Path integral virial estimator based on the scaling of fluctuation   coordinates: Application to quantum clusters with fourth-order propagators
  We first show that a simple scaling of fluctuation coordinates defined in terms of a given reference point gives the conventional virial estimator in discretized path integral, where different choices of the reference point lead to different forms of the estimator (e.g., centroid virial). The merit of this procedure is that it allows a finite difference evaluation of the virial estimator with respect to temperature, which totally avoids the need of higher-order potential derivatives. We apply this procedure to energy and heat capacity calculation of the (H_2)_22 and Ne_13 clusters at low temperature using the fourth-order Takahashi-Imada and Suzuki propagators. This type of calculation requires up to third-order potential derivatives if analytical virial estimators are used, but in practice only first-order derivatives suffice by virtue of the finite difference scheme above. From the application to quantum clusters, we find that the fourth-order propagators do improve upon the primitive approximation, and that the choice of the reference point plays a vital role in reducing the variance of the virial estimator. 
Takeshi M. Yamamoto