Effective Exchange Integral

Abstract

The magnetic properties of a linear chain of monovalent atoms are investigated from the point of view of perturbation theory. The many-electron wave functions for the system are expanded as linear combinations of determinantal functions which are eigenfunctions of $S^2$ and $S_z$. These determinantal functions are constructed from orthonormal one-electron orbitals of the Wannier type so that the nearest neighbor exchange integral is positive definite and approaches zero at large lattice spacings. The secular equation is set up using the method of the Dirac vector model. By means of the Kramers perturbation technique, the interaction of ionic states with those arising from the ground configuration is represented by means of an effective Hamiltonian operator with its associated matrix. The results of this treatment are analogous to those obtained by Paul in that an analytic expression is found for an effective nearest neighbor exchange integral $J^{\prime }$. This quantity is represented as the difference between the positive definite exchange integral and additional terms from ionic states. The present treatment defines in a fairly precise manner the type configurations which contribute to this effective exchange integral and the limits for which this parameterization is valid. The results of this analysis are compared with those obtained from recent calculations on a system of six hydrogen atoms.