On the interpolation of eigenvalues and a resultant integration scheme

The interpolation of one-electron energy band eigenvalues by way of an interpolating function expanded as a linear sum of star functions with expansion coefficients determined via a spline-like variational scheme is described. We discuss the practical aspects of such a scheme (first, set up by Shankland) and establish that it is a viable one; we then observe that it can serve as the basis of a very useful integration scheme. The viability is established through a discussion of the choice of the interpolating function and the associated problems of point set selection and truncation of the expansion. The proposed integration scheme arises from the crucial observations that the interpolation formalism can be described in terms of a transfer matrix which operates on the data to yield the expansion coefficients; this matrix depends on the choice of the variational, the point set, and the truncation but not the data. This allows one to produce a very powerful integration scheme which is applicable to a wide class of problems. copyright Academic Press, Inc