Monte Carlo eigenvalue and variance estimates from several functional optimizations

Abstract

Abstract Optimization of trial functions by minimization of the variance in local energies began with early variational calculations by Frost,a Conroy/ and later by Coldwell.c Minimizing the variance in local energies clearly leads to VQMC and DQMC energies with a lowered statistical error, but as pointed out in this paper, there may be better choices of functionals to optimize. Ten different functionals, assembled from local energy and weighting terms, all giving emphasis in varying degrees to the variance in local energies and the average of local energies, were investigated, with low-lying states of He, H2, and Hj as example systems. The trial wavefunctions were simple compact functions including explicit r12 terms, with as many as 36 parameters total for He 3 S and 122 for Ht. Optimization for a fixed set of configurations was followed by VQMC calculations for the most promising trial functions generated. Minimizing the variance alone in the optimizations gave the lowest variance in energies for the VQMC results. Minimizing a combination of energy and variance in a form used by Conroyb gave the lowest energies in most but not all cases. The choice of functional was found to have a significant effect on the accuracy of VQMC calculations, and the best choice was found to be system dependent.