Use of Non-orthogonal Wave Functions in the Treatment of Solids, with Applications to Ferromagnetism

It is proved by a rearrangement of L\"owdin's solution to the many-body problem that a vector model type energy expression is valid for a solid provided (in the simplest case of one electron per atom) the number of nearest neighbors times the overlap integral between them is small compared with unity.The approximation is also carried to include third order permutation terms, and the spin coupling via closed shells these "triple exchange" terms can produce is discussed.General expressions are given for the exchange integrals to be used in a vector model solution. These under the proper conditions reduce to the conventional integrals except for a slight modification, which helps to make the integrals negative.