Exchange interactions in magnetic insulators

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Abstract
The problem of incorporating crystal field ideas into the description of a system with a regular lattice is considered using a recently developed method, and applied to NiO. The basic idea is to define an unperturbed hamiltonian which is symmetric with respect to interchanges of electrons, which has the lattice symmetries, and which is conceptually simple. The difference between it and the full hamiltonian can then be treated by perturbation theory. The unperturbed hamiltonian has energies and states close to those expected for the full hamiltonian, with one major difference, that no propagation properties are contained in it. These are found in the perturbation, which has matrix elements between unperturbed states in which excitations have moved. A rather surprising result is that the crystal field splittings, at least in the unperturbed hamiltonian, can be calculated by conventional methods and no account need be taken of overlap, covalency, etc. The states of the electrons must, however, be orthogonal functions.