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Journal of Chemical Theory and Computation

Two-Tier Modular Anharmonic Small Matrix Path Integral with Composite Spin-Boson Baths

Nancy Makri

Published Sep 20, 2024• 6 citations
Publisher

Abstract

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.

DOI
10.1021/acs.jctc.4c00980

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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.
The influence functional (IF) encodes all of the information required for calculating the dynamical properties of a system in contact with its environment. A direct and simple procedure is introduced for extracting from a few numerical evaluations of the IF, without computing time correlation functions or evaluating integrals, the parameters required for path integral calculations, either within or beyond the harmonic mapping, and for assessing the accuracy of the harmonic bath approximation. In addition, the small matrix decomposition of the path integral (SMatPI) is extended to anharmonic environments and the required matrices are constructed directly from the IF.
The small matrix decomposition of the quasi-adiabatic propagator path integral (SMatPI) for a system coupled to a harmonic bath, which accounts for multitime memory correlations in the influence functional without the use of tensors, is extended to include a time-dependent term that drives the system. In the case of a periodic field, the algorithm requires the construction of SMatPI matrices initialized over a short time interval. The SMatPI algorithm circumvents the large array storage of tensor-based iterative path integral decompositions and, in the case of a periodic field, also eliminates the demanding tensor multiplication at each time step, leading to dramatic savings which allow the fully quantum mechanical treatment of multistate systems and long-memory environments.
A small matrix decomposition of the path integral expression (SMatPI) that yields the reduced density matrix of a system interacting with a dissipative harmonic bath is obtained by recursively spreading the entangled influence functional terms over longer time intervals while simultaneously decreasing their magnitude until these terms become negligible. This allows summation over the path integral variables one by one through multiplication of small matrices with dimension equal to that of the bare system. The theoretical framework of the decomposition is described using a diagrammatic approach. Analytical and numerical calculations show that the necessary time length for the temporal entanglement to become negligible is practically the same as the bath-induced memory. The properties and structure of the propagator matrices are discussed, and applications to multistate systems are presented.
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.
This paper reports the release of PathSum, a new software suite of state-of-the-art path integral methods for studying the dynamics of single or extended systems coupled to harmonic environments. The package includes two modules, suitable for system-bath problems and extended systems comprising many coupled system-bath units, and is offered in C++ and Fortran implementations. The system-bath module offers the recently developed small matrix path integral (SMatPI) and the well-established iterative quasi-adiabatic propagator path integral (i-QuAPI) method for iteration of the reduced density matrix of the system. In the SMatPI module, the dynamics within the entanglement interval can be computed using QuAPI, the blip sum, time evolving matrix product operators, or the quantum-classical path integral method. These methods have distinct convergence characteristics and their combination allows a user to access a variety of regimes. The extended system module provides the user with two algorithms of the modular path integral method, applicable to quantum spin chains or excitonic molecular aggregates. An overview of the methods and code structure is provided, along with guidance on method selection and representative examples.