Normalizing Cluster Wavefunctions in the Interstitial Region Within the Muffin-tin Approximation

In multiple-scattering methods, overlap integrals of cluster wavefunctions for
an interstitial region, which lies inside the Watson sphere and outside all the
enclosed ionspheres, are discussed. When the potential inside this
interstitial region is taken to be constant, and hence, obeys the Helmholtz
equation, the resulting solutions are known to involve Bessel
functions. Normalizing the cluster wavefunctions inside this intricate region
naturally leads to two-center integrals of various Bessel-type functions. In
this article, all of the numerous types of integrals that might arise are
exhaustively presented. Analytical expressions that are suitable for efficient
computation of the overlap integrals are worked out by employing
known addition theorems for the Bessel functions. These integrals require
careful attention as they otherwise lead to an artificial singularity that
may not cancel. Results for sample implementations will be discussed.