Molecular Algebraic Geometry: Electronic Structure of H$_3^+$ as Algebraic Variety

In this article, we demonstrate the restricted Hartree-Fock electronic structure computation of the molecule $H_3^+$ through computational algebra. We approximate the Hartree-Fock total energy by a polynomial composed of LCAO coefficients and atomic distances so that the minimum is determined by a set of polynomial equations. We get the roots of this set of equations through the techniques of computational algebraic geometry, namely, the Gr\"obner basis and primary ideal decomposition. This treatment enables us to describe the electronic structures as algebraic varieties in terms of polynomials.