Histogram filtering as a tool in variational Monte Carlo optimization

Optimization of wave functions in quantum Monte Carlo is a difficult task because the statistical uncertainty inherent to the technique makes the absolute determination of the global minimum difficult. To optimize these wave functions we generate a large number of possible minima using many independently generated Monte Carlo ensembles and perform a conjugate gradient optimization. Then we construct histograms of the resulting nominally optimal parameter sets and "filter" them to identify which parameter sets "go together" to generate a local minimum. We follow with correlated-sampling verification runs to find the global minimum. We illustrate this technique for variance and variational energy optimization for a variety of wave functions for small systellls. For such optimized wave functions we calculate the variational energy and variance as well as various non-differential properties. The optimizations are either on par with or superior to determinations in the literature. Furthermore, we show that this technique is sufficiently robust that for molecules one may determine the optimal geometry at tIle same time as one optimizes the variational energy.

On the path integral formulation and the evaluation of quantum statistical averages

Four problems of physical interest have been solved in this thesis using the path integral formalism. Using the trigonometric expansion method of Burton and de Borde (1955), we found the kernel for two interacting one dimensional oscillators• The result is the same as one would obtain using a normal coordinate transformation, We next introduced the method of Papadopolous (1969), which is a systematic perturbation type method specifically geared to finding the partition function Z, or equivalently, the Helmholtz free energy F, of a system of interacting oscillators. We applied this method to the next three problems considered• First, by summing the perturbation expansion, we found F for a system of N interacting Einstein oscillators^ The result obtained is the same as the usual result obtained by Shukla and Muller (1972) • Next, we found F to 0(Xi)f where A is the usual Tan Hove ordering parameter* The results obtained are the same as those of Shukla and Oowley (1971), who have used a diagrammatic procedure, and did the necessary sums in Fourier space* We performed the work in temperature space• Finally, slightly modifying the method of Papadopolous, we found the finite temperature expressions for the Debyecaller factor in Bravais lattices, to 0(AZ) and u(/K/ j,where K is the scattering vector* The high temperature limit of the expressions obtained here, are in complete agreement with the classical results of Maradudin and Flinn (1963) .