Approximating Densities of States with Gaps

TL;DR

This paper introduces a generalized boundary condition for approximating densities of states with multiple bands and gaps, improving accuracy over previous methods and applicable to complex atomic chains.

Contribution

It extends a quadratic boundary condition to an energy-independent form for multi-band densities with gaps, enabling better approximations in complex systems.

Findings

The new boundary condition performs favorably compared to maximum entropy methods.

It accurately reproduces densities with multiple bands separated by gaps.

Generalized hypergeometric functions satisfy the boundary condition exactly.

Abstract

Reconstructing a density of states or similar distribution from moments or continued fractions is an important problem in calculating the electronic and vibrational structure of defective or non-crystalline solids. For single bands a quadratic boundary condition introduced previously [Phys. Rev. B 74, 205121 (2006)] produces results which compare favorably with maximum entropy and even give analytic continuations of Green functions to the unphysical sheet. In this paper, the previous boundary condition is generalized to an energy-independent condition for densities with multiple bands separated by gaps. As an example it is applied to a chain of atoms with s, p, and d bands of different widths with different gaps between them. The results are compared with maximum entropy for different levels of approximation. Generalized hypergeometric functions associated with multiple bands satisfy the new boundary condition exactly.