On energy estimators in path integral Monte Carlo simulations:  Dependence of accuracy on algorithm

Two energy estimators, the Barker estimator and the Berne virial estimator, commonly used in path integral simulations of quantum systems are compared with respect to statistical accuracy. It is found that the accuracy of these estimators is strongly affected by the algorithm used. Four common algorithms are considered here: (1) the pure primitive algorithm, (2) the primitive algorithm augmented by whole chain moves, (3) the normal-mode algorithm, and (4) the staging algorithm. The error of the mean of the Barker energy estimator is found to grow as (P)1/2, where P is the number of discretization points of the quantum paths (or the number of chain particles in the isomorphic classical chain), for all of the algorithms above. The error of the mean of the Berne virial energy estimator is independent of P for algorithms 2, 3, and 4, and increases as (P)1/2 for algorithm 1. It is concluded that the virial estimator is far more accurate than the Barker estimator for algorithms 2, 3, and 4, and is at least as accurate for algorithm 1. Because the error analysis depends strongly on the temporal correlations in the sequence of values of the energy estimator generated during Monte Carlo or molecular-dynamics simulations, we review the general question of error analysis in simulations.