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

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.

arXiv commentThis document is the unedited Author's version of a Submitted Work that was subsequently accepted for publication in Journal of Chemical Theory and Computation, copyright © American Chemical Society after peer review. To access the final edited and published work see http://pubs.acs.org/journal/jctcce

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