Derivation of the Dirac Vector Model for a Solid

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Abstract
A cluster expansion derivation of the Dirac vector model, which has important applications in the theory of ferromagnetic and antiferromagnetic materials, is given for the case of a solid with a wave function which is a linear combination of Slater determinants, each corresponding to a different spin configuration. It is assumed that the single-particle wave functions are nonorthogonal, with one particle per lattice site. It is shown that there is no "nonorthogonality catastrophe" and that Carr's condition for the validity of the vector model applies in this case, as he has conjectured. Expressions are given for the direct energy and exchange integral.