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International Journal of Quantum Chemistry

Particle-hole gel'fand states and quasidegenerate perturbation theory

F. A. Matsen

Published Jun 19, 2009• 0 citations
Publisher

Abstract

The unitary group formulation of many-body theory is based on particle-hole Gel'fand states, states labeled only by the excitation process: h(hole orbitals)-p(particle orbitals). The particle-hole Gel'fand states are constructed from generator states, states obtained by applying weight-lowering products of generators to the highest weight state and then evaluating the matrix elements Lie algebraically. We make application to spin-projected, quasidegenerate perturbation theory.

DOI
10.1002/qua.560200817

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The unitary group formulation of many-body theory is based on particle-hole Gel'fand states, states labeled only by the excitation process: h(hole orbitals)-p(particle orbitals). The particle-hole Gel'fand states are constructed from generator states, states obtained by applying weight-lowering products of generators to the highest weight state and then evaluating the matrix elements Lie algebraically. We make application to spin-projected, quasidegenerate perturbation theory.
The representation of an N-particle Hamiltonian on an Nth rank orbital product space is exactly modeled by a second-degree polynomial in the infinitesimal generators of the unitary group. Symmetry adaptation of the product space with respect to the symmetric group yields Gel'fand states which are bases for the irreducible representations of the unitary group. These exist in closed form, as, in consequence, does the representation of the model Hamiltonian. The exclusion principle is imposed by restricting the irreducible representations of the unitary group. The unitary group formulation is equivalent to standard many-body theory for the particle-conserving case.
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.
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.
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.
Coupled-cluster many-electron theory (CCMET) has been developed in the second-quantized formulation with and without spin projection. In this article CCMET is developed in the unitary-group formulation where the group is U(p) and p is the number of orbitals in the basis set. The zero-order ground state is the highest-weight state of an irreducibly invariant subspace of U(p) and the excitation operators are infinitesimal generators. These irreducible spaces are uniquely labeled by Young diagrams which the Pauli principle limits to no more than one column for fermion orbitals and no more than two columns for freeon (spin-free) orbitals. In the latter case the spin is one-half the difference in the lengths of the two columns. We employ both the Gel'fand and the generator bases for the irreducible invariant vector spaces. Matrix elements for the former are evaluated by techniques due to Gel'fand, Biedenharn, Louck, Paldus, and Shavitt and for the latter directly by Lie algebraic and diagrammatic techniques as a simple function of the weight components of the highest-weight state. For freeon or fermion orbitals the results are equivalent to those obtained by the second-quantized formulation with or without spin projection.
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.