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Molecular Physics

Variational localized-site cluster expansions

Daniel Klein

Published Mar 1, 1976• 11 citations
Publisher

Abstract

The simplest localized-site cluster expansion for a system of doublet sites described within the context of a valence bond or Heisenberg model is considered. Exact matrix element formulae are obtained for a variety of systems, including some with fused rings. Numerical results for the linear chain case suggest that this simple ansatz improves over the common resonance theory presumption of Kekulé structures only.

DOI
10.1080/00268977600100621

Related papers

Variational localized-site cluster expansions

Daniel Klein, M. A. García‐Bach

The simplest variational localized-site cluster expansions for Hubbard and Pariser-Parr-Pople (PPP) models are considered and explicit matrix element formulae are developed. Numerical calculations for infinite chains and finite cycles are reported. It is found that even the simplest ansatz yields results in the atomic limit which are better than other conventional techniques. Near the band limit its accuracy decreases, as expected, but the next most simple ansatz does better.

Molecular Physics•Mar 1, 1976•7 citations

Variational localized-site cluster expansions. II. Trees and near trees

Douglas J. Klein, M. A. García‐Bach

Cluster expanded wavefunctions involving localized-site states and only pair excitations between nearest neighbor sites are considered. In the case that the bonds connecting these nearest neighbor sites form a tree graph (without rings), exact matrix element formulas are obtained in terms of the inverse of a modified type of topological matrix associated with this tree graph. The treatment of some near-tree cases, without any touching rings, is also discussed.

The Journal of Chemical Physics•Jun 15, 1976•8 citations

Variational localized-site cluster expansions. X. Dimerization in linear Heisenberg chains

Daniel Klein, M. A. García‐Bach

Ground-state instability to bond alternation in long linear chains is considered from the point of view of valence-bond (VB) theory. This instability is viewed as the consequence of a long-range order (LRO) which is expected if the ground state is reasonably described in terms of the Kekul\'e states (with nearest-neighbor singlet pairing). It is argued that the bond alternation and associated LRO predicted by this simple, VB picture is retained for certain linear Heisenberg models; many-body VB calculations on spin $s=\frac{1}{2}$ and $s=1$ chains are carried out in a test of this argument.

Physical review. B, Condensed matter•Jan 15, 1979•61 citations

Variational localized-site cluster expansions. IX. Many-body valence-bond theory

Douglas J. Klein

A cluster-expansion approach to constructing valence-bond wave functions is described, such that calculations even on many-body systems are possible. In addition, the valence-bond idea of electron pairing is extended to site pairing, or site coupling, even though the sites may involve many electrons and be of a higher spin multiplicity.

Physical review. B, Condensed matter•Jan 15, 1979•17 citations

Variational localized-site cluster expansions. XI. Coherent state derivation of effective heisenberg hamiltonians

Douglas J. Klein, W. A. Seitz, M. A. García‐Bach +2 more

A variational derivation of Heisenberg or valence-bond models from a Hubbard (or Pariser-Parr-Pople) Hamiltonian is described. Calculations within a low-order approximation are carried out for linear-chain and graphite lattices. The results seem to be quite accurate with little computational effort when compared with other approaches.

International Journal of Quantum Chemistry•Jul 9, 2009•1 citation

Variational localized‐site cluster expansions. VI. General theory revisited

Douglas J. Klein

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.

International Journal of Quantum Chemistry•Aug 1, 1977•11 citations
The simplest localized-site cluster expansion for a system of doublet sites described within the context of a valence bond or Heisenberg model is considered. Exact matrix element formulae are obtained for a variety of systems, including some with fused rings. Numerical results for the linear chain case suggest that this simple ansatz improves over the common resonance theory presumption of Kekulé structures only.
Ground-state instability to bond alternation in long linear chains is considered from the point of view of valence-bond (VB) theory. This instability is viewed as the consequence of a long-range order (LRO) which is expected if the ground state is reasonably described in terms of the Kekul\'e states (with nearest-neighbor singlet pairing). It is argued that the bond alternation and associated LRO predicted by this simple, VB picture is retained for certain linear Heisenberg models; many-body VB calculations on spin s=12s=\frac{1}{2}s=21​ and s=1s=1s=1 chains are carried out in a test of this argument.
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.