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Progress of Theoretical Physics Supplement

Path Integral Centroid Molecular Dynamics Simulation Extended to Bose and Fermi Statistics

Kenichi Kinugawa, Hidemi Nagao, Koji Ohta

Published Jan 1, 2000• 3 citations
Publisher

Abstract

We explicitly propose a practical technique for the path integral centroid molecular dynamics (CMD) extended to Bose/Fermi statistics and present a concrete set of equations of motion to be solved in the simulations with computational efficiency.
DOI
10.1143/ptps.138.531

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Numerical algorithms are developed for the centroid molecular dynamics (centroid MD) method to calculate dynamical time correlation functions for general many-body quantum systems. Approaches based on the normal mode path integral molecular dynamics and staging path integral Monte Carlo methods are described to carry out a direct calculation of the force on the centroid variables in the centroid MD algorithm. A more efficient, but approximate, scheme to compute the centroid force is devised which is based on the locally optimized harmonic approximation for the centroid potential. The centroid MD equations in the latter method can be solved with the help of an iterative procedure or through extended Lagrangian dynamics. A third algorithm introduces an effective centroid pseudopotential to approximate the full many-body centroid mean force potential by effective pairwise centroid interactions. Numerical simulations for both prototype models and more realistic many-body systems are performed to explore the feasibility and limitations of each algorithm.
We propose a method which uses centroid molecular dynamics (CMD) [J. Cao and G. A. Voth, J. Chem. Phys. 100, 5106 (1994)] real-time data in conjunction with the imaginary-time data generated using path integral Monte Carlo simulations in a numerical analytic continuation scheme based on the maximum entropy approach. We show that significant improvement is achieved by including short-time CMD data with the imaginary-time data. In particular, for a particle bilinearly coupled to a harmonic bath, these methods lead to significant improvements over previous calculations and even allow accurate determination of transport coefficients such as the diffusion coefficient and mobility for this system. In addition we show how maximum entropy method can be used to extract accurate dynamic information from short-time CMD data, and that this approach is superior to the direct Fourier transform of long-time data for systems characterized by broad, featureless spectral distributions.
A quantum dynamics approach, called Gaussian molecular dynamics, is introduced. As in the centroid molecular dynamics, the NNN-body quantum system is mapped to an NNN-body classical system with an effective Hamiltonian arising within the variational Gaussian wave-packet approximation. The approach is exact for the harmonic oscillator and for the high-temperature limit, accurate in the short-time limit and is computationally very efficient.
We present a formalism of the path integral centroid molecular dynamics (CMD) extended to Bose and Fermi statistics as a semiclassical approach to explore the dynamics of quantum many-body systems. The validity of the method is examined in relation to the time correlation functions. The presently proposed scheme, refined from our previous derivation [Chem. Phys. Lett. 307, 187 (1999)], is aimed at the calculations of not the exact quantum-mechanical dynamics but the semiclassical dynamics under certain approximations. The formalism is based on the projection operator with which the Bose/Fermi system is mapped onto a particular type of pseudo-Boltzmann system. In the pseudo-Boltzmann system the correlation due to the Bose/Fermi statistics is introduced via an extra pseudopotential called the permutation potential and its relevant operator. Using the present semiclassical formalism, the time correlation function of centroid position, which is evaluated from the CMD trajectories in the pseudo-Boltzmann system, is an approximation to the Kubo canonical correlation function of position operator of the exact quantum-statistical system composed of bosons or fermions. There is no such apparent relation between the momentum operator and the corresponding momentum centroid.