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Journal of the Physical Society of Japan

Quantum Transfer Monte Carlo Method for Finite Temperature Properties and Quantum Molecular Dynamics Method for Dynamical Correlation Functions

Masatoshi Imada, Minoru Takahashi

Published Oct 15, 1986• 47 citations
Publisher

Abstract

A quantum transfer matrix method is proposed and examined. To obtain finite temperature properties, a small number of Monte Carlo samples for the trace summation is taken without the Monte Carlo sampling of the path integral. We introduce the method of a random orthonormal base in the Monte Carlo sampling. This makes it possible to investigate larger size systems than the exact diagonalization. An advantage of this method is that it does not have negative sign difficulty in the path integral as contrast to the usual quantum Monte Carlo method. A quantum molecular dynamics method is also proposed to investigate dynamical correlation functions.

DOI
10.1143/jpsj.55.3354

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We present a numerically exact procedure for the calculation of an important class of finite temperature quantum mechanical time correlation functions. The present approach is based around the stationary phase Monte Carlo (SPMC) method, a general mathematical tool for the calculation of high dimensional averages of oscillatory integrands. In the present context the method makes possible the direct numerical path integral calculation of real-time quantum dynamical quantities for times appreciably greater than the thermal time (βℏ). Illustrative applications involving finite temperature anharmonic motion are presented. Issues of importance with respect to future applications are identified and discussed.
We formulate a new quantum many body simulation method for a general quantum fluid at any given temperature. Unlike path integral Monte Carlo our method evolves in imaginary time the density matrix from its initial delta function condition to its final thermal form at time time equal to the inverse temperature. It does this with a molecular dynamics scheme on a classical Hamiltonian that has the same functional form as the one of the quantum mechanical Hamiltonian according to the properties of the continuous representation of John R. Klauder. We then end up with the thermal density matrix that can be used to extract thermal averages of observables using the Monte Carlo method equally well in any statistics.
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This thesis describes several novel approaches in quantum molecular dynamics for obtaining properties of molecular systems in different regimes. We investigate ground state properties of chains of linear rotors with dipole–dipole interactions via the density matrix renormalization group (DMRG), by deriving the appropriate form of the interaction operator and implementing it in ITensor. This provides us with further evidence of a quantum phase transition in this system. We also improve the sampling of Gaussian mixture distributions for finite temperature path integral Monte Carlo (PIMC) of vibronic Hamiltonians. To do this, we replace random sampling by quasi-random sampling, and improve sampling distributions by optimizing their parameters. Finally, we introduce estimators and integrators for constrained free energy simulations in path integral molecular dynamics (PIMD). This method is applied to the study of a water dimer, for which we obtain a quantum potential of mean force.
This thesis presents and develops the path integral simulation techniques in application to small quantum systems at finite temperatures. The first goal is to obtain exact thermodynamic expressions for systems of noninteracting The rest and the major part of the thesis is dedicated to the development and testing of Bead-Fourier path integral molecular dynamics. Although, path integral molecular dynamics as well as path integral Monte Carlo are well First, molecular dynamics under Bead-Fourier scheme was developed and tested on the examples of quantum harmonic oscillator and Hydrogen atom. The main attention was paid to ergodicity problems. Then we addressed the question, Later, the formalism for identical particles was developed. Finally, the question of molecular dynamics efficacy was raised. It was shown, that formalisms for identical and distinguishable particles, both, can be reformulated into a more efficient ones, providing all dynamical variables