On Symmetry and the Reality of Holomorphic Hartree--Fock Wavefunctions

TL;DR

This paper investigates how spatial symmetry influences the existence and reality of holomorphic Hartree--Fock solutions, addressing solution disappearance issues in HF theory through symmetry classification and analysis.

Contribution

It introduces a symmetry-based framework to analyze holomorphic HF solutions and demonstrates how symmetry constraints affect solution reality and existence.

Findings

Symmetry imposes strict requirements on the reality of solutions.

Certain holomorphic HF solutions are governed by symmetry constraints.

Implications for HF algorithms are discussed.

Abstract

The coalescence and disappearance of Hartree--Fock (HF) solutions as the molecular structure varies have been a common source of criticism for the breakdown of the HF approximation to the potential energy surfaces. However, recent developments in holomorphic HF theory show that this disappearing behavior is only a manifestation of the way conventional HF equations prevent solutions from being analytically continued, but it is unclear what factors govern the existence and the locations of these disappearances. In this work, we explore some of these factors from the perspective of spatial symmetry by introducing a classification for symmetry constraints on electronic-structure calculations. This forms a framework for us to systematically investigate several analytic holomorphic HF solutions of a model $\textrm{[H}_4\textrm{]}^{2+}$ system in STO-3G and demonstrate that, under appropriate conditions, spatial symmetry imposes strict requirements on the reality of certain solutions. The implications for self-consistent-field HF search algorithms are then discussed. Throughout this article, the term reality means the quality of a holomorphic HF solution having real molecular orbitals.