Monte Carlo calculation of the quantum partition function via path integral formulations

Abstract

Using Bennett’s Monte Carlo (MC) method, we calculate the quantum partition functions of path integral formulations. First, from numerically exact results for a harmonic oscillator and a double-well potential, we discuss how fast each approximate partition function converges to the exact value as the number of integral variables involved in the formulation is increased. It turns out that most effective and most suitable for the MC simulation is Takahashi and Imada’s path integral fomulation based on a modified Trotter formula in which the original potential is replaced with an effective one. This formulation is well balanced between the following two factors: the effect of zero potential energy is underestimated, resulting in an improper increase in the partition function; and, on the other hand, effective potential restricts the motion of fictitious particles born in the formulation so that the partition function value tends to be smaller. Fictitious particles can be treated as classical ones. We therefore can apply Bennett’s MC method to calculating the ratio of two quantum partition functions (of a system under consideration and a reference system). As the number of fictitious particles N is increased, choice of reference system becomes less and less important and multistage sampling becomes dispensable. This, to some extent, compensates for the expense that N is larger than the real particle number. The tunneling mechanism of fictitious particles in the simulation is discussed.

Interplay between shell effects and electron correlations in quantum dots

“…A number of various different energy estimators has been discussed in the past. [16][17][18] To avoid these difficulties we developed a procedure which allows the calculation of all energy expectation values from the knowledge of the pair correlation functions…”

A new Fourier path integral method, a more general scheme for extrapolation, and comparison of eight path integral methods for the quantum mechanical calculation of free energies

“…Considerable attention has been given to the calculation of higher order corrections to the Trotter approximation. 22,28,[31][32][33][34][35] One of the most widely used expressions, 22,32,33 which we will simply refer to as the Takahashi-Imada ͑TI͒ approximation, can be implemented by replacing the potential in Eq. ͑17͒ with an effective potential…”

“…In the present paper, we use this approach to transform five DPI methods into analog FPI schemes, and we compare these methods to the C-FPI and PA-FPI methods as well as to a new method introduced below. The relative efficiency of various DPI and FPI methods has already been widely studied 9,14,[22][23][24][25] ͑and debated!͒. By using the Fourier analogs of the DPI methods we can employ essentially the same Monte Carlo sampling scheme for all eight methods, and this permits a more even-handed comparison of relative efficiencies than has been available previously.…”

The partial averaging Fourier path Integral approach based on the harmonic reference path

“…(1) and (15), are plotted in Fig. (16) is actually the trajectory of a harmonic oscillator that propagates in imaginary time and satis®es the boundary conditions: x0 x i and x" hb x f .…”