Unique Solutions to Hartree-Fock Equations for Closed Shell Atoms

TL;DR

This paper investigates the uniqueness of solutions to Hartree-Fock equations in atoms, showing that for large atomic numbers, the ground state is unique, with specific results for two-electron atoms and phase segregation phenomena.

Contribution

It establishes conditions under which the Hartree-Fock ground state is unique for closed shell atoms, especially for large atomic numbers, and identifies exceptions for smaller Z.

Findings

Hartree-Fock ground state is unique for large Z in closed shell atoms

Two-electron atoms with Z ≥ 35 have a unique ground state with specific orbital structure

Phase segregation occurs for some Z > 1, breaking uniqueness

Abstract

In this paper we study the problem of uniqueness of solutions to the Hartree and Hartree-Fock equations of atoms. We show, for example, that the Hartree-Fock ground state of a closed shell atom is unique provided the atomic number $Z$ is sufficiently large compared to the number $N$ of electrons. More specifically, a two-electron atom with atomic number $Z\geq 35$ has a unique Hartree-Fock ground state given by two orbitals with opposite spins and identical spatial wave functions. This statement is wrong for some $Z>1$, which exhibits a phase segregation.