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

Using an isomorphism of Coalson, we transform five different discretized path integral (DPI) methods into Fourier path integral (FPI) schemes. This allows an even-handed comparison of these methods to the conventional and partially averaged FPI methods as well as a new FPI method. It also allows us to apply to DPI methods a simple and highly effective perturbative correction scheme (previously presented for FPI methods) to account for the error due to retaining only a finite number of terms in the numerical evaluation of the propagator. We find that in all cases the perturbative corrections can be extrapolated to the convergence limit with high accuracy by using a correlated sequence of affordable calculations. The Monte Carlo sampling variances of all eight methods studied are very similar, but the variance of the perturbative corrections varies markedly with method. The efficiencies of the new FPI method (called rescaled fluctuation FPI) and one of Fourier analog methods compare favorably with that of the original FPI method. The rescaled fluctuation method not only proves practically successful, but it also gives insight into the origin of the dominant error in the conventional FPI scheme.