Monte Carlo algorithms for expectation values of coordinate operators

Abstract

Abstract In diffusion QMC with importance sampling, walkers provide samples of configurations with probabilities proportional to the product of the true wave function ‘¢ and a trial wavefunction t/Jr. The expectation value of the energy (very fortunately) may be calculated from the local energies of these samples as E = (‘l/JH’l/Jr)/(‘l/Jt/Jr) = (‘l/JE10ct/Jr)/(‘¢’¢r). But for an operator A which does not commute with the Hamiltonian H, the mixed expectation value (‘¢At/Jr) is only an approximation to the “pure” expectation value (‘ljJA’¢). Thus, for most quantities of interest other than the energy, an extension to standard diffusion QMC methods is required. This paper describes two approaches to the problem. Both are based on an earlier development by Liu, Kalos, and Chester0 involving the tracking of descendents of walkers to obtain pure expectation values. The first uses simple DQMC and the second uses VQMC with DQMC “side walkers.” These methods were tested in their application to the model systems H and H2, and both were found effective in yielding accuracies and precisions correct within 0.5% for the quantities (r) for H and (r2) and (z2) for H and H2. The large differences in electron and proton masses led to extensive computation requirements due to the slow equilibration and serial correlation induced by the heavier protons. The calculations were executed on one of the first massively parallel computers, a Thinking Machines CM-2 with 65,536 processors. Frequent communication among the processors was required to balance the number of walkers treated in each.