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The Journal of Chemical Physics

Alternancy symmetry: A unified viewpoint

Jaroslav Koutecký, Josef Paldus, Jiřı́ Čı́žek

Published Aug 15, 1985• 50 citations

Abstract

A general formulation of the alternancy symmetry adaptation for the semiempirical Pariser–Parr–Pople (PPP) type Hamiltonians is presented at both the spin-orbital and spin-adapted many-electron levels. The derivation of the general form of the alternancy symmetry conjugation operators is based solely on the tight-binding approximation for the short range one-particle part of the Hamiltonian considered. It starts by a simple formulation of the desired invariance properties of the PPP type Hamiltonian. Using algebraic properties of the unitary group generators and of their particle number nonconserving extensions, it leads to a completely explicit and general form for the alternancy symmetry conjugation operators. In this way the prior descriptions, which become special cases of this general formulation, are interrelated and unified. The spin and quasispin character of certain components of these operators are also pointed out and explicitly derived. The spin-adapted version is based on the unitary group formulation of the valence bond-type approach. A completely general formulation is given which applies to many-electron states of an arbitrary multiplicity of neutral systems, either of the normal kind with an identical number of starred and nonstarred sites, or of the radicaloid character with different numbers of starred and nonstarred sites. An explicit form of the dependence of the relative phase factor of the alternancy symmetry conjugation operator on the total spin, total electron number and on the number of starred and nonstarred sites is also given. General rules for the construction of spin and alternancy symmetry-adapted states are illustrated on a few simple examples. Finally, a brief discussion of the implications of the alternancy symmetry is presented.

DOI: 10.1063/1.449359

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A general formulation of the alternancy symmetry adaptation for the semiempirical Pariser–Parr–Pople (PPP) type Hamiltonians is presented at both the spin-orbital and spin-adapted many-electron levels. The derivation of the general form of the alternancy symmetry conjugation operators is based solely on the tight-binding approximation for the short range one-particle part of the Hamiltonian considered. It starts by a simple formulation of the desired invariance properties of the PPP type Hamiltonian. Using algebraic properties of the unitary group generators and of their particle number nonconserving extensions, it leads to a completely explicit and general form for the alternancy symmetry conjugation operators. In this way the prior descriptions, which become special cases of this general formulation, are interrelated and unified. The spin and quasispin character of certain components of these operators are also pointed out and explicitly derived. The spin-adapted version is based on the unitary group formulation of the valence bond-type approach. A completely general formulation is given which applies to many-electron states of an arbitrary multiplicity of neutral systems, either of the normal kind with an identical number of starred and nonstarred sites, or of the radicaloid character with different numbers of starred and nonstarred sites. An explicit form of the dependence of the relative phase factor of the alternancy symmetry conjugation operator on the total spin, total electron number and on the number of starred and nonstarred sites is also given. General rules for the construction of spin and alternancy symmetry-adapted states are illustrated on a few simple examples. Finally, a brief discussion of the implications of the alternancy symmetry is presented.

Static-spin-correlation functions 4〈

{s}_{\mathrm{pz}}

{s}_{\mathrm{qz}}

〉 are evaluated via diagrammatic valence-bond theory for exact states of finite Hubbard, Pariser-Parr-Pople (PPP), and related quantum cell models. Several properties of 〈

{s}_{\mathrm{pz}}

{s}_{\mathrm{qz}}

〉 that are independent of intersite potentials

{V}_{\mathrm{pp}\mathcal{'}}

in the Hartree-Fock (HF) approximation for half-filled alternant systems are shown to be sensitive to correlations in the ground state of N\ensuremath{\le}12 Hubbard and PPP chains. Accurate N\ensuremath{\rightarrow}\ensuremath{\infty} extrapolations are found for near neighbors. The correct ground-state charge density 〈

{n}_{p}

〉=1 is a property of half-filled alternant systems and thus not a test of HF or other one-electron approximations. Spin correlations in PPP models with standard molecular parameters and long-range Coulomb interactions are approximately given by on-site interactions of U\ensuremath{\simeq}2\ensuremath{\Vert}t\ensuremath{\Vert} in Hubbard models, although the general semiempirical PPP approach to conjugated molecules is preferable for wide-band systems.

Symmetry-adapted linear combinations of valence-bond (VB) diagrams are constructed for arbitrary point groups and total spin S using diagrammatic VB methods. VB diagrams are related uniquely to invariant subspaces whose size reflects the number of group elements; their nonorthogonality leads to sparser matrices and is fully incorporated into a binary integer representation. Symmetry-adapated linear combinations of VB diagrams are constructed for the 1764 singlets of a half-filled cube of eight sites, the 2.8 million π-electron singlets of anthracene, and for illustrative S≥0 systems.

A general formulation of multiconfigurational self-consistent-field theory is given. The fundamental condition to be satisfied by the

M

occupied orbitals ${\ensuremath{\psi}}_{\ensuremath{\mu}}$ of an $N(\ensuremath{\le}M)$-body system, in order that the total energy of one state be extremalized, is expressed in terms of the fundamental invariant $\mathbf{\ensuremath{\varrho}}=\ensuremath{\Sigma}{\ensuremath{\mu}=1}^{M}|{\ensuremath{\psi}}_{\ensuremath{\mu}}〉〈{\ensuremath{\psi}}_{\ensuremath{\mu}}|$. The condition placed on $\mathbf{\ensuremath{\varrho}}$ is of the same form as the condition on the fundamental invariant of the Hartree-Fock theory. Setting

M=N

, we derive the unrestricted Hartree-Fock equations from the condition on $\mathbf{\ensuremath{\varrho}}$. For

M>N

there are three freedoms one may take with the fundamental condition on $\mathbf{\ensuremath{\varrho}}$. Exploiting any or all of these freedoms yields alternative forms of the fundamental condition. This enables us to derive an effective one-body Hamiltonian which is the sum of a Hartree-type Hamiltonian and a correlation and exchange operator. For finite

M>N

, the one-body Hamiltonian contains a nonlocal exchange and correlation operator. This operator is defined in terms of the one- and two-body density matrices. The connection between the orbitals of this theory and those of the Hartree-Fock theory is explored. The theory as outlined here is applicable to any system of

N

identical particles, but our discussion is oriented towards electronic systems. The theory contains most self-consistent-field theories as special cases, and gives a basis for the self-consistent-field formulation of others.

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.