Cluster Expansion in the Heitler-London Approach to Many-Electron Problems

Abstract

The Heitler-London method based on nonorthogonal atomic orbitals is applied to arrays of an infinitely large number of atoms. The electronic energy is given by a quotient with strongly divergent numerator and denominator, which until now has defied correct computation. By noticing some resemblance of this problem to the linked-cluster expansion in many-body problems, we have now developed a new method to compute the Heitler-London energy. Here, the numerator and denominator are divided simultaneously by a common factor, which leads to a set of recurrence relations between the normalization matrices F. The matrices F are essential parts of the quotient to be calculated. When an overlap integral $〈k|h〉$ is represented by a line starting from $h$ and ending at $k$, the calculation of F using the recurrence relations is carried out systematically by drawing diagrams consisting of connected loops. Since our present aim in applying the Heitler-London method is to compute spin-wave spectra, the calculation is carried out in the complete space of spin waves and the energy expression is given by a Hermitian matrix. This introduces additional matrices A. The computation of A is also carried out by the diagram technique, since A can be expanded into an asymptotic series using F. Finally the energy matrix is written as a sum over connected diagrams, in accordance with the speculation obtained from the linked-cluster expansion. An error introduced by truncating the series of diagrams is also calculated. The present method not only ensures that the energy density in the Heitler-London method is finite, but also provides an accurate and practical way to compute the Heitler-London energy, which has never been accomplished previously. To calculate the ground-state energy the technique is simplified since the computation of A can be completely eliminated.