Practical evaluation of condensed phase quantum correlation functions: A Feynman–Kleinert variational linearized path integral method

Abstract

We report a new method for calculating the Wigner transform of the Boltzmann operator in the canonical ensemble. The transform is accomplished by writing the Boltzmann operator in a semiharmonic form, utilizing the variational centroid effective frequencies introduced by Feynman and Kleinert (FK). The approximate many-body Wigner transformed Boltzmann operator is then utilized with a linearized path integral (LPI) representation for correlation functions. It is shown that this new FK-LPI method is capable of calculating quite accurately the short time behavior of linear and highly nonlinear correlation functions for low temperature Lennard-Jones model systems and that it is vastly superior to classical dynamics. The feasibility of the FK-LPI method for large systems is illustrated by considering a model liquid composed of 32 oxygen molecules with periodic boundary conditions. Initial conditions for molecular dynamics are obtained from its Boltzmann Wigner transform and the FK-LPI method is shown to describe correctly the zero-point motion of the liquid. The effective frequency representation of the Wigner transformed thermal density operator provides an efficient way of sampling nonclassical initial conditions for molecular-dynamics simulations more generally. Applications to vibrational energy relaxation and rate constant calculations in complex molecular systems are discussed.

Including quantum effects in the dynamics of complex (i.e., large) molecular systems

“…(5) has the same form as the purely classical correlation function, the only difference being that the Wigner functions for operators Heller 54 discussed the approximation many years ago (including an illuminating discussion of its limitations), and it was used by Lee and Scully 55 to describe quantum effects in a collinear model of inelastic scattering. More recently it has been obtained form a different approach by Pollak, 56 and also by Rossky et al 57 directly from a path integral representation of the two time evolution operators in Eq. (1) (again by linearizing in the difference between the two paths).…”

Relation of exact Gaussian basis methods to the dephasing representation: Theory and application to time-resolved electronic spectra

“…Vectors and matrices in the D-dimensional vector space of nuclei are denoted by italics: e.g., q or p. The inner product and contraction of tensors in this space are denoted by · , as in q T · p. The DR (9) can be derived 9,10 by linearization of the semiclassical propagator and improves on a previous method 41 inspired by the semiclassical perturbation theory of Miller and co-workers. 42 Shi and Geva 14 derived the DR without invoking the semiclassical propagator-by linearizing 43 the path integral quantum propagator.…”

Test of the consistency of various linearized semiclassical initial value time correlation functions in application to inelastic neutron scattering from liquid para-hydrogen

“…The classical Wigner model is an old idea, but it is important to realize that it is contained within the SC-IVR approach, as a well-defined approximation to it 28,29 . There are other ways to derive the classical Wigner model (or one may simply postulate it) 9,35,40,41 , and we also note that the 'forward-backward semiclassical dynamics' (FBSD) approximation of Makri et al 32,[42][43][44][45][46][47][48][49][50][51][52][53][54][55][56] is very similar to it. The LSC-IVR/classical Wigner model cannot describe true quantum coherence effects in time correlation functions-more accurate SC-IVR approaches, such as the Fourier transform forward-backward IVR (FB-IVR) approach 22,57 (or the still more accurate generalized FB-IVR 58 ) of Miller et al, are needed for this-but it does describe some aspects of the quantum dynamics very well 26,[30][31][32]34,[59][60][61][62] .…”

“…The simplest (and most approximate) version of the SC-IVR is its 'linearized' approximation (LSC-IVR) 9,26,[28][29][30][31][32][33][34][35] , which leads to the classical Wigner model [36][37][38][39] for time correlation functions; see Section IIB for a summary of the LSC-IVR. The classical Wigner model is an old idea, but it is important to realize that it is contained within the SC-IVR approach, as a well-defined approximation to it 28,29 .…”