New implementation of the trajectory surface hopping method with use of the Zhu–Nakamura theory. II. Application to the charge transfer processes in the 3D DH2+ system

Deciphering the Internal Conversion Processes Involved in the Photochemical Ring-Opening of 1,3-Cyclohexadiene by Symmetric sp2-Carbon Substitutions

Chemical substituents hold the potential to markedly influence the photochemical behavior in molecular systems and assist in gaining a comprehensive understanding of nonadiabatic phenomena. In this study, we have conducted a comparative analysis of the influence of chemical substituents on the photochemical ring-opening of 1,3-cyclohexadiene (CHD), considering four systems: CHD, 2,3-dimethylcyclohexadiene (CHD-Me2-1), 1,4-dimethylcyclohexadiene (CHD-Me2-2), and 1,2,3,4-tetramethylcyclohexadiene (CHD-Me4), using electronic structure theory calculations and nonadiabatic molecular dynamics simulations. Employing extended multistate complete active space second-order perturbation (XMS-CASPT2) theory, we optimized reactants, S1 states, conical intersections (CIns), and products, revealing structural and energetic variations consistent with prior research. Nonadiabatic molecular dynamics simulation was used to gain insights into photochemical dynamics at state-averaged complete active space self-consistent field (SA-CASSCF) theory. CHD-Me4 exhibited reduced carbon-carbon single bond rupture rates, responsible for ring-opening, due to substituent proximity. Further, CHD-Me2-2 and CHD-Me4 displayed prolonged excited-state relaxation times, highlighting notable substituents' impact. Analysis of kinetic energy profiles of specific carbon atoms also revealed restrained atomic displacements, particularly in CHD-Me2-2 and CHD-Me4. These findings advance our understanding of how substituents modulate photochemical reactions in cyclohexadiene derivatives, guiding new molecular design and future research.

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.

Machine Learning for Electronically Excited States of Molecules

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

Machine Learning for Electronically Excited States of Molecules

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

Combining SchNet and SHARC: The SchNarc Machine Learning Approach for Excited-State Dynamics

In recent years, deep learning has become a part of our everyday life and is revolutionizing quantum chemistry as well. In this work, we show how deep learning can be used to advance the research field of photochemistry by learning all important properties-multiple energies, forces, and different couplings-for photodynamics simulations. We simplify such simulations substantially by (i) a phase-free training skipping costly preprocessing of raw quantum chemistry data; (ii) rotationally covariant nonadiabatic couplings, which can either be trained or (iii) alternatively be approximated from only ML potentials, their gradients, and Hessians; and (iv) incorporating spin-orbit couplings. As the deep-learning method, we employ SchNet with its automatically determined representation of molecular structures and extend it for multiple electronic states. In combination with the molecular dynamics program SHARC, our approach termed SchNarc is tested on two polyatomic molecules and paves the way toward efficient photodynamics simulations of complex systems.

Machine learning enables long time scale molecular photodynamics simulations

Photo-induced processes are fundamental in nature but accurate simulations of their dynamics are seriously limited by the cost of the underlying quantum chemical calculations, hampering their application for long time scales. Here we introduce a method based on machine learning to overcome this bottleneck and enable accurate photodynamics on nanosecond time scales, which are otherwise out of reach with contemporary approaches. Instead of expensive quantum chemistry during molecular dynamics simulations, we use deep neural networks to learn the relationship between a molecular geometry and its high-dimensional electronic properties. As an example, the time evolution of the methylenimmonium cation for one nanosecond is used to demonstrate that machine learning algorithms can outperform standard excited-state molecular dynamics approaches in their computational efficiency while delivering the same accuracy.

Development of semiclassical molecular dynamics simulation method

Various quantum mechanical effects such as nonadiabatic transitions, quantum mechanical tunneling and coherence play crucial roles in a variety of chemical and biological systems.

Non-Born–Oppenheimer stereodynamics study for the D+ + H2 (v, j = 0) reaction using coherent switching with decay of mixing method

A non-Born–Oppenheimer (BO) investigation using coherent switching with decay of mixing method is carried out for both reactive non-charge transfer and charge transfer channels of reaction D+ + H2 (v, j = 0). For the non-charge transfer channel, it is found that the non-BO effect has a small influence on the differential cross section, but it has a large impact on the rotational polarization of the HD product, which is reflected by the joint distribution P(θr, ϕr). And it is also found that the polarization of the charge transfer products shows a quite different form. The rotational polarizations are also investigated at various collision energies and initial vibrational states for both reactive channels.

Probing the π→π* photoisomerization mechanism of cis-azobenzene by multi-state ab initio on-the-fly trajectory dynamics simulation

Based on a newly developed algorithm to compute global nonadiabatic switching probability by using only electronic adiabatic potential energy surfaces and gradients, we performed on-the-fly, trajectory-surface hopping simulations at the 5SA-CASSCF(6,6)/6-31G quantum level to probe the π→π* photoisomerization mechanism of the azobenzene within four singlet low-lying electronic states (S0, S1, S2, and S3) coupled with a complicated conical intersection network. We found that four conical intersections between the S1 and S2 states (one is near the cis-isomer region, another near the trans-isomer region, and two others between cis and trans) play the most important roles for understanding the photoisomerization mechanism of azobenzene upon S2 and S3ππ* excitation. We studied six cases to demonstrate the photoisomerization mechanism in detail by choosing eight (six) typical reactive (nonreactive) trajectories, namely, two-step fast-fast processes having lifetimes of several tenths to one hundred femtoseconds and two-step, fast-slow and slow-slow processes having lifetimes of several hundred to one thousand femtoseconds. We found for the first time from simulation that once a trajectory visits the conical intersection near the trans-isomer after ππ* excitation, it could rapidly go through the inversion pathway to trans-azobenzene, and confirms the most recent experimental observations. We performed 536 sampling trajectories (336 from S2 and 200 from S3), initially starting from the Franck-Condon region of cis-azobenzene, and obtained a total reactive quantum yield of 0.3-0.45 in very good agreement with recent experimental results of 0.24-0.50. Moreover, the current method can estimate overall nonadiabatic transition probability for each sampling trajectory from beginning to end. This can greatly accelerate convergence of nonadiabatic molecular dynamic simulation, and, for instance, results in a quantum yield of 0.53 estimated from only eight typical reactive trajectories.

Trajectory-based nonadiabatic molecular dynamics without calculating nonadiabatic coupling in the avoided crossing case: trans↔cis photoisomerization in azobenzene

We develop a novel method to simulate analytical nonadiabatic switching probability based on effective coupling and effective collision energy by using only electronic adiabatic potential energy surfaces and its gradients in the case of avoided crossing types of nonadiabatic transitions. In addition, the present method can keep the same time step for computing both on-the-fly trajectory and nonadiabatic transitions accurately. The present method is most useful for localized nonadiabatic transitions induced by conical intersection. We employ the on-the-fly surface hopping algorithm with an ab initio quantum chemistry calculation to demonstrate a dynamic simulation for photoisomerization in azobenzene. Simulated quantum yield and lifetime converge to 0.39 and 53 femtosecond, respectively (0.33 and 0.81 picosecond) for cis-to-trans (trans-to-cis) photoisomerization with up to 800 (600) sampling trajectories. The present results agree well with those of the experiment, as well as results simulated with use of nonadiabatic coupling within Tully's fewest switching method. The present trajectory-based nonadiabatic molecular dynamics free from nonadiabatic coupling greatly enhances the simulation power of molecular dynamics for large complex chemical systems.

ELECTRON TRANSFER RATE UNIFORMLY VALID FROM NONADIABATIC TO ADIABATIC REGIME BASED ON THE ZHU–NAKAMURA THEORY

On the basis of the generalized nonadiabatic transition state theory recently introduced to remedy the crucial deficiencies of the conventional transition state theory, we have presented a new formula for electron transfer rate, which can cover the whole range from adiabatic to nonadiabatic regime in the absence of solvent dynamics control. The rate is expressed as a product of the well-known Marcus theory and a new coefficient that represents the effects of nonadiabatic transition at the crossing seam surface. The numerical comparisons are performed with different approaches and the present approach shows an excellent agreement with the quantum mechanical numerical solutions from weak to strong electronic coupling. The explanation of the experimental data of Nelsen et al. manifests the potential applicability of the present theory.

Quantum and semiclassical theories for nonadiabatic transitions based on overlap integrals related to fast degrees of freedom

Alternative treatments of quantum and semiclassical theories for nonadiabatic dynamics are presented. These treatments require no derivative couplings and instead are based on overlap integrals between eigenstates corresponding to fast degrees of freedom, such as electronic states. Derived from mathematical transformations of the Schrödinger equation, the theories describe nonlocal characteristics of nonadiabatic transitions. The idea that overlap integrals can be used for nonadiabatic transitions stems from an article by Johnson and Levine [Chem. Phys. Lett. 13, 168 (1972)]. Furthermore, overlap integrals in path-integral form have been recently made available by Schmidt and Tully [J. Chem. Phys. 127, 094103 (2007)] to analyze nonadiabatic effects in thermal equilibrium systems. The present paper expands this idea to dynamic problems presented in path-integral form that involve nonadiabatic semiclassical propagators. Applications to one-dimensional nonadiabatic transitions have provided excellent results, thereby verifying the procedure. In principle these theories that are presented can be applied to multidimensional systems, although numerical costs could be quite expensive.

Coriolis coupling and nonadiabaticity in chemical reaction dynamics

The nonadiabatic quantum dynamics and Coriolis coupling effect in chemical reaction have been reviewed, with emphasis on recent progress in using the time-dependent wave packet approach to study the Coriolis coupling and nonadiabatic effects, which was done by K. L. Han and his group. Several typical chemical reactions, for example, H+D(2), F+H(2)/D(2)/HD, D(+)+H(2), O+H(2), and He+H(2)(+), have been discussed. One can find that there is a significant role of Coriolis coupling in reaction dynamics for the ion-molecule collisions of D(+)+H(2), Ne+H(2)(+), and He+H(2)(+) in both adiabatic and nonadiabatic context.

Semiclassical Treatment of Thermally Activated Electron Transfer in the Inverted Region under the Fast Dielectric Relaxation

The previously formulated semiclassical theory (Zhao, Liang, and Nakamura, J. Phys. Chem. A 2006, 110, 8204) is used to study electron transfer in the Marcus inverted case by considering multidimensional potential energy surfaces of donor and acceptor. The Zhu-Nakamura formulas of nonadiabatic transition in the case of Landau-Zener type are incorporated into the approach. The theory properly takes into account the nonadiabatic transition coupled with the nuclear tunneling and can cover the whole range from weak to strong coupling regime uniformly under the assumption of fast solvent relaxation. The numerical calculations are performed for the 12-dimensional model of shifted harmonic oscillators and demonstrate that the reaction rate with respect to the electronic coupling shows a maximum, confirming the adiabatic suppression in the strong coupling limit. The adiabatic suppression is dramatically reduced by the effect of nuclear tunneling compared to the case that the Landau-Zener formula is used. The possible extension and applications to the case of the slow solvent dynamics are discussed.

Semiclassical theory of electronically nonadiabatic chemical dynamics: incorporation of the Zhu-Nakamura theory into the frozen Gaussian propagation method

The title theory is developed by combining the Herman-Kluk semiclassical theory for adiabatic propagation on single potential-energy surface and the semiclassical Zhu-Nakamura theory for nonadiabatic transition. The formulation with use of natural mathematical principles leads to a quite simple expression for the propagator based on classical trajectories and simple formulas are derived for overall adiabatic and nonadiabatic processes. The theory is applied to electronically nonadiabatic photodissociation processes: a one-dimensional problem of H2+ in a cw (continuous wave) laser field and a two-dimensional model problem of H2O in a cw laser field. The theory is found to work well for the propagation duration of several molecular vibrational periods and wide energy range. Although the formulation is made for the case of laser induced nonadiabatic processes, it is straightforwardly applicable to ordinary electronically nonadiabatic chemical dynamics.

Inelastic scattering matrix elements for the nonadiabatic collision B(P1∕22)+H2(Σg+1,j)↔B(P3∕22)+H2(Σg+1,j′)

Inelastic scattering matrix elements for the nonadiabatic collision B(2P1/2)+H2(1Sigmag+,j)<-->B(2P3/2)+H2(1Sigmag+,j') are calculated using the time dependent channel packet method (CPM). The calculation employs 1 2A', 2 2A', and 1 2A" adiabatic electronic potential energy surfaces determined by numerical computation at the multireference configuration-interaction level [M. H. Alexander, J. Chem. Phys. 99, 6041 (1993)]. The 1 2A' and 2 2A', adiabatic electronic potential energy surfaces are transformed to yield diabatic electronic potential energy surfaces that, when combined with the total B+H2 rotational kinetic energy, yield a set of effective potential energy surfaces [M. H. Alexander et al., J. Chem. Phys. 103, 7956 (1995)]. Within the framework of the CPM, the number of effective potential energy surfaces used for the scattering matrix calculation is then determined by the size of the angular momentum basis used as a representation. Twenty basis vectors are employed for these calculations, and the corresponding effective potential energy surfaces are identified in the asymptotic limit by the H2 rotor quantum numbers j=0, 2, 4, 6 and B electronic states 2Pja, ja=1/2, 3/2. Scattering matrix elements are obtained from the Fourier transform of the correlation function between channel packets evolving in time on these effective potential energy surfaces. For these calculations the H2 bond length is constrained to a constant value of req=1.402 a.u. and state to state scattering matrix elements corresponding to a total angular momentum of J=1/2 are discussed for j=0<-->j'=0,2,4 and 2P1/2<-->2P1/2, 2P3/2 over a range of total energy between 0.0 and 0.01 a.u.

Generalized trajectory surface hopping method based on the Zhu-Nakamura theory

We present a generalized formulation of the trajectory surface hopping method applicable to a general multidimensional system. The method is based on the Zhu-Nakamura theory of a nonadiabatic transition and therefore includes the treatment of classically forbidden hops. The method uses a generalized recipe for the conservation of angular momentum after forbidden hops and an approximation for determining a nonadiabatic transition direction which is crucial when the coupling vector is unavailable. This method also eliminates the need for a rigorous location of the seam surface, thereby ensuring its applicability to a wide class of chemical systems. In a test calculation, we implement the method for the DH(2) (+) system, and it shows a remarkable agreement with the previous results of C. Zhu, H. Kamisaka, and H. Nakamura, [J. Chem. Phys. 116, 3234 (2002)]. We then apply it to a diatomic-in-molecule model system with a conical intersection, and the results compare well with exact quantum calculations. The successful application to the conical intersection system confirms the possibility of directly extending the present method to an arbitrary potential of general topology.

Unified semiclassical theory for the two-state system: An analytical solution for general nonadiabatic tunneling

Unified semiclasical solution for general nonadiabatic tunneling between two adiabatic potential energy surfaces is established by employing unified semiclassical solution for pure nonadiabatic transition [C. Zhu, J. Chem. Phys. 105, 4159 (1996)] with the certain symmetry transformation. This symmetry comes from a detailed analysis of the reduced scattering matrix for Landau-Zener type of crossing as a special case of nonadiabatic transition and nonadiabatic tunneling. Traditional classification of crossing and noncrossing types of nonadiabatic transition can be quantitatively defined by the rotation angle of adiabatic-to-diabatic transformation, and this rotational angle enters the analytical solution for general nonadiabatic tunneling. The certain two-state exponential potential models are employed for numerical tests, and the calculations from the present general nonadiabatic tunneling formula are demonstrated in very good agreement with the results from exact quantum mechanical calculations. The present general nonadiabatic tunneling formula can be incorporated with various mixed quantum-classical methods for modeling electronically nonadiabatic processes in photochemistry.

Semiclassical Treatment of Thermally Activated Electron Transfer in the Intermediate to Strong Electronic Coupling Regime under the Fast Dielectric Relaxation

The generalized nonadiabatic transition-state theory (NA-TST) (Zhao, Y.; et al. J. Chem. Phys. 2004, 121, 8854) is used to study electron transfer with use of the Zhu-Nakamura (ZN) formulas of nonadiabatic transition in the case of fast dielectric relaxation. The rate constant is expressed as a product of the well-known Marcus formula and a coefficient which represents the correction due to the strong electronic coupling. In the case of general multidimensional systems, the Monte Carlo approach is utilized to evaluate the rate by taking into account the multidimensionality of the crossing seam surface. Numerical demonstration is made by using a model system of a collection of harmonic oscillators in the Marcus normal region. The results are naturally coincident with the perturbation theory in the weak electronic coupling limit; while in the intermediate to strong electronic coupling regime where the perturbation theory breaks down the present results are in good agreement with those from the quantum mechanical flux-flux correlation function within the model of effective one-dimensional mode.

Evaluation of canonical and microcanonical nonadiabatic reaction rate constants by using the Zhu–Nakamura formulas

We consider a problem of calculating both thermal and microcanonical rate constants for nonadiabatic chemical reactions. Instead of using the conventional transition state theory, we use a generalized seam surface and introduce a concept of a coordinate dependent effective nonadiabatic transition probability based on the Zhu-Nakamura theory which can treat the nonadiabatic tunneling properly. The present approach can be combined with Monte Carlo method so as to be applicable to chemical reactions in complicated systems. The method is demonstrated to work well in wide energy and temperature range. Numerical tests also show that it is very essential for accurate evaluation of the thermal rate constant to use the generalized seam surface and take into account the nonadiabatic tunneling effect.