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

Area of Science:

• Computational Chemistry
• Quantum Chemistry
• Theoretical Chemistry

Background:

• Restricted Hartree-Fock (RHF) theory is a cornerstone of electronic structure calculations.
• Understanding the behavior of solutions, especially complex ones, is crucial for accurate modeling.
• Holomorphic solutions offer a unique perspective on the electronic structure of molecules.

Purpose of the Study:

• To rigorously determine the existence and number of holomorphic restricted Hartree-Fock (h-RHF) solutions for two-electron systems.
• To investigate the behavior of these solutions under changes in molecular geometry and atomic charges.
• To analyze the nature of complex solutions and their relationship to molecular symmetry and system isomorphisms.

Main Methods:

• Application of algebraic geometry to count the exact number of h-RHF solutions.
• Detailed analysis of h-RHF states for specific molecular systems (HZ, HHeH2+, HHeH, ethene) using STO-3G basis set.
• Utilizing catastrophe theory to describe coalescence points and NOCI (Non-orthogonal Configuration Interaction) for ethene's π electrons.

Main Results:

• The exact number of h-RHF solutions for n basis functions is identified as 1/2(3^n - 1), confirming existence for all geometries.
• Holomorphic solutions are conserved across variations in geometry and atomic charges, with complex solutions appearing at coalescence points.
• Isomorphism between two-electron and two-electron hole systems is demonstrated, and a singlet-triplet state crossing is observed in ethene.

Conclusions:

• Holomorphic restricted Hartree-Fock solutions are fundamental and exist for all molecular configurations in two-electron systems.
• Complex solutions and their behavior at coalescence points provide insights into molecular symmetry and electronic structure.
• The study establishes a framework for understanding complex solutions and system equivalences in quantum chemistry.