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The Journal of Chemical Physics

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

Eva Martín-Fierro, Eli Pollak

Published Oct 25, 2006• 34 citations
Publisher

Abstract

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.

DOI
10.1063/1.2358985

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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.
A recently formulated continuum limit semiclassical initial value series representation (SCIVR) of the quantum dynamics of dissipative systems is applied to the study of vibrational relaxation of model harmonic and anharmonic oscillator systems. As is well known, the classical dynamics of dissipative systems may be described in terms of a generalized Langevin equation. The continuum limit SCIVR uses the Langevin trajectories as input, albeit with a quantum noise rather than a classical noise. Combining this development with the forward-backward form of the prefactor-free propagator leads to a tractable scheme for computing quantum thermal correlation functions. Here we present the first implementation of this continuum limit SCIVR series method to study two model problems of vibrational relaxation. Simulations of the dissipative harmonic oscillator system over a wide range of parameters demonstrate that at most only the first two terms in the SCIVR series are needed for convergence of the correlation function. The methodology is then applied to the vibrational relaxation of a dissipative Morse oscillator. Here, too, the SCIVR series converges rapidly as the first two terms are sufficient to provide the quantum mechanical relaxation with an estimated accuracy on the order of a few percent. The results in this case are compared with computations obtained using the classical Wigner approximation for the relaxation dynamics.
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.
The forward–backward (FB) version of the semiclassical (SC) initial value representation (IVR) is used to study quantum coherence effects in the time-dependent probability distribution of an anharmonic vibrational coordinate and its quenching when coupled to a thermal bath. It is shown that the FB-IVR accurately reproduces the detailed quantum coherent structure in the weak coupling regime, and also describes how this coherence is quenched with an increase of the system–bath coupling and/or the bath temperature. Comparisons are made with other approximations and the physical implications are discussed.
The recently derived exact representation of the quantum propagator in terms of semiclassical initial value representations (SCIVR) is used to optimize the width parameter in the SCIVR. Minimization of the expectation value of the correction operator related to the SCIVR leads to improved convergence of the representation. A test on a model one-dimensional double-well potential demonstrates that this optimization can give essentially exact results using only the first two terms in the SCIVR expansion of the exact propagator.
The forward–backward semiclassical dynamics methodology [J. Phys. Chem. 103, 7753, 9479 (1999)] is reformulated in the interaction representation. The new version of the method allows for a fully quantum mechanical description of a low-dimensional subsystem of interest, along with a semiclassical forward–backward treatment of the solvent coordinates and their coupling to the reference subsystem. Application to the long-time tunneling dynamics in a symmetric double-well system coupled to a harmonic bath shows that the interaction FBSD is capable of capturing quantitatively the tunneling and decoherence effects induced by weakly dissipative environments.
A rigorous, yet practical semiclassical formulation of time correlation functions or expectation values is presented. The main idea is to combine the forward and backward propagation steps into a single semiclassical propagator for those degrees of freedom that are not probed in a calculation, while retaining an explicit two-propagator description of the observable low-dimensional system. The combined forward-backward treatment of the environment naturally leads to extensive cancellation and results in an action that is small on the scale of Planck's constant. As a consequence, the integrand is smooth and thus amenable to Monte Carlo sampling. At the same time, important nonclassical effects terms, arising from interference among multiple bounce solutions of the system component, are fully accounted for.