A density matrix approach to the convergence of the self-consistent field iteration

TL;DR

This paper analyzes the local convergence of the self-consistent field iteration using a density matrix approach, deriving bounds and conditions that relate eigenvalue gaps to convergence behavior.

Contribution

It introduces a density matrix-based fixed-point analysis for SCF convergence, providing new bounds and insights linked to eigenvalue gaps.

Findings

Derived upper bounds for convergence based on spectral radius

Established conditions for local convergence of SCF

Compared bounds with actual convergence behavior in numerical examples

Abstract

In this paper, we present a local convergence analysis of the self-consistent field (SCF) iteration using the density matrix as the state of a fixed-point iteration. Sufficient and almost necessary conditions for local convergence are formulated in terms of the spectral radius of the Jacobian of a fixed-point map. The relationship between convergence and certain properties of the problem is explored by deriving upper bounds expressed in terms of higher gaps. This gives more information regarding how the gaps between eigenvalues of the problem affect the convergence, and hence these bounds are more insightful on the convergence behaviour than standard convergence results. We also provide a detailed analysis to describe the difference between the bounds and the exact convergence factor for an illustrative example. Finally we present numerical examples and compare the exact value of the convergence factor with the observed behaviour of SCF, along with our new bounds and the characterization using the higher gaps. We provide heuristic convergence factor estimates in situations where the bounds fail to well capture the convergence.