Monte Carlo algorithms for expectation values of coordinate operators

Abstract

Abstract Two Monte Carlo algorithms for computing quantum mechanical expectation values of coordinate operators, i.e., multiplicative operators that do not commute with the Hamiltonian, are presented and compared. The first employs a single quantum Monte Carlo (QMC) random walk, while the second involves a variational Monte Carlo (VMC) random walk with auxiliary QMC “side walks.” The tagging algorithm used for efficiently tracking descendants of a walker is described in detail for each approach. For the single-walk algorithm it is found that carrying weights together with branching significantly improves efficiency. Exploitation of the correlation between VMC and QMC expectation values is also considered. Large increases in efficiency in the second approach are found when such correlations are incorporated. It is found that both approaches readily yield accuracies and precisions of better than 0.5% for the model systems treated here, namely, H and H 2 . The second method, involving a VMC walk with auxiliary QMC walks, is the more efficient for these systems.