Anisotropy in Two-Center Exchange Interactions

Abstract

The exchange interaction between electrons on two different centers is developed in a bipolar series of angular momentum operators. In contrast to the isotropic Hamiltonian commonly used to represent this interaction, we find that all forms of anisotropic terms are needed, antisymmetric as well as symmetric ones. The degrees of the anisotropy assume all values commensurate with the angular momenta of the electrons. Beside the Dzyaloshinsky-Moriya term ${\mathrm{S}}_1$×${\mathrm{S}}_2$, two-center exchange interactions contain antisymmetric anisotropies of higher degrees. For ions with strong orbital contributions to their magnetic moments, anisotropic exchange does not enter merely as a perturbation, and we find that the magnitudes of the symmetric and antisymmetric anisotropies are as large as the isotropic part of the exchange interaction. Explicit expressions containing radial integrals are derived for the coefficients representing the anisotropy in the exchange interaction. These anisotropy coefficients are related to the more conventional exchange constants; we also find the number of independent parameters needed to describe two-center exchange interactions for various situations. The coefficients of anisotropy representing the exchange interaction between ions with $N$ equivalent electrons are related to the coefficients for the interaction between ions with one electron. Although anisotropic superexchange is not considered in detail, both the form of the Hamiltonian representing this interaction, and also the number of independent parameters in the Hamiltonian, immediately follow from the Hamiltonian for the two-center exchange interaction.