Diagrammatic valence‐bond theory for finite model Hamiltonians

<jats:title>Abstract</jats:title><jats:p>Valence bond (<jats:sc>VB</jats:sc>) diagrams form a complete basis for model Hamiltonians that conserve total spin, <jats:italic>S</jats:italic>, and have one valence state, ϕ<jats:sub><jats:italic>p</jats:italic></jats:sub>, per site. Hubbard and Pariser–Parr–Pople (<jats:sc>PPP</jats:sc>) models illustrate ionic problems, with zero, one, or two electrons in each ϕ<jats:sub><jats:italic>p</jats:italic></jats:sub>, while isotropic Heisenberg models illustrate spin problems, with only purely covalent <jats:sc>VB</jats:sc> diagrams. The difficulty of nonorthogonal <jats:sc>VB</jats:sc> diagrams is by‐passed by exploiting the finite dimensionality of the complete basis and working with unsymmetric sparse matrices. We introduce efficient bit manipulations for generating, storing, and handling <jats:sc>VB</jats:sc> diagrams as integers and describe a new coordinate relaxation method for the ground and lowest excited states of unsymmetric sparse matrices. Antiferromagnetic spin‐½ Heisenberg rings and chains of <jats:italic>N</jats:italic> ⩽ 20 spins, or 2<jats:sup><jats:italic>N</jats:italic></jats:sup> spin functions, are solved in <jats:italic>C</jats:italic><jats:sub>2</jats:sub> symmetry as illustrative examples. The lowest <jats:italic>S</jats:italic> = 1 and 0 excitations are related to domain walls, or spin solitons, and studied for alternations corresponding to polyacetylene. <jats:sc>VB</jats:sc> diagrams with arbitrary <jats:italic>S</jats:italic> and nonneighbor interactions are constructed for both spin and ionic problems, thus extending diagrammatic <jats:sc>VB</jats:sc> theory to other topologies.</jats:p>