Heat capacity estimators for random series path-integral methods by finite-difference schemes

Abstract

Previous heat capacity estimators used in path integral simulations either have large variances that grow to infinity with the number of path variables or require the evaluation of first-and second-order derivatives of the potential. In the present paper, we show that the evaluation of the total energy by the T-method estimator and of the heat capacity by the TT-method estimator can be implemented by a finite difference scheme in a stable fashion. As such, the variances of the resulting estimators are finite and the evaluation of the estimators requires the potential function only. By comparison with the task of computing the partition function, the evaluation of the estimators requires kϩ1 times more calls to the potential, where k is the order of the difference scheme employed. Quantum Monte Carlo simulations for the Ne 13 cluster demonstrate that a second order central-difference scheme should suffice for most applications.

Three applications of path integrals: equilibrium and kinetic isotope effects, and the temperature dependence of the rate constant of the [1,5] sigmatropic hydrogen shift in (Z)-1,3-pentadiene

“…22 for several types of constraints. These estimators are derived in a similar manner, based on the rescaling of coordinates and a finite difference evaluation of the derivative with respect to β that was used for the first time by Predescu and coworkers for reactant energies and heat capacities [27]. The statistical error of these estimators showed a surprising behavior in comparison with the estimators for E r .…”

Lévy-Ciesielski random series as a useful platform for Monte Carlo path integral sampling

“…Therefore, the thermodynamic estimators obtained from formal differentiation against the inverse temperature have finite variance in the limit of large number of path variables [13,32,33]. As for random series, Eq.…”

“…(48). The estimators have been reviewed elsewhere [32,33]. In the case where all path variables are updated simultaneously, we have employed the same maximal displacements for all variables a l,j and b l,j .…”

Comment on “Path-integral virial estimator based on the scaling of fluctuation coordinates: Application to quantum clusters with fourth-order propagators” [J. Chem. Phys. 123, 104101 (2005)]

“…1, while referring to the previous work of the present author and collaborators, [2][3][4] Yamamoto asserts that "Thus, we think that the dominant factor that made a large difference in their calculations is the choice of the reference point in the virial estimator, rather than whether the estimator was evaluated analytically or numerically via finite difference…." Neither the present author nor his collaborators have suggested that the technique utilized to compute the derivatives might be the cause of the differences in the behavior of the estimators.…”

“…Neither the present author nor his collaborators have suggested that the technique utilized to compute the derivatives might be the cause of the differences in the behavior of the estimators. Rather, they have pointed out 3,4 that there might be intrinsic differences in the estimators, differences that Yamamoto correctly identifies in Ref. 1.…”