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.     

Wavepacket propagation using time-sliced semiclassical initial value methods

A new semiclassical initial value representation (SC-IVR) propagator and a SC-IVR propagator originally introduced by Kay [J. Chem. Phys. 100, 4432 (1994)], are investigated for use in the split-operator method for solving the time-dependent Schrodinger equation. It is shown that the SC-IVR propagators can be derived from a procedure involving modified Filinov filtering of the Van Vleck expression for the semiclassical propagator. The two SC-IVR propagators have been selected for investigation because they avoid the need to perform a coherent state basis set expansion that is necessary in other time-slicing propagation schemes. An efficient scheme for solving the propagators is introduced and can be considered to be a semiclassical form of the effective propagators of Makri [Chem. Phys. Lett. 159, 489 (1989)]. Results from applications to a one-dimensional, two-dimensional, and three-dimensional Hamiltonian for a double-well potential are presented.     

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.     

Semiclassical calculation of nonadiabatic thermal rate constants: application to condensed phase reactions

The nonadiabatic transition state theory proposed recently by Zhao et al. [J. Chem. Phys. 121, 8854 (2004)] is extended to calculate rate constants of complex systems by using the Monte Carlo and umbrella sampling methods. Surface hopping molecular dynamics technique is incorporated to take into account the dynamic recrossing effect. A nontrivial benchmark model of the nonadiabatic reaction in the condensed phase is used for the numerical test. It is found that our semiclassical results agree well with those produced by the rigorous quantum mechanical method. Comparing with available analytical approaches, we find that the simple statistical theory proposed by Straub and Berne [J. Chem. Phys. 87, 6111 (1987)] is applicable for a wide friction region although their formula is obtained using Landau-Zener [Phys. Z. Sowjetunion 2, 46 (1932); Proc. R. Soc. London, Ser. A 137, 696 (1932)] nonadiabatic transition probability along a one-dimensional diffusive coordinate. We also investigate how the nuclear tunneling events affect the dependence of the rate constant on the friction.     

Herman-Kluk semiclassical dynamics of molecular rotations in laser fields

The action-angle mapping algorithm [R. Saha and M. Ovchinnikov, J. Chem. Phys. 124, 204112 (2006)] is utilized to provide a Herman-Kluk semiclassical initial value representation (SC-IVR) treatment of quantum dynamics of systems with non-Cartesian degrees of freedom. The non-Cartesian system under investigation is a linear rotor molecule in static electric and pulsed laser field. The results demonstrate that the SC-IVR procedure described in this work provides an accurate representation of quantum rotational dynamics of the system.     

Second virial coefficients of asymmetric top molecules

A short self-contained derivation is given for the second virial coefficient B2(T) of a gas consisting of identical interacting asymmetric rigid rotors. The resulting expression is correct through variant Planck's over h2. First, the canonical partition function is derived by means of an variant Planck's over h expansion of exp[-H/(k(B)T)] due to Friedmann [Adv. Chem. Phys. 4, 225 (1962)]. The present work applies angular momentum operators and known facts from angular momentum theory. It is considerably more accessible than Friedmann's exposition, which is not based on angular momentum operators, but instead on explicit derivatives with respect to Euler angles. The partition function obtained from the variant Planck's over h expansion is applied to the derivation of an expression for B2(T) that is identical in appearance to the expression for symmetric rotors of T Pack [J. Chem. Phys. 78, 7217 (1983)]. The final equation in this work is valid for rigid rotors of any symmetry.     

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.     

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.     

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.     

Long-time and unitary properties of semiclassical initial value representations

We numerically compare the semiclassical "frozen Gaussian" Herman-Kluk propagator [Chem. Phys. 91, 27 (1984)] and the "thawed Gaussian" propagator put forward recently by Baranger et al. [J. Phys. A 34, 7227 (2001)] by studying the quantum dynamics in some nonlinear one-dimensional potentials. The reasons for the lack of long-time accuracy and norm conservation in the latter method are uncovered. We amend the thawed Gaussian propagator with a global harmonic approximation for the stability of the trajectories and demonstrate that this revised propagator is a true alternative to the Herman-Kluk propagator with similar accuracy.     

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 Schr?dinger 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.     

Combined nonadiabatic transition-state theory and ab initio molecular dynamics study on selectivity of the alpha and beta bond fissions in photodissociation of bromoacetyl chloride

The selectivity of the alpha C-Cl and beta C-Br bond fissions upon n-->pi(*) excitation of bromoacetyl chloride has been investigated with combined nonadiabatic Rice-Ramsperger-Kassel-Marcus theory and ab initio molecular dynamics calculations, which are based on the potential energy profiles calculated with the complete active space self-consistent field and multireference configuration interaction methods. The Zhu-Nakamura [J. Chem. Phys. 101, 10630 (1994); 102, 7448 (1995)] theory is chosen to calculate the nonadiabatic hopping probability. It is found that nonadiabatic effect plays an important role in determining selective dissociations of the C-Cl and C-Br bonds. The calculated rate constants are close to those from experimentally inferred values, but the branching ratio of the alpha C-Cl and beta C-Br bond fissions is different from the experimental findings. The direct molecular dynamics calculations predict that fission of the C-Cl bond occurs on a time scale of picoseconds and cleavage of the beta C-Br bond proceeds with less probability within the same period. This reveals that the initial relaxation dynamics is probably another important factor that influences the selectivity of the C-Cl and C-Br bond fissions in photodissociation of BrCH(2)COCl at 248 nm.     

Reconciling semiclassical and Bohmian mechanics. V. Wavepacket dynamics

In previous articles [B. Poirier J. Chem. Phys. 121, 4501 (2004); C. Trahan and B. Poirier, ibid. 124, 034115 (2006); 124, 034116 (2006); B. Poirier and G. Parlant, J. Phys. Chem. A 111, 10400 (2007)] a bipolar counterpropagating wave decomposition, psi = psi(+) + psi(-), was presented for stationary states psi of the one-dimensional Schrodinger equation, such that the components psi(+/-) approach their semiclassical Wentzel-Kramers-Brillouin analogs in the large action limit. The corresponding bipolar quantum trajectories are classical-like and well behaved, even when psi has many nodes, or is wildly oscillatory. In this paper, the method is generalized for time-dependent wavepacket dynamics applications and applied to several benchmark problems, including multisurface systems with nonadiabatic coupling.     

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.     

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.     

New Energy-Based Decoherence Correction Approaches for Trajectory Surface Hopping

Inspired by the branching corrected surface hopping (BCSH) method [J. Xu and L. Wang, J. Chem. Phys. 150 , 164101 (2019)], we present two new decoherence time formulas for trajectory surface hopping. Both the proposed linear and exponential formulas characterize the decoherence time as functions of the energy difference between adiabatic states and correctly capture the decoherence effect due to wave packet reflection as predicted by BCSH. The relevant parameters are trained in a series of 200 diverse models with different initial nuclear momenta, and the exact quantum solutions are utilized as references. As demonstrated in the three standard Tully models, the two new approaches exhibit significantly higher reliability than the widely used counterpart algorithm while holding the appealing efficiency, thus promising for nonadiabatic dynamics simulations of general systems.     

Methane-water cross second virial coefficient with quantum corrections from an ab initio potential

We present our calculations of the cross second virial coefficient (B12) and of a related quantity, phi 12 = B12-TdB12/dT, for the methane-water system in the temperature range T = 200-1000 K. These calculations were performed using one of the ab initio potentials developed in previous work [Akin-Ojo and Szalewicz, J. Chem. Phys. 123, 134311 (2005)]. Quantum corrections of order variant Planck's over 2pi(2) were added to the computed classical values. We have estimated the uncertainties in our computed B12 and phi 12(T). This allowed evaluation of the quality of the experimental data to which we compare our results. We also provide an analytical expression for B12(T) as a function of the temperature T obtained by fitting the calculated values. This formula also predicts values of phi12(T) consistent with the directly calculated values.     

Semiclassical tunneling splittings from short time dynamics: Herman-Kluk-propagation and harmonic inversion

We investigate a recently proposed method [J. Chem. Phys. 108, 9206 (1998)] to obtain tunneling splittings from short time cross-correlation matrices that were propagated according to the semiclassical propagator of Herman and Kluk. The energy levels were extracted by harmonic inversion of the cross-correlation matrix using the filter diagonalization technique. The aim of this study is twofold: First, the short time behavior of the Herman-Kluk-propagator and the meaning of using cross-correlation matrices rather than autocorrelation functions is addressed. Numerical examples are given for one- and two-dimensional model potentials. Second, the performance of the method is investigated for a system with considerable anharmonicity and coupling. Here the proton transfer in 3,7-dichlorotropolone is considered using an ab initio reaction surface Hamiltonian approach. For this example also the extension to more dimensions is critically discussed.