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Physical Review A

Particle-hole formulation of the unitary group approach to the many-electron correlation problem. II. Matrix element evaluation

M. J. Boyle, Josef Paldus

Published Dec 1, 1980• 49 citations
Publisher

Abstract

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 $m\ensuremath{\ne}n$, $m$, $n=0$, 1, and 2, is discussed.

DOI
10.1103/physreva.22.2316

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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 U(n\ensuremath′+n\ensuremath′\ensuremath′)\ensuremath⊃U(n\ensuremath′)\ensuremath⨂U(n\ensuremath′\ensuremath′)\mathrm{U}({n}^{\ensuremath{'}}+{n}^{\ensuremath{'}\ensuremath{'}})\ensuremath{\supset}\mathrm{U}({n}^{\ensuremath{'}})\ensuremath{\bigotimes}\mathrm{U}({n}^{\ensuremath{'}\ensuremath{'}})U(n\ensuremath′+n\ensuremath′\ensuremath′)\ensuremath⊃U(n\ensuremath′)\ensuremath⨂U(n\ensuremath′\ensuremath′), where n\ensuremath′{n}^{\ensuremath{'}}n\ensuremath′ and n\ensuremath′\ensuremath′{n}^{\ensuremath{'}\ensuremath{'}}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 mp\ensuremath−nhm\mathrm{p}\ensuremath{-}n\mathrm{h}mp\ensuremath−nh propagators with m\ensuremath≠nm\ensuremath{\ne}nm\ensuremath=n, mmm, n=0n=0n=0, 1, and 2, is discussed.
A hole-particle version of the unitary group approach to the many-electron correlation problem is given. To achieve a unique state labeling, the subgroup chain of Flores and Moshinsky for the nuclear many-body problem is modified, and the rules for the construction of relevant irreducible representations and of the orthonormal spinadapted states are given. The structure of these hole-particle formalism spin-adapted bases is illustrated on several examples for all the three possible types of the hole-particle defect pertaining to the excitation, electron attachment, or detachment processes relative to the reference state used. The tensor character of the particle formalism electron-number-preserving generators is pointed out, and their basic classification in the hole-particle formalism is given.
A detailed exposition of explicit formulas used in the evaluation of raising-lowering forms of two-body-operator matrix elements is presented. The methods are based on the use of the unitary group distinct row table and graphical representation of the many-particle basis. All matrix elements are expressible in terms of scalar and simple matrix factors. In order to facilitate the derivations of simple, computationally efficient forms for the matrix factors, we utilize a recently developed calculus based on elementary graphs. The methods are applicable to systems of particles involving spins greater than (1/2 and reduce to previously known results for the case of spin (1/2.
A detailed exposition of explicit formulas used in the evaluation of raising-raising forms of two-body-operator matrix elements is presented. The methods are based on the use of the unitary-group distinct-row tabular-graphical representation of the many-particle basis. All matrix elements are expressible in the form of scalar and 2\ifmmode\times\else\texttimes\fi{}2 matrix factors. In order to facilitate the derivations of simple, computationally efficient forms for the matrix factors we develop a calculus based on elementary graphs. The methods are applicable to systems of particles involving spins greater than (1/2 and reduce to previously known results for the case of spin (1/2.
A new spin-dependent unitary group approach to the many-electron correlation problem is investigated. It is demonstrated that the matrix elements of the U(2n) generators, in the U(n)×U(2) adapted electronic Gel’fand basis, are determined by the matrix elements of a single U(n) adjoint tensor operator, herein denoted by Δij(1≤i, j≤n), where Δ is a polynomial of degree two in the U(n) matrix E=[Eij]. The method is then applied, in the second paper of the series, to derive a simple segment level formula for the matrix elements of all U(2n) generators. The advantages of this new procedure for computer implementation are discussed and the central role played by the matrix Δ for the determination of molecular spin densities is highlighted.
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.