Linear aspects of the KKR formalism

TL;DR

This paper introduces a one-dimensional KKR method emphasizing its linear features to improve numerical algorithms for energy band calculations, with potential applications to complex materials.

Contribution

The paper develops a nearly linear energy-dependent secular matrix and a one-dimensional analog of the Lloyd formula, enhancing the KKR formalism's computational efficiency.

Findings

Eigenvalue functions exhibit quasi-linear behavior after normalization and integration.

The approach simplifies the KKR equations, making them more suitable for complex materials.

The method can be extended to higher dimensions for advanced band structure calculations.

Abstract

We present one-dimensional KKR method with the aim to elucidate its linear features, particularly important in optimizing the numerical algorithms in energy bands computations. The conventional KKR equations based on the multiple scattering theory as well as novel forms of the secular matrix with nearly linear energy dependency of the eigenvalues are presented. The quasi-linear behaviour of these eigenvalue functions appears after (i) re-normalizing the wave functions in such a way that 'irregular' solutions vanish on the boundary of the 'muffin-tin' segments and (ii) integrating the full Green function over the whole Wigner-Seitz cell. In addition, using the aforementioned approach we derive one-dimensional analog of the generalized Lloyd formula. The novel KKR approach illustrated in one-dimension can be almost directly applied to the higher dimensional cases. This should open prospects for the accurate KKR band structure computations of very complex materials.