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.     

On the properties of a primitive semiclassical surface hopping propagator for nonadiabatic quantum dynamics

A previously developed nonadiabatic semiclassical surface hopping propagator [M. F. Herman J. Chem. Phys. 103, 8081 (1995)] is further studied. The propagator has been shown to satisfy the time-dependent Schrodinger equation (TDSE) through order h, and the O(h2) terms are treated as small errors, consistent with standard semiclassical analysis. Energy is conserved at each hopping point and the change in momentum accompanying each hop is parallel to the direction of the nonadiabatic coupling vector resulting in both transmission and reflection types of hops. Quantum mechanical analysis and numerical calculations presented in this paper show that the h2 terms involving the interstate coupling functions have significant effects on the quantum transition probabilities. Motivated by these data, the h2 terms are analyzed for the nonadiabatic semiclassical propagator. It is shown that the propagator can satisfy the TDSE for multidimensional systems by including another type of nonclassical trajectories that reflect on the same surfaces. This h2 analysis gives three conditions for these three types of trajectories so that their coefficients are uniquely determined. Besides the nonadiabatic semiclassical propagator, a numerically useful quantum propagator in the adiabatic representation is developed to describe nonadiabatic transitions.     

Toward an accurate and efficient semiclassical surface hopping procedure for nonadiabatic problems

The derivation of a semiclassical surface hopping procedure from a formally exact solution of the Schrodinger equation is discussed. The fact that the derivation proceeds from an exact solution guarantees that all phase terms are completely and accurately included. Numerical evidence shows the method to be highly accurate. A Monte Carlo implementation of this method is considered, and recent work to significantly improve the statistical accuracy of the Monte Carlo approach is discussed.     

Continuum limit semiclassical initial value representation for dissipative systems

In this paper, we consider a dissipative system in which the system is coupled linearly to a harmonic bath. In the continuum limit, the bath is defined via a spectral density and the classical system dynamics is given in terms of a generalized Langevin equation. Using the path integral formulation and factorized initial conditions, it is well known that one can integrate out the harmonic bath, leaving only a path integral over the system degrees of freedom. However, the semiclassical initial value representation treatment of dissipative systems has usually been limited to a discretized treatment of the bath in terms of a finite number of bath oscillators. In this paper, the continuum limit of the semiclassical initial value representation is derived for dissipative systems. As in the path integral, the action is modified with an added nonlocal term, which expresses the influence of the bath on the dynamics. The first order correction term to the semiclassical initial value approximation is also derived in the continuum limit.     

A prefactor free semiclassical initial value series representation of the propagator

A new class of prefactor free semiclassical initial value representations (SCIVR) of the quantum propagator is presented. The derivation is based on the physically motivated demand, that on the average in phase space and in time, the propagator obey the exact quantum equation of motion. The resulting SCIVR series representation of the exact quantum propagator is also free of prefactors. When using a constant width parameter, the prefactor free SCIVR propagator is identical to the frozen Gaussian propagator of Heller [J. Chem. Phys. 75, 2923 (1981)]. A numerical study of the prefactor free SCIVR series is presented for scattering through a double slit potential, a system studied extensively previously by Gelabert et al. [J. Chem. Phys. 114, 2572 (2001)]. As a basis for comparison, the SCIVR series is also computed using the optimized Herman-Kluk SCIVR. We find that the sum of the zeroth order and the first order terms in the series suffice for an accurate determination of the diffraction pattern. The same exercise, but using the prefactor free propagator series needs also the second order term in the series, however the numerical effort is not greater than that needed for the Herman-Kluk propagator, since one does not need to compute the monodromy matrix elements at each point in time. The numerical advantage of the prefactor free propagator grows with increasing dimensionality of the problem.     

Using the thermal Gaussian approximation for the Boltzmann operator in semiclassical initial value time correlation functions

The thermal Gaussian approximation (TGA) recently developed by Frantsuzov et al. [Chem. Phys. Lett. 381, 117 (2003)] has been demonstrated to be a practical way for approximating the Boltzmann operator exp(-betaH) for multidimensional systems. In this paper the TGA is combined with semiclassical (SC) initial value representations (IVRs) for thermal time correlation functions. Specifically, it is used with the linearized SC-IVR (LSC-IVR, equivalent to the classical Wigner model), and the "forward-backward semiclassical dynamics" approximation developed by Shao and Makri [J. Phys. Chem. A 103, 7753 (1999); 103, 9749 (1999)]. Use of the TGA with both of these approximate SC-IVRs allows the oscillatory part of the IVR to be integrated out explicitly, providing an extremely simple result that is readily applicable to large molecular systems. Calculation of the force-force autocorrelation for a strongly anharmonic oscillator demonstrates its accuracy, and calculation of the velocity autocorrelation function (and thus the diffusion coefficient) of liquid neon demonstrates its applicability.     

Real time correlation function in a single phase space integral beyond the linearized semiclassical initial value representation

It is shown how quantum mechanical time correlation functions [defined, e.g., in Eq. (1.1)] can be expressed, without approximation, in the same form as the linearized approximation of the semiclassical initial value representation (LSC-IVR), or classical Wigner model, for the correlation function [cf. Eq. (2.1)], i.e., as a phase space average (over initial conditions for trajectories) of the Wigner functions corresponding to the two operators. The difference is that the trajectories involved in the LSC-IVR evolve classically, i.e., according to the classical equations of motion, while in the exact theory they evolve according to generalized equations of motion that are derived here. Approximations to the exact equations of motion are then introduced to achieve practical methods that are applicable to complex (i.e., large) molecular systems. Four such methods are proposed in the paper--the full Wigner dynamics (full WD) and the second order WD based on "Wigner trajectories" [H. W. Lee and M. D. Scully, J. Chem. Phys. 77, 4604 (1982)] and the full Donoso-Martens dynamics (full DMD) and the second order DMD based on "Donoso-Martens trajectories" [A. Donoso and C. C. Martens, Phys. Rev. Lett. 8722, 223202 (2001)]--all of which can be viewed as generalizations of the original LSC-IVR method. Numerical tests of the four versions of this new approach are made for two anharmonic model problems, and for each the momentum autocorrelation function (i.e., operators linear in coordinate or momentum operators) and the force autocorrelation function (nonlinear operators) have been calculated. These four new approximate treatments are indeed seen to be significant improvements to the original LSC-IVR approximation.     

Forward-backward semiclassical initial value series representation of quantum correlation functions

The forward-backward (FB) approximation as applied to semiclassical initial value representations (SCIVR's) has enabled the practical application of the SCIVR methodology to systems with many degrees of freedom. However, to date a systematic representation of the exact quantum dynamics in terms of the FB-SCIVR has proven elusive. In this paper, we provide a new derivation of a forward-backward phase space SCIVR expression (FBPS-SCIVR) derived previously by Thompson and Makri [Phys. Rev. E 59, R4729 (1999)]. This enables us to represent quantum correlation functions exactly in terms of a series whose leading order term is the FBPS-SCIVR expression. Numerical examples for systems with over 50 degrees of freedom are presented for the spin boson problem. Comparison of the FBPS-SCIVR with the numerically exact results of Wang [J. Chem. Phys. 113, 9948 (2000)] obtained using a multiconfigurational time dependent method shows that the leading order FBPS-SCIVR term already provides an excellent approximation.     

Chemical reaction rates using the semiclassical Van Vleck initial value representation

A semiclassical initial value representation formulation using the Van Vleck [Proc. Natl. Acad. Sci. U.S.A. 14, 178 (1928)] propagator has been used to calculate the flux correlation function and thereby reaction rate constants. This Van Vleck formulation of the flux-flux correlation function is computationally as simple as the classical Wigner [Trans. Faraday Soc. 34, 29 (1938)] model. However, unlike the latter, it has the ability to capture quantum interference/coherence effects. Classical trajectories are evolved starting from the dividing surface that separates reactants and products, and are evolved negatively in time. This formulation has been tested on model problems ranging from the Eckart barrier, double well to the collinear H+H2.     

A semiclassical initial-value representation for quantum propagator and boltzmann operator

Starting from the position-momentum integral representation, we apply the correction operator method to the derivation of a uniform semiclassical approximation for the quantum propagator and then extend it to approximate the Boltzmann operator. In this approach, the involved classical dynamics is determined by the method itself instead of given beforehand. For the approximate Boltzmann operator, the corresponding classical dynamics is governed by a complex Hamiltonian, which can be described as a pair of real Hamiltonian systems. It is demonstrated that the semiclassical Boltzmann operator is exact for linear systems. A quantum propagator in the complex time is thus proposed and preliminary numerical results show that it is a reasonable approximation for calculating thermal correlation functions of general systems. © 2018 Wiley Periodicals, Inc.     

A surface hopping algorithm for nonadiabatic minimum energy path calculations

The article introduces a robust algorithm for the computation of minimum energy paths transiting along regions of near‐to or degeneracy of adiabatic states. The method facilitates studies of excited state reactivity involving weakly avoided crossings and conical intersections. Based on the analysis of the change in the multiconfigurational wave function the algorithm takes the decision whether the optimization should continue following the same electronic state or switch to a different state. This algorithm helps to overcome convergence difficulties near degeneracies. The implementation in the MOLCAS quantum chemistry package is discussed. To demonstrate the utility of the proposed procedure four examples of application are provided: thymine, asulam, 1,2‐dioxetane, and a three‐double‐bond model of the 11‐cis‐retinal protonated Schiff base. © 2015 Wiley Periodicals, Inc.     

Higher order phase corrected transition amplitudes for time dependent semiclassical surface hopping calculations

A trajectory based, surface hopping expansion of the time dependent quantum propagator has recently been shown to satisfy the multi-state Schrodinger equation to all orders in . Higher order transition amplitudes for hops between states within an interval of fixed length along the trajectory are presented. These amplitudes include contributions from terms corresponding to any number of hops in the interval. They also account for the dependence of the phase associated with the trajectory and the time taken to cross the interval on the location of the hops within the interval. The higher order amplitudes allow for the use of wider intervals in numerical surface hopping calculations. More of the interference between different hopping trajectories is analytically accounted for when the higher order amplitudes are used with wider intervals. Monte Carlo procedures must generally be employed in deciding whether to hop or not in each interval for multi-dimensional problems. Numerical calculations on a model system indicate that the use of the higher order amplitudes can significantly improve the efficiency and accuracy of these methods.     

New coherent state representation for the imaginary time propagator with applications to forward-backward semiclassical initial value representations of correlation functions

There have been quite a few attempts in recent years to provide an initial value coherent state representation for the imaginary time propagator exp(-betaH). The most notable is the recent time evolving Gaussian approximation of Frantsuzov and Mandelshtam [J. Chem. Phys. 121, 9247 (2004)] which may be considered as an expansion of the imaginary time propagator in terms of coherent states whose momentum is zero. In this paper, a similar but different expression is developed in which exp(-betaH) is represented in a series whose terms are weighted phase space averages of coherent states. Such a representation allows for the formulation of a new and simplified forward-backward semiclassical initial value representation expression for thermal correlation functions.     

Conditions for the equivalence of the “surface hopping” and complex-valued time methods to affect nonadiabatic transitions

In this paper we examine two semiclassical theories used to describe collision induced electronic transitions, namely the surface hopping method of Tully and Preston and the “exact” semiclassical method of Miller and George, and give conditions such that the two methods are equivalent. An example using a DIM potential energy for H⁺₃ is discussed.     

An analysis of the accuracy of an initial value representation surface hopping wave function in the interaction and asymptotic regions

The behavior of an initial value representation surface hopping wave function is examined. Since this method is an initial value representation for the semiclassical solution of the time independent Schrodinger equation for nonadiabatic problems, it has computational advantages over the primitive surface hopping wave function. The primitive wave function has been shown to provide transition probabilities that accurately compare with quantum results for model problems. The analysis presented in this work shows that the multistate initial value representation surface hopping wave function should approach the primitive result in asymptotic regions and provide transition probabilities with the same level of accuracy for scattering problems as the primitive method.     

Simultaneous-trajectory surface hopping: a parameter-free algorithm for implementing decoherence in nonadiabatic dynamics

In this paper, we introduce a trajectory-based nonadiabatic dynamics algorithm which aims to correct the well-known overcoherence problem in Tully's popular fewest-switches surface hopping algorithm. Our simultaneous-trajectory surface hopping algorithm propagates a separate classical trajectory on each energetically accessible adiabatic surface. The divergence of these trajectories generates decoherence, which collapses the particle wavefunction onto a single adiabatic state. Decoherence is implemented without the need for any parameters, either empirical or adjustable. We apply our algorithm to several model problems and find a significant improvement over the traditional algorithm.     

Initial value representation for the SU(n) semiclassical propagator

The semiclassical propagator in the representation of SU(n) coherent states is characterized by isolated classical trajectories subjected to boundary conditions in a doubled phase space. In this paper, we recast this expression in terms of an integral over a set of initial-valued trajectories. These trajectories are monitored by a filter that collects only the appropriate contributions to the semiclassical approximation. This framework is suitable for the study of bosonic dynamics in n modes with fixed total number of particles. We exemplify the method for a Bose-Einstein condensate trapped in a triple-well potential, providing a detailed discussion on the accuracy and efficiency of the procedure.     

Determination of molecular vibrational state energies using the ab initio semiclassical initial value representation: application to formaldehyde

We have demonstrated the use of ab initio molecular dynamics (AIMD) trajectories to compute the vibrational energy levels of molecular systems in the context of the semiclassical initial value representation (SC-IVR). A relatively low level of electronic structure theory (HF/3-21G) was used in this proof-of-principle study. Formaldehyde was used as a test case for the determination of accurate excited vibrational states. The AIMD-SC-IVR vibrational energies have been compared to those from curvilinear and rectilinear vibrational self-consistent field/vibrational configuration interaction with perturbation selected interactions-second-order perturbation theory (VSCF/VCIPSI-PT2) and correlation-corrected vibrational self-consistent field (cc-VSCF) methods. The survival amplitudes were obtained from selecting different reference wavefunctions using only a single set of molecular dynamics trajectories. We conclude that our approach is a further step in making the SC-IVR method a practical tool for first-principles quantum dynamics simulations.