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The Journal of Chemical Physics

Propagator expansions for softly coupled potentials: A model for complex reaction dynamics

Steven D. Schwartz

Published Dec 15, 1989• 16 citations
Publisher

Abstract

In this paper we investigate a new approach to reduced dimensionality descriptions of quantum mechanical systems resident in a bath. We study physical situations in which the coupling between the system and the bath is slowly varying. Our method involves an operator expansion of the Feynman propagator following the Zassenhaus theorem. From this general expansion we are able to derive an especially simple special case in which the coupling is a slowly varying function of the position operators of the system and the bath. From the approximate propagator after tracing over bath degrees, we are able to derive a short time propagator which yields both a form for efficient numerical calculation and an effective Schrödinger equation for the evolution of the system under the average influence of the bath. This theory is then applied to tunneling rearrangement in mixed crystals of benzoic acid. We find that independent of potential energy perturbations, dynamic system bath couplings increase the rate of tunneling. A central goal of this type of approach is to model the increasingly complex experimental data for large (often biological) systems.

DOI
10.1063/1.457232

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In this paper we investigate a new approach to reduced dimensionality descriptions of quantum mechanical systems resident in a bath. We study physical situations in which the coupling between the system and the bath is slowly varying. Our method involves an operator expansion of the Feynman propagator following the Zassenhaus theorem. From this general expansion we are able to derive an especially simple special case in which the coupling is a slowly varying function of the position operators of the system and the bath. From the approximate propagator after tracing over bath degrees, we are able to derive a short time propagator which yields both a form for efficient numerical calculation and an effective Schrödinger equation for the evolution of the system under the average influence of the bath. This theory is then applied to tunneling rearrangement in mixed crystals of benzoic acid. We find that independent of potential energy perturbations, dynamic system bath couplings increase the rate of tunneling. A central goal of this type of approach is to model the increasingly complex experimental data for large (often biological) systems.
This paper presents a new approach to quantum evolution in the presence of a quantum bath. We develop an equation of motion for an observed system evolving under the influence of an unobserved quantum bath. The methodology we follow uses operator expansions of the Feynman propagator. Corrections to the zeroth order approximation are corrections to an adiabatic approximation. In this paper we explicitly develop an approximation which is infinite order in bath and system coupling, but first order in system degree of freedom. This infinite order approximation is found through a resummation of an infinite class of terms in the operator expansion. We first present a simplified single time (Markovian) version of the theory. We then present a derivation for including the effects of memory. The approach developed in this paper also has the potential for systematic improvement. In other words, while the bath and system coupling in this calculation is treated to infinite order, the system itself is only treated to first order. We will briefly discuss how these higher order corrections can be found. Finally, we present a test calculation of the our approach with comparison to exact results. For a two-dimensional test problem with potential much like that for collinear H+H2 the effective one-dimensional approximation we apply produces essentially exact results.
In this paper we report a new method for resummation of operator expansions of the evolution operator. These resummation techniques are applied to the Feynman propagator and then to the derivation of an effective Schrödinger equation for general system–bath problems. This paper presents a significant advance over work previously reported by us [J. Chem. Phys. 96, 5952 (1992)], and it provides a highly accurate way to calculate quantum mechanical data for many dimensional systems. We then apply this new analytic resummation technique to the calculation of flux–flux correlation functions for two, three, and four dimensional problems in order to study the range of applicability of the approach. For moderate frequencies and coupling strengths, not only is the resummed operator calculational method essentially exact, it also requires a fraction of the computer time that the exact calculation consumes. For truly many dimensional problems this approach should provide the first accurate quantum mechanics of rate processes.
For common condensed phase problems described by a low-dimensional system coupled to a harmonic bath, Feynman’s path integral formulation of time-dependent quantum mechanics leads to expressions for the reduced density matrix of the system where the effects of the harmonic environment enter through an influence functional that is nonlocal in time. In a recent Letter [Chem. Phys. Lett. 221, 482 (1994)], we demonstrated that the range of the nonlocal interactions is finite even at zero temperature, such that the nonlocal kernel extends over only a few time steps if the path integral is expressed in terms of accurate quasiadiabatic propagators. This feature arises from disruption of phase coherence in macroscopic environments and leads to Markovian dynamics for an augmented reduced density tensor, permitting iterative time evolution schemes. In the present paper we analyze the structure and properties of the relevant tensor propagator. Specifically, we show that the tensor multiplication scheme rigorously conserves the trace of the reduced density matrix, and that in cases of short-range nonlocality it leads to Redfield-type equations which are correct to all orders in perturbation theory and which take into account memory effects. We also argue that a simple eigenvector analysis reveals (without actual iteration) the nature of the dynamics and of the equilibrium state, and directly yields quantum reaction or relaxation rates.
An algorithm for the simulation of quantum-classical dynamics is presented. Quantum-classical evolution is effected by a propagator exp(i (L) over capt) involving the quantum classical Liouville operator (L) over cap that describes the evolution of a quantum subsystem coupled to a classical bath. Such a mixed description provides a means to study the dynamics of complex many-body systems where certain degrees of freedom are treated quantum mechanically. The algorithm is constructed by decomposing the time interval Deltat into small segments of length At and successively applying the propagator in the short time segments to obtain the evolution for long times. The algorithm is shown to be a discretization of the iterated Dyson form of the propagator whose direct solution is vexatious. The sequential short-time propagation algorithm is applied to the spin-boson model for a range of values of the Kondo parameter and shown to be effective.
A hybrid method is proposed to propagate system-bath quantum dynamics that use both basis functions and coupled quantum trajectories. In it, the bath is represented with an ensemble of Bohmian trajectories while the system degrees of freedom are accounted through reduced density matrices. By retaining the Hilbert space structure for the system, the method is able to capture interference processes that are challenging to describe in Bohmian dynamics due to singularities that these processes introduce in the quantum potential. By adopting quantum trajectories to represent the bath, the method beats the exponential scaling of the computational cost with the bath size. This combination makes the method suitable for large-scale ground and excited state fully quantum molecular dynamics simulations. Equations of motion for the quantum trajectories and reduced density matrices are derived from the Schrodinger equation and a computational algorithm to solve these equations is proposed. Through computations in two-dimensional model systems, the method is shown to offer an accurate description of subsystem observables and of quantum decoherence, which is difficult to obtain when the quantum nature of the bath is ignored. The scaling of the method is demonstrated using a model with 21 degrees of freedom. The limit of independent trajectories is recovered when the mass of bath degrees of freedom is much larger than the one of the system, in agreement with mixed quantum-classical descriptions.
A hybrid method is proposed to propagate system-bath quantum dynamics that use both basis functions and coupled quantum trajectories. In it, the bath is represented with an ensemble of Bohmian trajectories while the system degrees of freedom are accounted through reduced density matrices. By retaining the Hilbert space structure for the system, the method is able to capture interference processes that are challenging to describe in Bohmian dynamics due to singularities that these processes introduce in the quantum potential. By adopting quantum trajectories to represent the bath, the method beats the exponential scaling of the computational cost with the bath size. This combination makes the method suitable for large-scale ground and excited state fully quantum molecular dynamics simulations. Equations of motion for the quantum trajectories and reduced density matrices are derived from the Schrödinger equation and a computational algorithm to solve these equations is proposed. Through computations in two-dimensional model systems, the method is shown to offer an accurate description of subsystem observables and of quantum decoherence, which is difficult to obtain when the quantum nature of the bath is ignored. The scaling of the method is demonstrated using a model with 21 degrees of freedom. The limit of independent trajectories is recovered when the mass of bath degrees of freedom is much larger than the one of the system, in agreement with mixed quantum-classical descriptions.