Sandor Kristyan
Research Centre for Natural Sciences (HUN-REN RCNS),
H-1117 Budapest, Magyar Tudósok körútja 2, Hungary
Publication Date: 20 December 2025
Corresponding author: kristyan.sandor@ttk.mta.hu, kristyan.sandor@ttk.hu
Abstract: The self-consistent field (SCF) part of the Hartree–Fock self-consistent field (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 fourth-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 usto find not only the
ground but also the lowest-lying excited states. Calculations on some molecular systems with different multiplicities are provided for demonstration. The relation of our algorithm to the Kohn-Sham density functional theory (KS-DFT) algorithm is also discussed 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.
© European Society of Computational Methods in Sciences and Engineering
Keywords: Hartree–Fock method, self-consistent field optimization, algebraic geometry, Buchberger’s algorithm,
multivariable Newton slope method, one-determinant approximation for ground and excited electronic states; ortho and para helium
Mathematics Subject Classification: 14-04, 14-06, 35-04, 35-06, 81Q05
PACS: 31.10.+z, 31.15.−p, 31.15.Ne,