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

TL;DR

This paper applies computational algebraic geometry to model the electronic structure of H₃⁺, representing it as an algebraic variety derived from polynomial equations obtained via Hartree-Fock calculations.

Contribution

It introduces a novel algebraic geometric approach to electronic structure computation, using polynomial approximations and algebraic methods like Gr"obner bases.

Findings

Electronic structures are described as algebraic varieties.

Polynomial equations derived from Hartree-Fock energy minimization.

Application of algebraic geometry techniques to quantum chemistry.

Abstract

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.