Approximations of the Green's Function in Multiple Scattering Theory for Crystalline Systems

TL;DR

The paper addresses rigorous analysis of Green's-function approximations in multiple scattering theory for disordered crystalline systems, focusing on the scattering path matrix (SPM) that encodes the Green's function in a local service representation. It develops a perturbation-based framework using a carefully chosen reference potential to obtain fast off-diagonal decay and proves exponential convergence of the diagonal SPM element with respect to the scattering region size $R$ and the scattering-path length $L$. Two practical numerical schemes are analyzed: truncation of the scattering region and an iterative (Born-series) approach with path-length truncation, both accompanied by explicit error bounds and reduced computational costs. Numerical experiments on 1D periodic, two-component alloy, and random potentials validate the theory, showing exponential convergence of local DoS and SPM quantities and demonstrating the viability of linear-scaling MST computations in crystalline disordered systems. Overall, the work provides a rigorous foundation for efficient MST computations with controlled accuracy in complex materials.

Abstract

The multiple scattering theory (MST) is a Green's function method that has been widely used in electronic structure calculations for crystalline disordered systems. The key property of the MST method is the scattering path matrix (SPM) that characterizes the Green's function within a local solution representation. This paper studies various approximations of the SPM, under the condition that an appropriate reference is used for perturbation. In particular, we justify the convergence of the SPM approximations with respect to the size of scattering region and the length of scattering path, which are the central numerical parameters to achieve a linear-scaling MST method. We present numerical experiments on several typical systems to support the theory.

Figures (12)

• Figure 2.1: Left: A triangular Bravais lattice. Middle: A space-filling decomposition with Voronoi cells. Right: A random two-component alloy configuration.
• Figure 2.2: A schematic illustration of a Cauchy contour. Left: a dumbbell-shaped contour. Right: an open contour.
• Figure 2.3: Left: An one-dimensional schematic illustration of rectangular reference potential. Right: A schematic illustration of Cauchy contour, where the black and red thick lines represent the spectrum of $\mathcal{H}$ and $\mathcal{H}^{\rm r}$, respectively.
• Figure 3.1: A schematic illustration of the scattering paths when $R=10, N_{\rm i}=5, L=5$. Path 1-3 satisfy (a)-(c), while path 4 and path 5 do not satisfy (b) and (c), respectively.
• Figure 4.1: Example 1: (a) Off-diagonal decay of reference SPM; (b) Convergence of SPM; (c) Convergence of local DoS.

Theorems & Definitions (10)

• Remark 2.1: Electron density as DoS with Fermi-Dirac function
• Remark 2.2: "Averaged" DoS for disorded systems
• Remark 2.3: Principle of choosing the reference
• Lemma 2.1
• Theorem 3.1
• proof
• Remark 3.1: Approximation of the electron density
• Remark 3.2: Born series expansion
• Theorem 3.2
• proof