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Physical Review

Improved Quantum Theory of Many-Electron Systems. I. Construction of Eigenfunctions of S^2 Which Satisfy Pauli's Principle

William A. Goddard

Published May 5, 1967• 239 citations
Publisher

Abstract

A general method of obtaining many-electron wave functions which are eigenfunctions of ${\stackrel{^}{S}}^{2}$ and which also satisfy Pauli's principle is developed. This method is based on the properties of the irreducible units of the group algebra of the symmetric group. Using this method, expansions of the exact wave function are found which will in succeeding articles form the basis of a powerful method of obtaining accurate and quite useful many-electron wave functions for atoms, molecules, and solids.

DOI
10.1103/physrev.157.73

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A general method of obtaining many-electron wave functions which are eigenfunctions of {\stackrel{^}{S}}^{2} and which also satisfy Pauli's principle is developed. This method is based on the properties of the irreducible units of the group algebra of the symmetric group. Using this method, expansions of the exact wave function are found which will in succeeding articles form the basis of a powerful method of obtaining accurate and quite useful many-electron wave functions for atoms, molecules, and solids.
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Quantum Monte Carlo (QMC) methods are a class of stochastic techniques that can be used to compute the properties of electronic systems accurately from first principles. This thesis is mainly concerned with the development of trial wave functions for QMC. An extension of the backflow transformation to inhomogeneous electronic systems is presented and applied to atoms, molecules and extended systems. The backflow transformation I have developed typically retrieves an additional 50% of the remaining correlation energy at the variational Monte Carlo level, and 30% at the diffusion Monte Carlo level; the number of parameters required to achieve a given fraction of the correlation energy does not appear to increase with system size. The expense incurred by the use of backflow transformations is investigated, and it is found to scale favourably with system size. Additionally, I propose a single wave function form for studying the electron-hole system which includes pairing effects and is capable of describing all of the relevant phases of this system. The effectiveness of this general wave function is demonstrated by applying it to a particular transition between two phases of the symmetric electron-hole bilayer, and it is found that using a single wave function form gives a more accurate physical description of the system than using a different wave function to describe each phase. Both of these developments are new, and they provide a powerful set of tools for designing accurate wave functions. Backflow transformations are particularly important for systems with repulsive interactions, while pairing wave functions are important for attractive interactions. It is possible to combine backflow and pairing to further increase the accuracy of the wave function. The wave function technology that I have developed should therefore be useful across a very wide range of problems.
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