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Journal of Media Literacy Education

Hierarchy of N-body correlations in different dimensions

Mervlyn Moodley

Published Jan 1, 2003• 1 citation
Publisher

Abstract

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.

DOI
10.23860/diss-1930

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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.
Since the form of PN is the same as that occurring in the statistical mechanics of the classical gas (replace f 2 ( r ij ) by exp[- V ( r ij /kT ]), we can use the same integration techniques that are used in the classical problem. W. L. McMillan, 1965 Summary Building on the random walk methods developed in the previous chapter, we show how to compute properties of many-body trial wavefunctions using a random walk. This method, called variational Monte Carlo, is the simplest stochastic quantum many-body technique. Whereas mean-field methods are usually limited to single determinants, variational Monte Carlo can treat any correlated trial function, as long as its values are computable. We discuss how to optimize such trial wavefunctions, how to compute their momentum distribution, how to use non-local pseudopotentials, how to compute excited states, and how to correct for the finite size of the simulation cell. Deterministic quantum methods have difficulties. For example, the Hartree–Fock method assumes the wavefunction is a single Slater determinant, neglecting correlation. If one expands as a sum of determinants, it is very difficult to have the results size-consistent since the number of determinants needed will grow exponentially with the system size. As we have seen, the DMFT method introduced in Ch. 16 assumes locality. In Ch. 6 we discussed general properties of many-body wavefunctions. Using Monte Carlo methods, we can directly incorporate correlation into a wavefunction, without having to make any further approximations other than the form of the correlation factors. In many cases the energy and other properties are very close to the exact results. Some of the usual restrictions on the form of the many-body wavefunction are not an issue in variational Monte Carlo. The most important generalization of the HF wavefunction is to put correlation directly into the wavefunction via the “Jastrow” factor. At next order, one can use the “backflow” wavefunction, in which correlation is also built into the determinant. The variational Monte Carlo method (VMC) was first used by McMillan [44] to calculate the ground-state properties of superfluid 4 He. One of the key problems at that time was whether the observed superfluid properties were a consequence of Bose condensation.
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.
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.
The path-integral renormalization group and direct energy minimization method of practical first-principles electronic structure calculations for multi-body systems within the framework of the real-space finite-difference scheme are introduced. These two methods can handle higher dimensional systems with consideration of the correlation effect. Furthermore, they can be easily extended to the multicomponent quantum systems which contain more than two kinds of quantum particles. The key to the present methods is employing linear combinations of nonorthogonal Slater determinants (SDs) as multi-body wavefunctions. As one of the noticeable results, the same accuracy as the variational Monte Carlo method is achieved with a few SDs. This enables us to study the entire ground state consisting of electrons and nuclei without the need to use the Born-Oppenheimer approximation. Recent activities on methodological developments aiming towards practical calculations such as the implementation of auxiliary field for Coulombic interaction, the treatment of the kinetic operator in imaginary-time evolutions, the time-saving double-grid technique for bare-Coulomb atomic potentials and the optimization scheme for minimizing the total-energy functional are also introduced. As test examples, the total energy of the hydrogen molecule, the atomic configuration of the methylene and the electronic structures of two-dimensional quantum dots are calculated, and the accuracy, availability and possibility of the present methods are demonstrated.