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Physical Review B

Molecular dynamics with quantum fluctuations

Ionuţ Georgescu, Vladimir A. Mandelshtam

Published Sep 9, 2010• 12 citations
Publisher

Abstract

A quantum dynamics approach, called Gaussian molecular dynamics, is introduced. As in the centroid molecular dynamics, the $N$-body quantum system is mapped to an $N$-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.

DOI
10.1103/physrevb.82.094305

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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.
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.
This paper presents a mathematical derivation of a model for quantum-classical molecular dynamics (QCMD) as a partial classical limit of the full Schrödinger equation. This limit is achieved in two steps: separation of the full wave function and short wave asymptotics for its ‘‘classical’’ part. Both steps can be rigorously justified under the same smallness assumptions. This throws some light on the time-dependent self-consistent-field method and on mixed quantum-semiclassical models, which also depend on the separation step. On the other hand, the theory leads to a characterization of the critical situations in which the QCMD model is in danger of largely deviating from the solution of full Schrödinger equation. These critical situations are exemplified in an illustrative numerical simulation: the collinear collision of a classical particle with a harmonic quantum oscillator.
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.
The motion of wave packets can be easily determined for any Hamiltonian that is quadratic in position and momentum, even if the coefficients of the terms in the Hamiltonian vary with time. The method is based on the existence of an invariant operator, linear in position and momentum, the coefficients of these operators being solutions of the corresponding classical system. This immediately yields a set of very simple wave packets whose evolution is easily determined, and whose magnitude has the same form as the energy eigenfunctions of the harmonic oscillator (i.e., Gaussian or Hermite–Gaussian). This set provides a complete basis for finding the evolution of any state and the exact propagator is readily determined. For the harmonic oscillator, this set includes coherent states, squeezed states, displaced number states, and squeezed number states as special cases. The following important properties hold for every quadratic Hamiltonian, no matter what time dependence is present in the Hamiltonian: (1) The motion of the centroid of any wave packet separates from that of any moments relative to the centroid. (2) Every detail of the evolution of the quantum system can be calculated from the solutions of the corresponding classical system. (3) Wave functions of Gaussian (or Hermite–Gaussian) form will retain that form as they evolve.
A new method, here called thermal Gaussian molecular dynamics (TGMD), for simulating the dynamics of quantum many-body systems has recently been introduced [I. Georgescu and V. A. Mandelshtam, Phys. Rev. B 82, 094305 (2010)]. As in the centroid molecular dynamics (CMD), in TGMD the N-body quantum system is mapped to an N-body classical system. The associated both effective Hamiltonian and effective force are computed within the variational Gaussian wave-packet approximation. The TGMD is exact for the high-temperature limit, accurate for short times, and preserves the quantum canonical distribution. For a harmonic potential and any form of operator \documentclass[12pt]{minimal}\begin{document}A^\hat{A}A^\end{document}Â, it provides exact time correlation functions CAB(t) at least for the case of \documentclass[12pt]{minimal}\begin{document}B^\hat{B}B^\end{document}B̂, a linear combination of the position, \documentclass[12pt]{minimal}\begin{document}x^\hat{x}x^\end{document}x̂, and momentum, \documentclass[12pt]{minimal}\begin{document}p^\hat{p}p^​\end{document}p̂, operators. While conceptually similar to CMD and other quantum molecular dynamics approaches, the great advantage of TGMD is its computational efficiency. We introduce the many-body implementation and demonstrate it on the benchmark problem of calculating the velocity time auto-correlation function for liquid para-hydrogen, using a system of up to N = 2592 particles.