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arXiv:physics/0102069 · Recent advances in computational

Trial function optimization for excited states of van der Waals clusters

M. P. Nightingale, V. Melik-Alaverdian

Published Feb 1, 2002• 0 citations
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Abstract

A method is introduced to optimize excited state trial wave functions. The method is applied to ground and vibrationally excited states of bosonic van der Waals clusters of upto seven particles. Employing optimized trial wavefunctions with three-body correlations, we use correlation function Monte Carlo to estimate the corresponding excited state energies.

DOI
10.1142/9789812775696_0009

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A method is introduced to optimize excited state trial wave functions. The method is applied to ground and vibrationally excited states of bosonic van der Waals clusters of upto seven particles. Employing optimized trial wavefunctions with three-body correlations, we use correlation function Monte Carlo to estimate the corresponding excited state energies.
The computational physics methods of various processes have become very popular during the last several years. Our interest in his field is predominantly motivated by the expectation that it will be possible to perform a detailed comparison between experimental and theoretical results obtained in the study of clusters of a variety of sizes. Algebraic invariants are employed to deal efficiently with the many-body correlations. We study the applicability of the developed trial wave functions over some range of particle masses, from Ar to He and lighter. We compute the ground and excited states energies and various geometric sizes of small clusters as a function of a continuously varied mass parameter. For sufficiently small masses the cluster can no longer support a bound state, and we are interested in particular in the behavior of the clusters in the vicinity of the critical mass, i.e., the point at which the binding energy approaches zero. We show that the use of bosonic polynomials provides enough variational freedom to approximate ground and excited states of clusters of any mass. We discuss how the properties of clusters change with mass.
In calculating energy spectra, unlimited accuracy is only obtained if N-body wavefunctions contain N-body correlations except in special cases. These wave-functions are however not practical both for analytic and computational considerations. To this extent, the possiblity of utilizing correlations less than N is investigated. These investigations are based on the hypothesis whereby in D-dimensions, correlations greater than (D + 1)-body correlations are considerably less important. To accomplish these investigations, we compare the quality Q of the wavefunctions, pertaining to two-, three- and four-body correlations, in various dimensions. The trial wavefunctions used in our Monte Carlo method are linear combinations of elementary basis functions that consist of non-linear variational parameters. This method entails a linear optimization nested in a non-linear one. Once these trial wavefunctions are optimized we employ them as basis functions in a correlation function Monte Carlo which reduces systematically the variational bias of the energy estimates. This method is used to calculate ground and excited state energies of small bosonic van der Waal clusters using wavefunctions with two-, three- and four-body correlations. In calculating energy spectra in different dimensions, it was observed that for N atoms there exists an inter-dimensional degeneracy in the energy spectrum for (N − 1)-dimensions and (N + 1)-dimensions. The finite difference method is also used to calculate 'exact' energy spectra of two-body clusters, and these results are used as a benchmark to compare results obtained by our Monte Carlo method. We also describe a method of obtaining better statistics from quantum Monte Carlo estimators based on the assumption that these estimators are afflicted with distributions that resemble the lognormal probability distribution.
Abstract This paper describes improved trial wavefunctions for helium clusters and their use in variational QMC calculations of their binding energies, structural properties, and breathing mode vibrational excited states, for clusters of 3 to 270 atoms. The wavefunctions developed were similar to those of earlier work but included convenient analytic functions for two- and three-body correlations. All terms were of the Jastrow exponential form, based on sums of one-, two-, and three-body terms t1(ri), t2(rij), and t3(rij, rjk,rki), expressed as functions of interparticle distances. Excited-state wavefunctions for the breathing vibrational mode were obtained with the aid of procedures adapted from treatment of the quantum mechanics of liquid drop models. The interactions of helium atoms were specified as pairwise-additive with an accurate semi-empirical pair potential. Optimizations of parameters in the trial wavefunctions were carried out for individual cluster sizes with an exhaustive search procedure in a series of trial variational calculations, in which the parameters were varied to minimize the energy. A partial correlation of results was obtained by use of identical initial coordinates and random number seeds in these optimization calculations. Once the wavefunctions were optimized, longer calculations were used to determine properties of the clusters. In addition, diffusion QMC calculations were used to obtain more accurate nonvariational energies for comparisons.
Over the past few decades many new and improved techniques have been developed to calculate ground and excited states of atomic clusters with varying degrees of accuracy. Together with progressively enhanced computer capabilities and the development of sophisticated techniques, the accuracy of results obtained from ab initio methods are closely approximating machine precision. In this paper, we use the finite difference method, whose precision is only governed by the machine precision, together with an optimization criterion to calculate the spectrum of two-body van der Waal clusters. This method is also shown to be robust enough to calculate energies close to the unbinding transition.