Characteristics of Two-Electron Atoms Examined Using the Hartree-Fock Approximation Difficulties

The investigation examines the presence and properties of Holomorphic restricted Hartree–Fock (h-RHF) solutions within (2-electron) systems. Through the utilization of (algebraic geometry), the specific ways of solutions associated with n basis functions is formally determined as ½(3ⁿ-1), providing clear evidence that states are compulsory for all molecular configurations. A thorough analysis of the h-RHF states of HZ (STO-3G) subsequently highlights both the continuation of Holomorphic solutions amidst variations in geometry or atomic charges, and the emergence of complex h-RHF solutions at points of convergence. By employing "catastrophe theory", these theory converging ways are elaborated upon, certainly the effect of molecular symmetry. We have been two ways getting of result the h-RHF states of HHeH²⁺ and HHeH (STO-3G), explanation the structural similarity between systems featuring 2-electrons and 2-holes. Eventually, a realization is executed on the h-RHF states of "ethane" (STO-3G) by considering the π-electrons as a two–electron problem, and utilizing NOCI to select an intersection between the ground state energy "singlet and triplet" states at vertical structure. The utilization of the "4-h-RHF" and "4-h-UHF" levels has been employed like a fundamental for "NOCI" in order to detect a junction between "the lowest energy singlet and triplet states" occurring at a curvature "angle of 90◦". This is achieved by approaching the "π electrons" in ethylene as a 2-electron dilemma. Subsequently, this outcome is validated within the context of the basis set approximation through variance within the precise "STO-3G energies" calculated using FCIQMC, showcasing the ability of integrating "Holomorphic Hartree–Fock theory and NOCI".