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International Journal of Quantum Chemistry

Variational localized‐site cluster expansions. VIII. Projection of spin symmetries

Daniel Klein, M. A. García‐Bach

Published Aug 1, 1977• 3 citations
Publisher

Abstract

Abstract We investigate the total spin structure of an approximate localized‐site wave function for a collection of paramagnetic sites interacting so as to favor a singlet ground state. As the number of sites becomes infinite we obtain the distribution of weights of the different symmetry components of the localized site wave function; further, although only a very small fraction of such nonsymmetric wave functions is actually singlet, we find that it generally yields the same bulk property expectation values as its singlet‐projected component.

DOI
10.1002/qua.560120206

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Abstract We investigate the total spin structure of an approximate localized‐site wave function for a collection of paramagnetic sites interacting so as to favor a singlet ground state. As the number of sites becomes infinite we obtain the distribution of weights of the different symmetry components of the localized site wave function; further, although only a very small fraction of such nonsymmetric wave functions is actually singlet, we find that it generally yields the same bulk property expectation values as its singlet‐projected component.
Abstract A general and systematic localized‐site cluster expansion is described. It provides simple physically appealing and computationally feasible wave function ansätze, as well as, via extension of the cluster expansion, a unique expression for general, possibly exact, wave functions. We show that a variety of matrix element evaluation techniques are available and that they are sometimes exact though relatively simple. These techniques often proceed in terms of computational time proportional to the number of different types of sites, rather than their total number. Thus it seems we have a new and viable many‐body theory, which is expected to be most accurate and most readily implementable when a physical picture with slightly perturbed localized sites is relevant.
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Recent work on approximating ground states of Heisenberg spin clusters by projected Hartree–Fock theory (PHF) is extended to a cluster-based ansatz (cPHF). Whereas PHF variationally optimizes a site–spin product state for the restoration of spin- and point-group symmetry, cPHF groups sites into discrete clusters and uses a cluster-product state as the broken-symmetry reference. Intracluster correlation is thus already included at the mean-field level, and intercluster correlation is introduced through symmetry projection. Variants of cPHF differing in the broken and restored symmetries are evaluated for ground states and singlet-triplet gaps of antiferromagnetic spin rings for various cluster sizes, where cPHF in general affords a significant improvement over ordinary PHF, although the division into clusters lowers the cyclical symmetry. In contrast, certain two- or three-dimensional spin arrangements permit cluster groupings compatible with the full spatial symmetry. We accordingly demonstrate that cPHF yields approximate ground states with correct spin- and point-group quantum numbers for honeycomb lattice fragments and symmetric polyhedra.
What is commonly called the Gutzwiller wave function $\ensuremath{\psi}$, an approximate ground state of the single-band Hubbard Hamiltonian, is considered here for NNN-site NNN-electron rings, when the number DDD of doubly occupied sites is small (N=6,10,14,18N=6, 10, 14, 18N=6,10,14,18). For $D\ensuremath{\rightarrow}0$ spin correlations in $\ensuremath{\psi}$ surprisingly close to the exact values at zero bandwidth are found (i.e., those of the antiferromagnetic Heisenberg-model ground state). But the energy is grossly in error. A simple modification of $\ensuremath{\psi}$ reproduces the exact energy with remarkable accuracy.
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