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Proceedings of the Physical Society

Construction of S2 Eigenfunctions by the Method of Spin Operators II: Six-electron Systems

F Berencz

Published Feb 1, 1958• 4 citations
Publisher

Abstract

In the first part of this paper a spin operator was introduced which when operating on the eigenfunctions of the total S z operator related to maximal, minimal and zero projections of total spin creates each eigenfunction of the total S 2 corresponding to its different eigenvalues. In this second part a brief discussion of the proposed operator technique developed for the problem of six-electron system is presented.

DOI
10.1088/0370-1328/71/2/302

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In the first part of this paper a spin operator was introduced which when operating on the eigenfunctions of the total S z operator related to maximal, minimal and zero projections of total spin creates each eigenfunction of the total S 2 corresponding to its different eigenvalues. In this second part a brief discussion of the proposed operator technique developed for the problem of six-electron system is presented.
This work reports on a new and effective method that simplifies the many-electron problem. Instead of a wave function, an operator is used to describe the symmetry of the many-electron system through a transformation. It is proven that the two approaches are physically equivalent, but the operator formalism works much more effectively than the wave function formalism. In the operator formalism, the wave function of an n-electron system can be separated into its spatial and spin parts, and the symmetries of identical particles can be correctly described. For example, it can provide the correct spin and spatial symmetry of many-electron atoms or molecules. For many-electron atoms, one can obtain the simultaneous eigenfunction of L̂2 and Ŝ2. Consequently, the spectroscopic terms of a configuration can be obtained, for example, the hydrogen atom in an energy formula with four quantum numbers (n, l, ml, and ms) for each electron in an atom, including the Hartree–Fock theory. We theoretically analyze and discuss the atomic structure of many-electron atoms using this formula.
The problem of construction of projection operators on eigensubspaces of symmetry operators is considered. This problem arises in many approximate methods for solving time-independent and time-dependent quantum problems, and its solution ensures proper physical symmetries in development of approximate methods. The projector form is sought as a function of symmetry operators and their eigenvalues characterizing the eigensubspace of interest. This form is obtained in two steps: (1) identification of algebraic structures within a set of symmetry operators (e.g., groups and Lie algebras) and (2) construction of the projection operators for individual symmetry operators. The first step is crucial for efficient projection operator construction because it allows for use of information on irreducible representations of the present algebraic structure. The discussed approaches have potential to stimulate further developments of variational approaches for electronic structure of strongly correlated systems and in quantum computing.