Replacing the Emblematic Self-Consistent-Field Part in Hartree-Fock-SCF Method with Algebraic Geometry Optimization for One-Determinant Approximation of Ground and Excited States in Molecular Structure Calculations

The SCF part of the HF-SCF method is responsible for finding the ground state as the global minimum of the one-determinant approximation of the electronic energy, which is a 4th order multivariable polynomial of the LCAO coefficients and Lagrange multipliers. In this work we replace this SCF part with algebraic geometry along with the multivariable Newton slope method, which allows us to find not only the ground but also the lowest-lying excited states. Calculations on some molecular systems with different multiplicities are provided for demonstrations. The relation of our algorithm to the KS-DFT algorithm is also commented on along with the correlation energy, as well as the opportunity to calculate, e.g., characteristic x-ray radiations, Auger electrons, laser- and hyper-excitations. For easier reading, some details are collected in the Appendix.