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The Journal of Chemical Physics

Quantum dynamics for a system coupled to slow baths: On-the-fly filtered propagator method

Eunji Sim

Published Sep 8, 2001• 92 citations
Publisher

Abstract

An on-the-fly filtered propagator functional path integral scheme is introduced as an efficient way of calculating the iterative dynamics of complex condensed systems. Time evolution of the reduced density matrix of a dissipative quantum system is evaluated iteratively by filtering the negligible propagator elements at each propagation step. This on-the-fly filtering along with the finiteness of the bath memory dramatically reduces the configuration space to be integrated without losing numerical accuracy of the results. The required computational storage space for the configuration and the weight of the survived path segments increases linearly with the bath memory length. Up to the bath memory time, it is found that a strikingly small fraction of the configurations survives the on-the-fly filtering process and the number of surviving configurations increases algebraically with the propagation time. At times longer than the bath memory time, the number of surviving configurations required for numerically accurate results is essentially saturated and is less than 0.1% of the total number of the configurations. This new scheme is extremely useful for the problems of low-frequency solvents for which the bath memory spans many time steps of the propagation.

DOI
10.1063/1.1394208

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An on-the-fly filtered propagator functional path integral scheme is introduced as an efficient way of calculating the iterative dynamics of complex condensed systems. Time evolution of the reduced density matrix of a dissipative quantum system is evaluated iteratively by filtering the negligible propagator elements at each propagation step. This on-the-fly filtering along with the finiteness of the bath memory dramatically reduces the configuration space to be integrated without losing numerical accuracy of the results. The required computational storage space for the configuration and the weight of the survived path segments increases linearly with the bath memory length. Up to the bath memory time, it is found that a strikingly small fraction of the configurations survives the on-the-fly filtering process and the number of surviving configurations increases algebraically with the propagation time. At times longer than the bath memory time, the number of surviving configurations required for numerically accurate results is essentially saturated and is less than 0.1% of the total number of the configurations. This new scheme is extremely useful for the problems of low-frequency solvents for which the bath memory spans many time steps of the propagation.
Iterative path integral methods provide a numerically exact approach to the dynamics of a quantum system coupled to a dissipative bath. These methods involve step-by-step propagation of a generalized multi-time reduced density matrix with a propagator that includes influence functional interactions that span the bath-induced memory length. Low-frequency baths lead to long memory that can span many time steps, necessitating the use of filtering procedures that select and store only statistically significant path segments, avoiding storage of the entire sparse propagator. The present paper further enhances the iterative path integral methodology through a deterministic procedure that identifies at the start of the calculation all memory-length path segments whose weights exceed a chosen threshold and builds the propagator matrix for use in all propagation steps. A binning procedure allows efficient pairing of path segments required in the construction of the propagator. Additional savings are achieved via a simple criterion for assessing the adequacy of a chosen path selection threshold prior to propagation. Application to a tight-binding model for charge transfer in extended systems reveals the important role of bath-induced memory on the dynamics of these systems.
The reduced density matrix of a system coupled to a harmonic dissipative bath can be obtained accurately using an iterative path integral methodology that involves a propagator tensor, whose rank is roughly equal to the number of time steps that span the bath-induced memory. The present work shows that even in situations where the bath is characterized by high-frequency modes (which necessitate a small time step) and slow oscillators (which give rise to very long memory), efficient propagation that takes into account the memory on all time scales is feasible by renormalizing the propagator matrix to include successively larger-length influence functional correlations. At its crudest level, the renormalized propagator is an ordinary matrix, thus the computational demands of the method are very modest, similar to those for propagating the bare system. Yet, test applications on dissipative two-level systems show that the method produces accurate results in demanding situations. By including memory on successively increasing time scales, the method is capable of capturing nonexponential kinetics induced by sluggish solvents.
The iterative decomposition of the blip-summed path integral [N. Makri, J. Chem. Phys. 141, 134117 (2014)] is described. The starting point is the expression of the reduced density matrix for a quantum system interacting with a harmonic dissipative bath in the form of a forward-backward path sum, where the effects of the bath enter through the Feynman-Vernon influence functional. The path sum is evaluated iteratively in time by propagating an array that stores blip configurations within the memory interval. Convergence with respect to the number of blips and the memory length yields numerically exact results which are free of statistical error. In situations of strongly dissipative, sluggish baths, the algorithm leads to a dramatic reduction of computational effort in comparison with iterative path integral methods that do not implement the blip decomposition. This gain in efficiency arises from (i) the rapid convergence of the blip series and (ii) circumventing the explicit enumeration of between-blip path segments, whose number grows exponentially with the memory length. Application to an asymmetric dissipative two-level system illustrates the rapid convergence of the algorithm even when the bath memory is extremely long.
Quantum mechanical effects play an important role in dynamics of condensed phases. It is also well known that the difficulty of solving the full Schrödinger equation grows exponentially with time. One approach to cope up with this difficulty is to restrict quantum treatment to a few particles (system) and treat the dynamics of the rest of particles (bath) using classical trajectories. Path integral methods due to their trajectory like nature provide excellent tool to develop quantum-classical methods, but to capture long time dynamics, the number of classical trajectories grow exponentially. A part of this work focuses on improving Quantum-Classical Path Integral (QCPI) treatments that allow for larger path integral time step by building “smarter” propagators. These improved solvent-driven reference propagators is developed by incorporating physically motivated approximations to the solvent. When used in QCPI expression they allow convergence with larger time steps leading to an exponential reduction of the number of trajectories required. Further advantages of these propagators include improvements in path filtering techniques, smaller number of path integral steps for achieving the memory decoherence time, and smoothing of the integrand which leads to convergence with fewer Monte Carlo sample points. These ideas have been validated on the spin boson model - a prototypical model to study condensed phase dynamics, which consists of a two-level system coupled to a harmonic bath. The new approach of building the reference propagators combined with their iterative evaluation and filtering is validated using parameters that mimic the first electron transfer in wild-type photosynthetic reaction centres. A real-world dynamical simulation contains various anharmonic effects, Ferrocene (donor) - Ferrocenium (acceptor) system in liquid Benzene was studied where the anharmonic effects of the bath of liquid Benzene on the dynamics of the system were treated using Ensemble Averaged Classical Path. The process of electron or proton transfer occurs between two species having different charges and fluctuations in the solvent plays a key role in determining the rates of such reaction dynamics. To correctly account for the effects of initial distribution of the solvent on the dynamics of the quantum system, the concept of equilibrating the bath to the donor state of the system is used. Two different approaches have been used within the influence functional framework.
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.
We propose an efficient path integral scheme for calculating the quantum dynamics of an arbitrary one-dimensional system coupled nonlinearly to many anharmonic noninteracting ‘‘bath’’ degrees of freedom. The starting point is an improved discretization of the path integral in terms of numerically constructed propagators [Chem. Phys. Lett. 193, 435 (1992)]. The resulting influence functional is comprised of one-dimensional correlation functions with step-structured time-dependent potentials and therefore is similar in structure to that employed in the spin-boson calculations of Coalson [J. Chem. Phys. 86, 995 (1987)]. In the present case, though, the influence functional is nonlinear and is computed using numerical iterative wave function propagation methods. Numerical tests on a system coupled to ten anharmonic oscillators demonstrate the efficiency of the proposed scheme, which requires numerical effort that scales only linearly with the number of anharmonic bath degrees of freedom.