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The Journal of Chemical Physics

A unitary group formulation of open-shell electron propagator theory

Gregory Born, Isaiah Shavitt

Published Jan 1, 1982• 31 citations
Publisher

Abstract

By replacing the usual annihilation and creation operators of second quantization by appropriately normalized fundamental Wigner operators of the unitary group U(n) and by representing the many-electron spin eigenfunctions in terms of the Gelfand–Tsetlin basis for the appropriate irreducible representation, we have succeeded in developing an attractive electron propagator formalism which incorporates closed-shell, open-shell, or multiconfigurational reference states. Matrix element evaluation for the fundamental U(n) Wigner operators is treated, and illustrative three-orbital examples involving reference states of doublet and triplet spin symmetry are presented.

DOI
10.1063/1.442702

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By replacing the usual annihilation and creation operators of second quantization by appropriately normalized fundamental Wigner operators of the unitary group U(n) and by representing the many-electron spin eigenfunctions in terms of the Gelfand–Tsetlin basis for the appropriate irreducible representation, we have succeeded in developing an attractive electron propagator formalism which incorporates closed-shell, open-shell, or multiconfigurational reference states. Matrix element evaluation for the fundamental U(n) Wigner operators is treated, and illustrative three-orbital examples involving reference states of doublet and triplet spin symmetry are presented.
Graphical methods of spin algebras are used to derive the expressions for the matrix elements of the total particle-number-conserving operators in the basis of hole-particle states, adapted to the chain $\mathrm{U}({n}^{\ensuremath{'}}+{n}^{\ensuremath{'}\ensuremath{'}})\ensuremath{\supset}\mathrm{U}({n}^{\ensuremath{'}})\ensuremath{\bigotimes}\mathrm{U}({n}^{\ensuremath{'}\ensuremath{'}})$, where ${n}^{\ensuremath{'}}$ and ${n}^{\ensuremath{'}\ensuremath{'}}$ designate dimensions of particle and hole subspaces, respectively. The matrix elements are expressed as a product of segment values, each associated with one orbital level as in the particle formalism, and of an additional segment value, representing a linkage of the hole and particle subspaces. It is shown that the particle formalism segment values can be used throughout except for the link segment, whose possible values are derived. An example of hole-particle bases and of their graphical representations is given and the advantages of the hole-particle formalism in shell-model calculations are outlined. An extension of this formalism to particle-number-nonconserving operators, needed in applications involving the $m\mathrm{p}\ensuremath{-}n\mathrm{h}$ propagators with $m\ensuremath{\ne}n$, mmm, n=0n=0n=0, 1, and 2, is discussed.
Abstract The representation of an N‐electron Schrödinger Hamiltonian on an orthonormal, spin‐free (freon) orbital product space is exactly modeled by a second‐degree polynomial in the infinitesimal generators of the unitary group. Symmetry adaptation of the orbital product space with respect to the symmetric group yields Gel'fand states which provide base for irreducible represntations of the unitary group. These exist in closed form as, in consequence, does the representation of the model Hamiltonian in this same basis. We retain as physical only those states characterized by tableaux with no more than two columns for which the spin labelling is one‐half the difference in the lengths of the two columns. The unitary group formulation is equivalent to standard, number‐conserving, second‐quantized, many‐body theory.
Abstract This paper lays the algebraic foundation of a unitary group approach to many‐body theory. We define a set of second quantized spin‐shift operators which are used to construct spin‐adapted many‐electron configuration functions. We investigate the particle‐hole transformation, normal ordering, and contraction of spin shifts. This gives an orbital Wick's theorem reflecting the permutational structure of the states spanning the irreducible representations of the spatial unitary group. We study the spinless Hamiltonian, and the relationship between spin shifts and unitary group generators.
Abstract This paper shows that the spin‐shift formalism developed in B. T. Pickup and A. Mukhopadhyay [Int. J. Quantum Chem. 26, 101 (1984)] supports a one‐component diagrammatics which has a systematics akin to that in the spin‐orbital many‐body theory. The diagrams are neither Goldstone nor Yutsis type, and characterize the chain U(2R) ⊃ U(R)⊗SU(2) on which the spin‐shift formalism is based. Accordingly, while the lines in such diagrams are labeled by the orbital indices, the diagram structure adequately reflects the irreducible representation of the group U(R). In this sense the paper presents a unitary group approach to the natural generalization of the usual many‐body theory for the spin‐adapted cases. A set of very simple rules is derived; their similarity with the corresponding rules in the ordinary many‐body theory and practical utility are discussed in connection with (a) matrix elements over many‐electron spin states and (b) closed‐ and open‐shell many‐body perturbation theory. A possibility of integral‐driven many‐body perturbation theory for open‐shells is indicated. Connections of this formalism with others are also discussed.
A fully spin-adapted approach to many-electron density matrices is developed in the context of the unitary group approach to many-electron systems. An explicit expression for the single-electron spin-density operator, as a polynomial of degree two in the orbital U(n) generators, is derived for the case of spin-independent systems. Extensions to spin-dependent systems are also considered, leading to the appearance of total-spin transition densities, whose general properties are investigated. A corresponding formalism for the two-electron density matrix, which is capable of further generalization, is also developed. The results of this paper, together with recent developments on the matrix elements of the U(2n) generators in the electronic Gel’fand basis, afford a versatile method for the direct calculation of one- and two-body density matrices in the unitary group approach framework.