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

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.