 1-methyl-propenyl substituted CI (MPCI), in order to investigate the influence of the position and number of substituents on the photochemical process of CI in S1 states. It is found that they play critical roles in the reactivity, dominant product channel, and mechanism of the CIs. More specifically, introducing a methyl group on either C1 (α-C) or C3 (γ-C) position of a vinyl-substituted CI (VCI) skeleton facilitates the rotation of the C1═O1 bond and leads to the formation of a three-membered dioxirane ring; meanwhile, it evidently enhances the reactively of the S1-state molecule. Meanwhile, methyl substitution on the vinyl moiety [i.e., C2 (β-C) and C3 (γ-C) positions] is beneficial for the rotation of the C2═C3 bond and thus facilitates the formation of the five-membered 1,2-dioxole ring, and the substitution on C2 site decreases the reactivity. The cosubstitution of C2 and C3 atoms by methyl groups well balances the features of VCI in the sense of high reactivity, consistently predominant channel, and possible dioxole side-product. The findings here not only deepen the knowledge on the photochemical processes of the CI but also inspire the rethinking of the "old" concept of substitution effect.

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

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

Nonadiabatic molecular dynamics with subsystem density functional theory: application to crystalline pentacene

In this work, we report the development and assessment of the nonadiabatic molecular dynamics approach with the electronic structure calculations based on the linearly scaling subsystem density functional method. The approach is implemented in an open-source embedded Quantum Espresso/Libra software specially designed for nonadiabatic dynamics simulations in extended systems. As proof of the applicability of this method to large condensed-matter systems, we examine the dynamics of nonradiative relaxation of excess excitation energy in pentacene crystals with the simulation supercells containing more than 600 atoms. We find that increased structural disorder observed in larger supercell models induces larger nonadiabatic couplings of electronic states and accelerates the relaxation dynamics of excited states. We conduct a comparative analysis of several quantum-classical trajectory surface hopping schemes, including two new methods proposed in this work (revised decoherence-induced surface hopping and instantaneous decoherence at frustrated hops). Most of the tested schemes suggest fast energy relaxation occurring with the timescales in the 0.7–2.0 ps range, but they significantly overestimate the ground state recovery rates. Only the modified simplified decay of mixing approach yields a notably slower relaxation timescales of 8–14 ps, with a significantly inhibited ground state recovery.

Vibrational predissociation study of NeIBr(A) using Quasiclassical Trajectories and Trajectory Surface Hopping methods

The lifetimes of NeIBr were obtained for the vibrational levels v=12,…,19 and for the van der Waals (vdW) level n=1. Was used the Quasi-Classical Trajectories (QCT) and the Trajectory Surface Hopping (TSH) under a diabatic representation. A kinetic mechanism was implemented to study the vibrational predissociation. For this purpose, a bunch of trajectories were taken in order to properly describe the energy transfer up to 5 quanta from the IBr diatom to vdW modes. The results obtained were generally consistent with previous experimental results for this system.