Chemical reaction rates using the semiclassical Van Vleck initial value representation

Which Algorithm Best Propagates the Meyer-Miller-Stock-Thoss Mapping Hamiltonian for Non-Adiabatic Dynamics?

A common strategy to simulate mixed quantum-classical dynamics is by propagating classical trajectories with mapping variables, often using the Meyer-Miller-Stock-Thoss (MMST) Hamiltonian or the related spin-mapping approach. When mapping the quantum subsystem, the coupled dynamics reduce to a set of equations of motion to integrate. Several numerical algorithms have been proposed, but a thorough performance comparison appears to be lacking. Here, we compare three time-propagation algorithms for the MMST Hamiltonian: the Momentum Integral (MInt) (J. Chem. Phys., 2018, 148, 102326), the Split-Liouvillian (SL) (Chem. Phys., 2017, 482, 124-134), and the algorithm in J. Chem. Phys., 2012, 136, 084101 that we refer to as the Degenerate Eigenvalue (DE) algorithm due to the approximation required during derivation. We analyze the accuracy of individual trajectories, correlation functions, energy conservation, symplecticity, Liouville's theorem, and the computational cost. We find that the MInt algorithm is the only rigorously symplectic algorithm. However, comparable accuracy at a lower computational cost can be obtained with the SL algorithm. The approximation implicitly made within the DE algorithm conserves energy poorly, even for small timesteps, and thus leads to slightly different results. These results should guide future mapping-variable simulations.

A Semiclassical Framework for Mixed Quantum Classical Dynamics

Semiclassical (SC) approximations for quantum dynamic simulations in complex chemical systems range from rigorously accurate methods that are computationally expensive to methods that exhibit near-classical scaling with system size but are limited in their ability to describe quantum effects. In practical studies of high-dimensional reactions, neither extreme is the best choice: frequently a high-level quantum mechanical description is only required for a handful of modes, while the majority of environment modes that do not play a key role in the reactive event of interest are well served with a lower level of theory. In this feature, we introduce modified Filinov filtration as a powerful tool to construct mixed quantum-classical SC theories where different subsystems can be quantized to different extents without introducing ad hoc intersubsystem interaction terms. We demonstrate that these Filinov-based SC methods can systematically tune between quantum and classical limit SC behavior, offering a practical way forward to accurate and computationally efficient simulations of high-dimensional quantum processes.

Semiclassical dynamics in the mixed quantum-classical limit

The semiclassical double Herman-Kluk initial value representation is an accurate approach to computing quantum real time correlation functions, but its applications are limited by the need to evaluate an oscillatory integral. In previous work, we have shown that this "sign problem" can be mitigated using the modified Filinov filtration technique to control the extent to which individual modes of the system contribute to the overall phase of the integrand. Here, we follow this idea to a logical conclusion: we analytically derive a general expression for the mixed quantum-classical limit of the semiclassical correlation function-analytical mixed quantum-classical-initial value representation (AMQC-IVR), where the phase contributions from the "classical" modes of the system are filtered while the "quantum" modes are treated in the full semiclassical limit. We numerically demonstrate the accuracy and efficiency of the AMQC-IVR formulation in calculations of quantum correlation functions and reaction rates using three model systems with varied coupling strengths between the classical and quantum subsystems. We also introduce a separable prefactor approximation that further reduces computational cost but is only accurate in the limit of weak coupling between the quantum and classical subsystems.

An open-chain imaginary-time path-integral sampling approach to the calculation of approximate symmetrized quantum time correlation functions

We introduce a scheme for approximating quantum time correlation functions numerically within the Feynman path integral formulation. Starting with the symmetrized version of the correlation function expressed as a discretized path integral, we introduce a change of integration variables often used in the derivation of trajectory-based semiclassical methods. In particular, we transform to sum and difference variables between forward and backward complex-time propagation paths. Once the transformation is performed, the potential energy is expanded in powers of the difference variables, which allows us to perform the integrals over these variables analytically. The manner in which this procedure is carried out results in an open-chain path integral (in the remaining sum variables) with a modified potential that is evaluated using imaginary-time path-integral sampling rather than requiring the generation of a large ensemble of trajectories. Consequently, any number of path integral sampling schemes can be employed to compute the remaining path integral, including Monte Carlo, path-integral molecular dynamics, or enhanced path-integral molecular dynamics. We believe that this approach constitutes a different perspective in semiclassical-type approximations to quantum time correlation functions. Importantly, we argue that our approximation can be systematically improved within a cumulant expansion formalism. We test this approximation on a set of one-dimensional problems that are commonly used to benchmark approximate quantum dynamical schemes. We show that the method is at least as accurate as the popular ring-polymer molecular dynamics technique and linearized semiclassical initial value representation for correlation functions of linear operators in most of these examples and improves the accuracy of correlation functions of nonlinear operators.

Wave packet propagation across barriers by semiclassical initial value methods

Semiclassical initial value representation (IVR) formulas for the propagator have difficulty describing tunneling through barriers. A key reason is that these formulas do not automatically reduce, in the classical limit, to the version of the Van Vleck-Gutzwiller (VVG) propagator required to treat barrier tunneling, which involves trajectories that have complex initial conditions and that follow paths in complex time. In this work, a simple IVR expression, that has the correct tunneling form in the classical limit, is derived for the propagator in the case of one-dimensional barrier transmission. Similarly, an IVR formula, that reduces to the Generalized Gaussian Wave Packet Dynamics (GGWPD) expression [D. Huber, E. J. Heller, and R. Littlejohn, J. Chem. Phys. 89, 2003 (1988)] in the classical limit, is derived for the transmitted wave packet. Uniform semiclassical versions of the IVR formulas are presented and simplified expressions in terms of real trajectories and WKB penetration factors are described. Numerical tests show that the uniform IVR treatment gives good results for wave packet transmission through the Eckart and Gaussian barriers in all cases examined. In contrast, even when applied with the proper complex trajectories, the VVG and GGWPD treatments are inaccurate when the mean energy of the wave packet is near the classical transmission threshold. The IVR expressions for the propagator and wave packet are cast as contour integrals in the complex space of initial conditions and these are generalized to potentially allow treatment of a larger variety of systems. A steepest descent analysis of the contour integral formula for the wave packet in the present cases confirms its relationship to the GGWPD method, verifies its semiclassical validity, and explains results of numerical calculations.

Wavepacket approach to the cumulative reaction probability within the flux operator formalism

Expressions for the singular flux operator eigenfunctions and eigenvalues are given in terms of the Dirac delta-function representable as a localized Gaussian wavepacket. This functional form enables computation of the cumulative reaction probability N(E) from the wavepacket time-correlation functions. The Gaussian based form of the flux eigenfunctions, which is not tied to a finite basis of a quantum-mechanical calculation, is particularly useful for approximate calculation of N(E) with the trajectory based wavepacket propagation techniques. Numerical illustration is given for the Eckart barrier using the conventional quantum-mechanical propagation and the quantum trajectory dynamics with the approximate quantum potential. N(E) converges with respect to the Gaussian width parameter, and the convergence is faster at low energy. The approximate trajectory calculation overestimates tunneling in the low energy regime, but gives a significant improvement over the parabolic estimate of the tunneling probability.