Holomorphic Hartree-Fock Theory: The Nature of Two-Electron Problems

TL;DR

This paper investigates holomorphic restricted Hartree-Fock solutions for two-electron systems, establishing their existence across all geometries and analyzing their behaviour using algebraic geometry and catastrophe theory.

Contribution

It rigorously determines the number of holomorphic Hartree-Fock solutions for two-electron problems and explores their behaviour across different molecular geometries and systems.

Findings

Number of solutions is (3^n - 1)/2 for n basis functions.

Holomorphic solutions are conserved as geometry varies.

Complex solutions emerge at coalescence points.

Abstract

We explore the existence and behaviour of holomorphic restricted Hartree-Fock (h-RHF) solutions for two-electron problems. Through algebraic geometry, the exact number of solutions with $n$ basis functions is rigorously identified as $\frac{1}{2}(3^n - 1)$, proving that states must exist for all molecular geometries. A detailed study on the h-RHF states of HZ (STO-3G) then demonstrates both the conservation of holomorphic solutions as geometry or atomic charges are varied and the emergence of complex h-RHF solutions at coalescence points. Using catastrophe theory, the nature of these coalescence points is described, highlighting the influence of molecular symmetry. The h-RHF states of HHeH$^{2+}$ and HHeH (STO-3G) are then compared, illustrating the isomorphism between systems with two electrons and two electron holes. Finally, we explore the h-RHF states of ethene (STO-3G) by considering the $\pi$ electrons as a two-electron problem, and employ NOCI to identify a crossing of the lowest energy singlet and triplet states at the perpendicular geometry.