Convergence of SCF sequences for the Hartree-Fock equation

TL;DR

This paper proves the convergence of the self-consistent field (SCF) method sequences for solving the Hartree-Fock equation, providing theoretical validation for its use in quantum chemistry.

Contribution

It establishes convergence conditions for SCF sequences and demonstrates their limit corresponds to a solution of the Hartree-Fock equation, enhancing the theoretical foundation of the method.

Findings

SCF sequences converge after unitary transformations.

A sufficient condition for the limit to solve Hartree-Fock is provided.

Convergence of density operators is proven.

Abstract

The Hartree-Fock equation is a fundamental equation in many-electron problems. It is of practical importance in quantum chemistry to find solutions to the Hartree-Fock equation. The self-consistent field (SCF) method is a standard numerical calculation method to solve the Hartree-Fock equation. In this paper we prove that the sequence of the functions obtained in the SCF procedure is composed of a sequence of pairs of functions that converges after multiplication by appropriate unitary matrices, which strongly ensures the validity of the SCF method. A sufficient condition for the limit to be a solution to the Hartree-Fock equation after multiplication by a unitary matrix is given, and the convergence of the corresponding density operators is also proved. The method is based mainly on the proof of approach of the sequence to a critical set of a functional, compactness of the critical set, and the proof of the \L ojasiewicz inequality for another functional near critical points.