Conical intersections and semiclassical trajectories: comparison to accurate quantum dynamics and analyses of the trajectories

Semiclassical trajectory methods are tested for electronically nonadiabatic systems with conical intersections. Five triatomic model systems are presented, and each system features two electronic states that intersect via a seam of conical intersections (CIs). Fully converged, full-dimensional quantum mechanical scattering calculations are carried out for all five systems at energies that allow for electronic de-excitation via the seam of CIs. Several semiclassical trajectory methods are tested against the accurate quantum mechanical results. For four of the five model systems, the diabatic representation is the preferred (most accurate) representation for semiclassical trajectories, as correctly predicted by the Calaveras County criterion. Four surface hopping methods are tested and have overall relative errors of 40%-60%. The semiclassical Ehrenfest method has an overall error of 66%, and the self-consistent decay of mixing (SCDM) and coherent switches with decay of mixing (CSDM) methods are the most accurate methods overall with relative errors of approximately 32%. Furthermore, the CSDM method is less representation dependent than both the SCDM and the surface hopping methods, making it the preferred semiclassical trajectory method. Finally, the behavior of semiclassical trajectories near conical intersections is discussed.     

Algorithmic decoherence time for decay-of-mixing non-Born-Oppenheimer dynamics

The performance of an analytical expression for algorithmic decoherence time is investigated for non-Born-Oppenheimer molecular dynamics. There are two terms in the function that represents the dependence of the decoherence time on the system parameters; one represents decoherence due to the quantum time-energy uncertainty principle and the other represents a back reaction from the decoherent force on the classical trajectory. We particularly examine the question of whether the first term should dominate. Five one-dimensional two-state model systems that represent limits of multidimensional nonadiabatic dynamics are designed for testing mixed quantum-classical methods and for comparing semiclassical calculations with exact quantum calculations. Simulations are carried out with the semiclassical Ehrenfest method (SE), Tully's fewest switch version (TFS) of the trajectory surface hopping method, and the decay-of-mixing method with natural switching, coherent switching (CSDM), and coherent switching with reinitiation (CSDM-D). The CSDM method is demonstrated to be the most accurate method, and it has several desirable features: (i) It behaves like the representation-independent SE method in the strong nonadiabatic coupling regions; (ii) it behaves physically like the TFS method in noninteractive region; and (iii) the trajectories are continuous with continuous momenta. The CSDM method is also demonstrated to balance coherence well with decoherence, and the results are nearly independent of whether one uses the adiabatic or diabatic representation. The present results provide new insight into the formulation of a physically correct decoherence time to be used with the CSDM method for non-Born-Oppenheimer molecular dynamic simulations.     

Non-Born-Oppenheimer trajectories with self-consistent decay of mixing

A semiclassical trajectory method, called the self-consistent decay of mixing (SCDM) method, is presented for the treatment of electronically nonadiabatic dynamics. The SCDM method is a modification of the semiclassical Ehrenfest (SE) method (also called the semiclassical time-dependent self-consistent-field method) that solves the problem of unphysical mixed final states by including decay-of-mixing terms in the equations for the evolution of the electronic state populations. These terms generate a force, called the decoherent force (or dephasing force), that drives the electronic component of each trajectory toward a pure state. Results for several mixed quantum-classical methods, in particular the SCDM, SE, and natural-decay-of-mixing methods and several trajectory surface hopping methods, are compared to the results of accurate quantum mechanical calculations for 12 cases involving five different fully dimensional triatomic model systems. The SCDM method is found to be the most accurate of the methods tested. The method should be useful for the simulation of photochemical reactions.     

Adiabatic states derived from a spin-coupled diabatic transformation: semiclassical trajectory study of photodissociation of HBr and the construction of potential curves for LiBr+

The development of spin-coupled diabatic representations for theoretical semiclassical treatments of photodissociation dynamics is an important practical goal, and some of the assumptions required to carry this out may be validated by applications to simple systems. With this objective, we report here a study of the photodissociation dynamics of the prototypical HBr system using semiclassical trajectory methods. The valence (spin-free) potential energy curves and the permanent and transition dipole moments were computed using high-level ab initio methods and were transformed to a spin-coupled diabatic representation. The spin-orbit coupling used in the transformation was taken as that of atomic bromine at all internuclear distances. Adiabatic potential energy curves, nonadiabatic couplings and transition dipole moments were then obtained from the diabatic ones and were used in all the dynamics calculations. Nonadiabatic photodissociation probabilities were computed using three semiclassical trajectory methods, namely, coherent switching with decay of mixing (CSDM), fewest switches with time uncertainty (FSTU), and its recently developed variant with stochastic decoherence (FTSU/SD), each combined with semiclassical sampling of the initial vibrational state. The calculated branching fraction to the higher fine-structure level of the bromine atom is in good agreement with experiment and with more complete theoretical treatments. The present study, by comparing our new calculations to wave packet calculations with distance-dependent ab initio spin-orbit coupling, validates the semiclassical trajectory methods, the semiclassical initial state sample scheme, and the use of a distance-independent spin-orbit coupling for future applications to polyatomic photodissociation. Finally, using LiBr(+) as a model system, it is shown that accurate spin-coupled potential curves can also be constructed for odd-electron systems using the same strategy as for HBr.     

Single switch surface hopping for a model of pyrazine

The single switch trajectory surface hopping algorithm is tested for numerical simulations of a two-state three-mode model for the internal conversion of pyrazine through a conical intersection of potential energy surfaces. The algorithm is compared to two other surface hopping approaches, namely, Tully's method of the fewest switches [J. Chem. Phys. 93, 1061 (1990)] and the method by Voronin et al. [J. Phys. Chem. A 102, 6057 (1998)]. The single switch algorithm achieves the most accurate results. Replacing its deterministic nonadiabatic branching condition by a probabilistic accept-reject criterion, one obtains the method of Voronin et al. without momentum adjustment. This probabilistic version of the single switch approach outperforms the considered algorithms in terms of accuracy, memory requirement, and runtime.     

Non-Born-Oppenheimer molecular dynamics of Na...FH photodissociation

The accuracy of non-Born-Oppenheimer (electronically nonadiabatic) semiclassical trajectory methods for simulations of "deep quantum" systems is reevaluated in light of recent quantum mechanical calculations of the photodissociation of the Na...FH van der Waals complex. In contrast to the conclusion arrived at in an earlier study, semiclassical trajectory methods are shown to be qualitatively accurate for this system, thus further validating their use for systems with large electronic energy gaps. Product branching in semiclassical surface hopping and decay-of-mixing calculations is affected by a region of coupling where the excited state is energetically forbidden. Frustrated hops in this region may be attributed to a failure of the treatment of decoherence, and a stochastic model for decoherence is introduced into the surface hopping method and is shown to improve the agreement with the quantum mechanical results. A modification of the decay-of-mixing method resulting in faster decoherence in this region is shown to give similarly improved results.     

NeI2 Photofragmentation Dynamics Through Quasiclassical and Semiclassical Studies

The triatomic system NeI₂ is studied under the consideration that the diatom is found in an excited electronic state (B). The vibrational levels (v=13, …, 23) are considered within two well-known theoretical procedures: quasi-classical trajectories (QCT), where the classical equations of motion for nuclei are solved on a single potential energy surface (PES), and the trajectory surface hopping (TSH) method, where the same are solved in a bunch of crossed vibrational PES (diabatic representation). The trajectory surface hopping fewest switches (TSHFS) is implemented to minimize the number of hoppings, thus allowing the calculations of hopping probability between the different PES's, and the kinetic mechanism to track the dissociation path. From these calculations, several observables such as, the lifetimes, vibrational and rotational energies (I₂), dissociation channels, are obtained. Our results are compared with previous experimental and theoretical work.     

Using semiclassical surface hopping for coupled partial wave calculations on systems with non-spherically symmetric potentials

It is shown that a semiclassical surface hopping (SH) approach provides a simple and efficient method for scattering calculations with non-spherically symmetric potentials. The calculations are performed by expanding the wave function in an angular momentum state basis. Since the potential is not spherically symmetric, the different angular states are coupled. The semiclassical SH method, which is typically used for problems with coupled electronic states, can, in principle, be employed for any coupled state problem. The particular SH method employed is known to provide highly accurate results for coupled electronic state problems. The method is tested on model two angular state problems using potential surfaces and couplings arising from a non-spherically symmetric scattering problem. The results for these model problems are in excellent agreement with exact quantum calculations. Full calculations, which are converged with regard to the number of angular basis states, are also performed for the non-spherically symmetric problem. It is shown that an approximation to the surface hopping amplitudes that simplifies the numerical implementation of the method provides results in excellent agreement with the full surface hopping calculation.     

A coherent state approach to semiclassical nonadiabatic dynamics

A semiclassical (SC) approximation to the quantum mechanical propagator for nonadiabatic systems is derived. Our derivation starts with an exact path integral expression that uses canonical coherent states for the nuclear degrees of freedom and spin coherent states for the electronic degrees of freedom. A stationary path approximation (SPA) is then applied to the path integral to obtain the SC approximation. The SPA results in complex classical trajectories of both nuclear and electronic degrees of freedom and a double ended boundary condition. The root search problem is solved using the previously proposed "real trajectory local search" algorithm. The SC approximation is tested on three simple one dimensional two-state systems proposed by Tully [J. Chem. Phys. 93, 1061 (1990)], and the SC results are compared to Ehrenfest and surface hopping predictions. Excellent agreement with quantum results is reached when the SC trajectory is far away from caustics. We discuss the origin of caustics in this SC formalism and the strengths and weaknesses of this approach.     

A new approach to decoherence and momentum rescaling in the surface hopping algorithm

As originally proposed, the fewest switches surface hopping (FSSH) algorithm does not allow for decoherence between wavefunction amplitudes on different adiabatic surfaces. In this paper, we propose an inexpensive correction to standard FSSH dynamics wherein we explicitly model the decoherence of nuclear wave packets on distinct electronic surfaces. Our augmented fewest switches surface hopping approach is conceptually simple and, thus far, it has allowed us to capture several key features of the exact quantum results. Two points in particular merit attention. First, we obtain the correct branching ratios when a quantum particle passes through more than one region of nonadiabatic coupling. Second, our formalism provides a new and natural approach for rescaling nuclear momenta after a surface hop. Both of these features should become increasingly important as surface hopping schemes are applied to higher-dimensional problems.     

Interpretation of Adiabatic and Diabatic Populations from Trajectories of Branching Corrected Surface Hopping

The branching corrected surface hopping (BCSH) has been demonstrated as a robust approach to improve the performance of the traditional fewest switches surface hopping (FSSH) for nonadiabatic dynamics simulations of standard scattering problems [J. Chem. Phys. 150 , 164101 (2019)]. Here, we study how reliable populations of both adiabatic and diabatic states can be interpreted from BCSH trajectories. Using exact quantum solutions and FSSH results as references, we investigate a series of one-dimensional two-level scattering models and illustrate that excellent time-dependent populations can be obtained by BCSH. Especially, we show that different trajectory analysis strategies produce noticeable differences in different representations. Namely, the method based on active states performs better to get populations of adiabatic states, while the method based on wavefunctions produces more reliable results for populations of diabatic states.     

A spherical electron cloud hopping model for studying product branching ratios of dissociative recombination

A spherical electron cloud hopping (SECH) model is proposed to study the product branching ratios of dissociative recombination (DR) of polyatomic systems. In this model, the fast electron-captured process is treated as an instantaneous hopping of a cloud of uniform spherical fractional point charges onto a target M+q ion (or molecule). The sum of point charges (-1) simulates the incident electron. The sphere radius is determined by a critical distance (Rc eM) between the incoming electron (e-) and the target, at which the potential energy of the e(-)-M+q system is equal to that of the electron-captured molecule M+q(-1) in a symmetry-allowed electronic state with the same structure as M(+q). During the hopping procedure, the excess energies of electron association reaction are dispersed in the kinetic energies of M+q(-1) atoms to conserve total energy. The kinetic energies are adjusted by linearly adding atomic momenta in the direction of driving forces induced by the scattering electron. The nuclear dynamics of the resultant M+q(-1) molecule are studied by using a direct ab initio dynamics method on the adiabatic potential energy surface of M+q(-1), or together with extra adiabatic surface(s) of M+q(-1). For the latter case, the "fewest switches" surface hopping algorithm of Tully was adapted to deal with the nonadiabaticity in trajectory propagations. The SECH model has been applied to study the DR of both CH+ and H3O+(H2O)2. The theoretical results are consistent with the experiment. It was found that water molecules play an important role in determining the product branching ratios of the molecular cluster ion.     

Mixed quantum-classical equilibrium: Surface hopping

We re-examine the analysis of the equilibrium limits of the fewest switches surface hopping algorithm for mixed quantum-classical dynamics. In contrast with previously reported results, we show that surface hopping does not, in general, exactly yield Boltzmann equilibrium, but that in practice the observed deviations are quite small. We also demonstrate that surface hopping does approach the exact equilibrium distribution in both the limits of small adiabatic splitting and/or strong nonadiabatic coupling. We verify these analytical results with numerical simulations for a simple two-level quantum system connected to a bath of classical particles.     

Mixed quantum/classical investigation of the photodissociation of NH3(A) and a practical method for maintaining zero-point energy in classical trajectories

The photodissociation dynamics of ammonia upon excitation of the out-of-plane bending mode (mode nu(2) with n(2)=0,[ellipsis (horizontal)],6 quanta of vibration) in the A electronic state is investigated by means of several mixed quantum/classical methods, and the calculated final-state properties are compared to experiments. Five mixed quantum/classical methods are tested: one mean-field approach (the coherent switching with decay of mixing method), two surface-hopping methods [the fewest switches with time uncertainty (FSTU) and FSTU with stochastic decay (FSTU/SD) methods], and two surface-hopping methods with zero-point energy (ZPE) maintenance [the FSTUSD+trajectory projection onto ZPE orbit (TRAPZ) and FSTUSD+minimal TRAPZ (mTRAPZ) methods]. We found a qualitative difference between final NH(2) internal energy distributions obtained for n(2)=0 and n(2)>1, as observed in experiments. Distributions obtained for n(2)=1 present an intermediate behavior between distributions obtained for smaller and larger n(2) values. The dynamics is found to be highly electronically nonadiabatic with all these methods. NH(2) internal energy distributions may have a negative energy tail when the ZPE is not maintained throughout the dynamics. The original TRAPZ method was designed to maintain ZPE in classical trajectories, but we find that it leads to unphysically high internal vibrational energies. The mTRAPZ method, which is new in this work and provides a general method for maintaining ZPE in either single-surface or multisurface trajectories, does not lead to unphysical results and is much less time consuming. The effect of maintaining ZPE in mixed quantum/classical dynamics is discussed in terms of agreement with experimental findings. The dynamics for n(2)=0 and n(2)=6 are also analyzed to reveal details not available from experiment, in particular, the time required for quenching of electronic excitation and the adiabatic energy gap and geometry at the time of quenching.     

Simulation of environmental effects on coherent quantum dynamics in many-body systems

In this paper we describe an application of the trajectory-based semiclassical Liouville method for modeling coherent molecular dynamics on multiple electronic surfaces to the treatment of the evolution and decay of quantum electronic coherence in many-body systems. We consider a model representing the coherent evolution of quantum wave packets on two excited electronic surfaces of a diatomic molecule in the gas phase and in rare gas solvent environments, ranging from small clusters to a cryogenic solid. For the gas phase system, the semiclassical trajectory method is shown to reproduce the evolution of the electronic-nuclear coherence nearly quantitatively. The dynamics of decoherence are then investigated for the solvated systems using the semiclassical approach. It is found that, although solvation in general leads to more rapid and extensive loss of quantum coherence, the details of the coupled system-bath dynamics are important, and in some cases the environment can preserve or even enhance quantum coherence beyond that seen in the isolated system.     

Nonadiabatic surface hopping Herman-Kluk semiclassical initial value representation method revisited: applications to Tully's three model systems

The nonadiabatic surface hopping Herman-Kluk (HK) semiclassical initial value representation (SC-IVR) method for nonadiabatic problems is reformulated. The method has the same spirit as Tully's surface hopping technique [J. Chem. Phys. 93, 1061 (1990)] and almost keeps the same structure as the original single-surface HK SC-IVR method except that trajectories can hop to other surfaces according to the hopping probabilities and phases, which can be easily integrated along the paths. The method is based on a rather general nonadiabatic semiclassical surface hopping theory developed by Herman [J. Chem. Phys. 103, 8081 (1995)], which has been shown to be accurate to the first order in h and through all the orders of the nonadiabatic coupling amplitude. Our simulation studies on the three model systems suggested by Tully demonstrate that this method is practical and capable of describing nonadiabatic quantum dynamics for various coupling situations in very good agreement with benchmark calculations.     

First principles semiclassical calculations of vibrational eigenfunctions

Vibrational eigenfunctions are calculated on-the-fly using semiclassical methods in conjunction with ab initio density functional theory classical trajectories. Various semiclassical approximations based on the time-dependent representation of the eigenfunctions are tested on an analytical potential describing the chemisorption of CO on Cu(100). Then, first principles semiclassical vibrational eigenfunctions are calculated for the CO(2) molecule and its accuracy evaluated. The multiple coherent states initial value representations semiclassical method recently developed by us has shown with only six ab initio trajectories to evaluate eigenvalues and eigenfunctions at the accuracy level of thousands trajectory semiclassical initial value representation simulations.