Recent advances in the linearized semiclassical initial value representation/classicalWigner model for the thermal correlation function

Abstract

This article focuses on most recent advances in the linearized semiclassical initial value representation (LSC-IVR)/classical Wigner model that includes quantum effects with classical trajectories and recovers exact thermal correlation functions (of even nonlinear operators, that is, nonlinear functions of position or momentum operators) in the classical, high temperature, and harmonic limits. Two methods for implementing the LSC-IVR/classical Wigner which are in principle feasible to be combined with general force fields or even ab initio electronic structure methods have been reviewed. One is the local Gaussian approximation with the imaginary time path integral approach, the other is the quantum thermal bath method. The article emphasizes on the theory and the algorithms for implementation, while it also covers recent applications and limitations of the LSC-IVR/classical Wigner.

Nonadiabatic semiclassical dynamics in the mixed quantum-classical initial value representation

“…[44][45][46][47][48][49] Unfortunately, much like exact quantum methods, the high computational cost of numerically converging oscillatory integrals has limited these methods to low-dimensional systems. Efforts to mitigate the sign problem have led to the development of more approximate methods such as the linearized (LSC)-IVR [50][51][52][53] that fail to capture quantum coherence effects, a) Electronic mail: na346@cornell.edu and various forward-backward (FB) methods that are either less accurate or computationally expensive. [54][55][56][57][58][59][60][61][62] The recently-introduced Mixed Quantum-Classical (MQC)-IVR method 63,64 employs a modified Filinov filtration (MFF) scheme 45,[63][64][65][66][67][68][69][70][71][72][73][74][75] to damp the oscillatory phase of the integrand and has been shown to improve numerical convergence without significant loss of accuracy.…”

Boltzmann-conserving classical dynamics in quantum time-correlation functions: “Matsubara dynamics”

“…The LSC-IVR retains the Boltzmann quantum statistics inside a Wigner transform, 26 is exact in the zero-time, harmonic and high-temperature limits, and has been developed into a practical method by several authors. [19][20][21][22][23] However, it has a serious drawback: the classical dynamics does not in general preserve the quantum Boltzmann distribution, and thus the quality of the statistics deteriorates over time.…”

Thermal quantum time-correlation functions from classical-like dynamics

“…The method is exact in the high-temperature limit, for harmonic systems (where the higher terms in the Moyal series vanish without approximation) and as t → 0 [32,46,47]. LSC-IVR gives fairly good short-time dynamics, though can miss interference effects in non-dissipative systems [6,48]. A more serious shortcoming is that the classical dynamics does not conserve the quantum Boltzmann distribution, leading to zero-point energy flowing from high-frequency modes to translations and giving spurious effects in simulations [49]; an effect sometimes called 'zero-point energy leakage'.…”

“…A more serious shortcoming is that the classical dynamics does not conserve the quantum Boltzmann distribution, leading to zero-point energy flowing from high-frequency modes to translations and giving spurious effects in simulations [49]; an effect sometimes called 'zero-point energy leakage'. Evaluating the Wigner-transformed Boltzmann distribution requires a multidimensional Fourier transform which is often approximated [6], and at low temperatures this distribution can have negative values [50]. Nevertheless, it has successfully been applied to reaction rates [51], vibrational energy relaxation and spectra [6,49].…”

“…Applications of many of the methods discussed here have already been extensively reviewed, including centroid molecular dynamics (CMD) [3], ring polymer molecular dynamics (RPMD) [4], RPMD rate theory [5] and the linearized semiclassical initial-value representation (LSC-IVR) [6]. Consequently, applications of these methods are only mentioned when pertinent.…”