PATH INTEGRAL MONTE CARLO METHOD

The method is based on Feynman path integral interpretation of Quantum mechanics. Each quantum particle is represented by a "ring polymer" whose elements (beads) are connected by harmonic forces ("springs"). Interparticle interactions are expressed as interactions between simultaneous beads of the polymers.

The Path Integral Monte Carlo (PIMC) method provides a direct, non-shifted statistical evaluation of quantum canonical averages:

< A > = Tr(A)

where = exp(-H) is the canonical density matrix, H is the Hamiltonian and is the reciprocal temperature.

The great advantage of the PIMC method is that it allows, in principle, really ab initio, without any assumptions beyound the Schrödinger equation, simulations of quantum systems. Still, an application of the PIMC method to a real system of quantum particles (e.g. electrons in the field of nuclei) at finite temperature remains a challenging problem. The main obstacles, both of principal and computational nature, come from the necessity of antisymmetrisation of the density matrix for a many-electron system.

The following paper provides a methodological basis for application of the PIMC method to the system of quantum identical particles with spin 1/2.

A.P.Lyubartsev, P.N.Vorontsov-Velyaminov. "Path Intergal Monte Carlo method in quantum statistics for a system of N identical fermions" Physical Review A, v.48(6), pp.4075-4083 (1993)

Abstract

A rigorous expression for the partition function of N identical fermions with spin 1/2 is obtained in a form suitable for Path Integral Monte Carlo (PIMC) simulation of a many electron system at finite temperature. A sum over permutations is reduced to a sum over classes with readily determinable factors. Distribution over projections and values of the total spin is also obtained enabling PIMC calculations of spin-dependent quantities. A new MC algorithm is proposed which accounts simultaneously for all types of trajectory linkages. A test calculation - four electrons in a spherical cavity with radius 1 nm at temperature 300 - 2000 K is performed.

See also:

P.N.Vorontsov-Velyaminov, M.O.Nesvit and R.I.Gorbunov, "Bead-Fourier path integral Monte Carlo method applied to systems of identical particles", Rhys.Rev.E, v.55(2),p.1979-1997 (1997)

Recent papers on the subject:

P.N.Vorontsov-Velyaminov and A.P.Lyubartsev, "Entropic sampling in the Path Integral Monte Carlo method", J. Phys. A: Math. Gen. v. 36, pp. 685-693 (2003)
Abstract; Text (PDF). This paper describes application of Entropy sampling approach to compute density of states and thermodynamic averages of a quantum system

S.D.Ivanov, A.P.Lyubartsev and A.Laaksonen "Bead-Fourier Path Integral Molecular Dynamics" Phys. Rev. E, v.67, 066710 (2003) Full text (APS)
Abstract
Molecular dynamics formulation of Bead-Fourier path integral method for simulation of quantum systems at finite temperatures is presented. Within this scheme, both the bead coordinates and Fourier coefficients, defining the path representing the quantum particle, are treated as generalized coordinates with corresponding generalized momenta and masses. Introduction of the Fourier harmonics together with the center-of-mass thermostating scheme is shown to remove the ergodicity problem, known to pose serious difficulties in standard path integral molecular dynamics simulations. The method is tested for quantum harmonic oscillator and hydrogen atom (Coulombic potential). The simulation results are compared with the exact analytical solutions available for these both systems. Convergence of the results with respect to the number of beads and Fourier harmonics is analyzed. It was shown that addition of a few Fourier harmonics already improves the simulation results substantially, even for a relatively small number of beads. The proposed Bead-Fourier Path-Integral molecular dynamics is a reliable and efficient alternative to simulations of quantum systems.