Maximum Ionization in Restricted and Unrestricted Hartree-Fock Theory

Hazel Cox,Ville J J Syrjanen,Michael Melgaard

doi:10.3390/atoms9010013

Abstract

In this paper, we investigate the maximum number of electrons that can be bound to a system of nuclei modelled by Hartree-Fock theory. We consider both the Restricted and Unrestricted Hartree-Fock models. We are taking a non-existence approach (necessary but not sufficient), in other words we are finding an upper bound on the maximum number of electrons. In giving a detailed account of the proof of Lieb’s bound [Theorem 1, Phys. Rev. A 29 (1984), 3018] for the Hartree-Fock models we establish several new auxiliary results, furthermore we propose a condition that, if satisfied, will give an improved upper bound on the maximum number of electrons within the Restricted Hartree-Fock model. For two-electron atoms we show that the latter condition holds.

Highlights

Quantum chemistry, the application of quantum mechanics to chemical systems, is used to understand stability, reactivity, and chemical/physical processes at the molecular level by solving the Schrödinger equation (SE)
In Hartree-Fock theory, the N-electron wave function from the Schrödinger equation is approximated by an anti-symmetrized product of N single-electron wave functions and the two-electron repulsion operator in the Hamiltonian is replaced by an effective one-electron operator which takes a mean field approach to electron-electron repulsion
As applications of the auxiliary results, we provide the proof of Theorem 1 for these Hartree-Fock methods in Sections 6–8, firstly for the atomic Restricted Hartree-Fock (RHF) case, for the molecular RHF case, and for the molecular unrestricted Hartree-Fock (UHF) case

Summary

Introduction

The application of quantum mechanics to chemical systems, is used to understand stability, reactivity, and chemical/physical processes at the molecular level by solving the Schrödinger equation (SE). The Kohn-Sham equations use the expression for the HF kinetic energy to obtain the exact KE of the non-interacting reference system with the same density as the real interacting system They were able to show that despite working with an orbital model and a single determinant wavefunction for a model system of non-interacting electrons they were still able to incorporate electron correlation. Numerical calculations and mathematical proofs agree that the many-body, non-relativistic, time-independent Schrödinger equation for the hydride ion supports a single bound state. No such result has been reported for the Hartree-Fock hydride ion. The mathematical proof of the Lieb criterion is extended to explicitly consider product spin orbitals with real basis functions within the restricted and unrestricted HF formalisms to mirror the details used in conventional HF numerical calculations. The proof of Theorem 2 is given in Section 9 and the proofs of the auxiliary results are provided in the Appendix A

Results

Auxiliary Results

Restricted Hartree-Fock and Unrestricted Hartree-Fock

Proof of Theorem 1 for the Molecular RHF Model

Proof of Theorem 1 for the UHF Model

Molecular UHF Model