Convergence of SCF sequences for the Hartree-Fock equation

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 t...

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Bibliographic Details
Published inarXiv.org
Main Author Ashida, Sohei
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 25.01.2023
Subjects
Convergence
Critical point
Hartree approximation
Multiplication
Operators (mathematics)
Quantum chemistry
Self consistent fields
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Summary: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.
ISSN:2331-8422