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The Journal of Chemical Physics

Calculation of correlated initial state in the hierarchical equations of motion method using an imaginary time path integral approach

Linze Song, Qiang Shi

Published Nov 20, 2015• 38 citations
Publisher

Abstract

Based on recent findings in the hierarchical equations of motion (HEOM) for correlated initial state [Y. Tanimura, J. Chem. Phys. 141, 044114 (2014)], we propose a new stochastic method to obtain the initial conditions for the real time HEOM propagation, which can be used further to calculate the equilibrium correlation functions and symmetrized correlation functions. The new method is derived through stochastic unraveling of the imaginary time influence functional, where a set of stochastic imaginary time HEOM are obtained. The validity of the new method is demonstrated using numerical examples including the spin-Boson model, and the Holstein model with undamped harmonic oscillator modes.

DOI
10.1063/1.4935799

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Based on recent findings in the hierarchical equations of motion (HEOM) for correlated initial state [Y. Tanimura, J. Chem. Phys. 141, 044114 (2014)], we propose a new stochastic method to obtain the initial conditions for the real time HEOM propagation, which can be used further to calculate the equilibrium correlation functions and symmetrized correlation functions. The new method is derived through stochastic unraveling of the imaginary time influence functional, where a set of stochastic imaginary time HEOM are obtained. The validity of the new method is demonstrated using numerical examples including the spin-Boson model, and the Holstein model with undamped harmonic oscillator modes.
We investigate the application of the imaginary time hierarchical equations of motion method to calculate real time quantum correlation functions. By starting from the path integral expression for the correlated system-bath equilibrium state, we first derive a new set of equations that decouple the imaginary time propagation and the calculation of auxiliary density operators. The new equations, thus, greatly simplify the calculation of the equilibrium correlated initial state that is subsequently used in the real time propagation to obtain the quantum correlation functions. It is also shown that a periodic decomposition of the bath imaginary time correlation function is no longer necessary in the new equations such that different decomposition schemes can be explored. The applicability of the new method is demonstrated in several numerical examples, including the spin-Boson model, the Holstein model, and the double-well model for proton transfer reaction.
A traditional stochastic hierarchy equations of motion method is extended into the correlated real-time and imaginary-time propagations, in this paper, for its applications in calculating the equilibrium correlation functions. The central idea is based on a combined employment of stochastic unravelling and hierarchical techniques for the temperature-dependent and temperature-free parts of the influence functional, respectively, in the path integral formalism of the open quantum systems coupled to a harmonic bath. The feasibility and validity of the proposed method are justified in the emission spectra of homodimer compared to those obtained through the deterministic hierarchy equations of motion. Besides, it is interesting to find that the complex noises generated from a small portion of real-time and imaginary-time cross terms can be safely dropped to produce the stable and accurate position and flux correlation functions in a broad parameter regime.
For a system strongly coupled to a heat bath, the quantum coherence of the system and the heat bath plays an important role in the system dynamics. This is particularly true in the case of non-Markovian noise. We rigorously investigate the influence of system-bath coherence by deriving the reduced hierarchal equations of motion (HEOM), not only in real time, but also in imaginary time, which represents an inverse temperature. It is shown that the HEOM in real time obtained when we include the system-bath coherence of the initial thermal equilibrium state possess the same form as those obtained from a factorized initial state. We find that the difference in behavior of systems treated in these two manners results from the difference in initial conditions of the HEOM elements, which are defined in path integral form. We also derive HEOM along the imaginary time path to obtain the thermal equilibrium state of a system strongly coupled to a non-Markovian bath. Then, we show that the steady state hierarchy elements calculated from the real-time HEOM can be expressed in terms of the hierarchy elements calculated from the imaginary-time HEOM. Moreover, we find that the imaginary-time HEOM allow us to evaluate a number of thermodynamic variables, including the free energy, entropy, internal energy, heat capacity, and susceptibility. The expectation values of the system energy and system-bath interaction energy in the thermal equilibrium state are also evaluated.
The hierarchical equations of motion (HEOM) method has recently been widely applied to many problems of quantum dynamics in condensed phase. It is now well known that the auxiliary density operators (ADOs) in the HEOM formalism contain system-bath correlations that are important in calculating various dynamical properties, yet quantitative relations to explicitly calculate such correlations from the ADOs are still scarce. This paper extends a previous study [Q. Shi et al., J. Chem. Phys. 130, 164518 (2009)] in investigating the physical meaning of ADOs to general spectral densities and lower temperature cases. Using the path-integral technique, we derive exact relations between the expectation values of the collective bath coordinate and the ADOs, which could be very useful in investigating the correlated system-bath dynamics directly with the HEOM formalism. Numerical examples concerning the evolution of the expectation values of the collective bath coordinate are also presented.
The hierarchical equations of motion (HEOMs) have developed into an important tool in simulating quantum dynamics in condensed phases. Yet, it has recently been found that the HEOM may become numerically unstable in simulations using discrete harmonic oscillator modes [I. S. Dunn, et al., J. Chem. Phys. 150, 184109 (2019)]. In this paper, a new set of equations of motion are obtained based on the equivalence between the HEOM for discrete harmonic oscillator modes and the mixed quantum-classical Liouville equation. The new set of equations can thus be regarded as the expansion of the same phase space partial differential equation using different basis sets. It is shown that they have similar structures as the original HEOM but are free from the problem of numerical instability. The new set of equations are also incorporated into the matrix product state method, where it is found that the trace of the reduced density operator is not well conserved during the propagation. A modified time-dependent variational principle is then proposed to achieve better trace conservation.
We investigated applications of the hierarchical equation of motion (HEOM) method to perform high order perturbation calculations of reduced quantum dynamics for a harmonic bath with arbitrary spectral densities. Three different schemes are used to decompose the bath spectral density into analytical forms that are suitable to the HEOM treatment: (1) The multiple Lorentzian mode model that can be obtained by numerically fitting the model spectral density. (2) The combined Debye and oscillatory Debye modes model that can be constructed by fitting the corresponding classical bath correlation function. (3) A new method that uses undamped harmonic oscillator modes explicitly in the HEOM formalism. Methods to extract system-bath correlations were investigated for the above bath decomposition schemes. We also show that HEOM in the undamped harmonic oscillator modes can give detailed information on the partial Wigner transform of the total density operator. Theoretical analysis and numerical simulations of the spin-Boson dynamics and the absorption line shape of molecular dimers show that the HEOM formalism for high order perturbations can serve as an important tool in studying the quantum dissipative dynamics in the intermediate coupling regime.