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The Journal of Chemical Physics

Modular path integral for finite-temperature dynamics of extended systems with intramolecular vibrations

Sohang Kundu, Nancy Makri

Published Jul 28, 2020• 23 citations
Publisher

Abstract

The modular decomposition of the path integral is a linear-scaling, numerically exact algorithm for calculating dynamical properties of extended systems composed of multilevel units with local couplings. In a recent article, we generalized the method to wavefunction propagation in aggregates characterized by non-diagonal couplings between adjacent units. Here, we extend the method to the calculation of reduced density matrices in aggregates where each unit includes an arbitrary number of coupled harmonic bath modes, which may describe intramolecular normal mode vibrations, at finite temperature. The effects of harmonic modes are included through influence functional factors, which involve analytical expressions that we derive. Representative applications to spin arrays described by the Heisenberg Hamiltonian with dissipative interactions and to J-aggregates of perylene bisimide, where all coupled normal modes are treated explicitly, are presented.

DOI
10.1063/5.0014838

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The modular decomposition of the path integral is a linear-scaling, numerically exact algorithm for calculating dynamical properties of extended systems composed of multilevel units with local couplings. In a recent article, we generalized the method to wavefunction propagation in aggregates characterized by non-diagonal couplings between adjacent units. Here, we extend the method to the calculation of reduced density matrices in aggregates where each unit includes an arbitrary number of coupled harmonic bath modes, which may describe intramolecular normal mode vibrations, at finite temperature. The effects of harmonic modes are included through influence functional factors, which involve analytical expressions that we derive. Representative applications to spin arrays described by the Heisenberg Hamiltonian with dissipative interactions and to J-aggregates of perylene bisimide, where all coupled normal modes are treated explicitly, are presented.
The modular path integral (MPI) formulation offers a numerically exact, versatile, and highly efficient approach to the quantum dynamics of extended systems characterised by a single-file arrangement of units with short-range interactions, such as hydrocarbons, molecular aggregates with exciton couplings, or spin chains. Rather than propagating the many-particle wavefunction or density matrix in time in the usual stepwise fashion, the MPI scheme proceeds by sequentially integrating over each unit after linking its quantum paths to those of its neighbour and leads to linear scaling with system length. This paper shows that the linking matrix can be further decomposed into a diagonal matrix and a product of low-dimensional matrices, which can be applied sequentially. This factorisation changes the cost scaling from L2 to Llog⁡L, where L is the number of quantum paths, leading to dramatic savings that are analogous to those attained by the fast Fourier transform algorithm and allowing application of the MPI procedure to systems with larger units and to longer propagation times. Applications to spin chains and to electron-vibration dynamics in a large molecular aggregate illustrate the efficiency of the algorithm.
, 101101 (2018)], it was shown that the locality of interactions in many systems of interest allows a decomposition of the path integral and its evaluation via sequential linking of the paths of relatively small "modules" (e.g., chemical groups or monomers). The present paper describes the modular path integral methodology for simulating dynamical properties by propagating the density matrix in real time. The procedure is first presented for the simple topology of a single-file arrangement of units interacting via nearest neighbor couplings and subsequently extended to the calculation of two-particle correlations in arrays that may also contain some long-range interactions, to the treatment of systems with side chains or cyclic structures, to the simulation of internal dynamics in long organic molecules, and to the modifications required for coupling of one or several units of a system to dissipative environments. Illustrative applications to the dynamics of interacting two-level-systems are presented.
The modular decomposition of the path integral, which leads to linear scaling with the system length, is extended to Hamiltonians with intermonomer couplings that are not diagonalizable in any single-particle basis. An optimal factorization of the time evolution operator is identified, which minimizes the number of path integral variables while ensuring high accuracy and preservation of detailed balance. The modular path integral decomposition is described, along with a highly efficient tensor factorization of the path linking process. The algorithm is illustrated with applications to a model of coupled spins and a Frenkel exciton chain.
We review an iterative path integral method that allows the computation of exact, long-time dynamics of small systems interacting with a dissipative bath. The method takes advantage of the finite memory of large environments at finite temperature to allow an iterative evaluation of the dynamics, thereby replacing an exponential scaling in simulation time with a linear scaling. This method is applied to calculate the dynamics of two model systems. The first consists of two-qubits interacting with a common bath. In this case we observe a variety of entanglement effects. If the qubits are initially separable, through an indirect coupling, the bath can create steady state entanglement between the qubits. This effect is due to the existence of a decoherence-free subspace as a result of the form of the qubit Hamiltonian and system-bath coupling. Entanglement created by the bath is shown to decrease with increasing temperature and system-bath coupling strength. However large system-bath coupling causes a faster increase in the entanglement. Initially entangled qubits lose their entanglement as a result of interactions with the bath, an effect that is heightened at higher temperatures. Direct coupling between the qubits is shown to slow the decay of entanglement and preserve some entanglement at long times; however at high temperatures this steady state entanglement becomes negligible. The second system we consider is vibrational degree of freedom coupled to a bath of harmonic oscillators or two-level systems. We compute four-time correlation functions which are used to calculate response functions relevant to third order infrared or seventh order Raman experiments for harmonic, Morse, and quadratic-quartic potentials. Our calculations reveal the role of potential features (anharmonicity and eigenvalue spectrum), both on short and long time scales, on the response function. Further, thermal excitation causes dramatic changes in the appearance of the response function, introducing symmetry with respect to the main diagonal. Finally, coupling to harmonic dissipative baths leads to decay of the response function (primarily along the t3 direction) and a broadening of the peaks in its Fourier transform. At high temperatures two-level-system baths are less efficient in destroying coherence than harmonic baths of similar parameters.
The anharmonic small matrix path integral (anh-SMatPI) algorithm is adapted to composite environments, where each anharmonic bath unit couples to the system and to its own harmonic bath. The propagation matrices are obtained from numerical evaluation of the composite influence functional, which is based on separate iterative path integral calculations for each bath module. The numerically exact nature of the algorithm is demonstrated on a two-level system (TLS) coupled to dissipative TLS baths of one, two, and 50 units.