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The Journal of Physical Chemistry Letters

Path Integral over Equivalence Classes for Quantum Dynamics with Static Disorder

Nancy Makri

Published Jan 31, 2024• 7 citations
Publisher

Abstract

An efficient, fully quantum mechanical, real-time path integral method for including the effects of static disorder in the dynamics of systems coupled to common or local harmonic baths is presented. Rather than performing a large number of demanding calculations for different realizations of the system Hamiltonian, the influence of the bath is captured through a single evaluation of the path sum by grouping the system paths into equivalence classes of fixed system amplitudes. The method is illustrated with several analytical and numerical examples that show a variety of nontrivial effects arising from the interplay among coherence, dissipation, thermal fluctuations, and geometric phases.

DOI
10.1021/acs.jpclett.3c03555

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An efficient, fully quantum mechanical, real-time path integral method for including the effects of static disorder in the dynamics of systems coupled to common or local harmonic baths is presented. Rather than performing a large number of demanding calculations for different realizations of the system Hamiltonian, the influence of the bath is captured through a single evaluation of the path sum by grouping the system paths into equivalence classes of fixed system amplitudes. The method is illustrated with several analytical and numerical examples that show a variety of nontrivial effects arising from the interplay among coherence, dissipation, thermal fluctuations, and geometric phases.
Recent progress in numerical methods for evaluating the real-time path integral in dissipative harmonic environments is reviewed. Quasi-adiabatic propagators constructed numerically allow convergence of the path integral with large time increments. Integration of the harmonic bath leads to path integral expressions that incorporate the exact dynamics of the quantum particle along the adiabatic path, with an influence functional that describes nonadiabatic corrections. The resulting quasi-adiabatic propagator path integral is evaluated by efficient system-specific quadratures in most regimes of parameter space, although some cases are handled by grid Monte Carlo sampling. Exploiting the finite span of nonlocal influence functional interactions characteristic of broad condensed phase spectra leads to an iterative scheme for calculating the path integral over arbitrary time lengths. No uncontrolled approximations are introduced, and the resulting methodology converges to the exact quantum result with modest amounts of computational power. Applications to tunneling dynamics in the condensed phase are described.
In this thesis the path integral formalism is applied to the calculation of the dynamics of dissipative quantum systems. The time evolution of a system of bilinearly coupled bosonic modes is treated using the real-time path integral technique in coherent-state representation. This method is applied to a damped harmonic oscillator within the Caldeira-Leggett model. In order to get the stationary trajectories the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system-bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In addition, it is shown that systems without initial system-bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations the classical trajectories for the system coordinate can be recovered. The path integral formalism in a combined phase-space and coherent-state representation is applied to the problem of curve-crossing dynamics. The system of interest is described by two coupled one-dimensional harmonic potential energy surfaces interacting with a heat bath. The mapping approach is used to rewrite the Lagrangian function of the electronic part of the system. Using the Feynman-Vernon influence-functional method the bath is eliminated whereas the non-Gaussian part of the path integral is treated using the perturbation theory in the small coordinate shift between potential energy surfaces. The vibrational and the population dynamics is considered in a lowest order of the perturbation. The dynamics of a Gaussian wave packet is analyzed along a one-dimensional reaction coordinate. Also the damping rate of coherence in the electronic part of the relevant system is evaluated within the ordinary and variational perturbation theory. The analytic expressions for the rate functions are obtained in the low and high temperature regimes.
Fully quantum mechanical real-time path integral calculations on a two-level system (TLS) interacting with a simple or composite anharmonic bath are presented. The bath consists of a few or many units, each comprising two or three electronic states coupled to secondary harmonic vibrational baths. The primitive harmonic bath approximation leads to qualitatively wrong system dynamics at high temperatures, while the effective harmonic bath mapping becomes exact in the macroscopic bath limit. Significant deviations are observed when the number of bath units is small, including skewing and blue shifts in the population oscillations and enhanced or suppressed coherence. These effects reflect the essential anharmonic character of the bath and cannot be captured by harmonic bath treatments.
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.
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.