An approach to first principles electronic structure calculation by symbolic-numeric computation

TL;DR

This paper introduces a novel symbolic-numeric approach to first principles electronic structure calculations using polynomial approximations and algebraic methods, enabling unified solutions for physical parameters and inverse problems.

Contribution

It presents a new algebraic method based on polynomial equations and Gröbner bases for electronic structure calculations, unifying forward and inverse problem solving.

Findings

Polynomial equations derived from Hartree-Fock-Roothaan are solvable via symbolic computation.

Gröbner bases simplify polynomial systems, revealing relationships between physical parameters.

The approach enables simultaneous optimization and inverse problem solutions.

Abstract

This article is an introduction to a new approach to first principles electronic structure calculation. The starting point is the Hartree-Fock-Roothaan equation, in which molecular integrals are approximated by polynomials by way of Taylor expansion with respect to atomic coordinates and other variables. It leads to a set of polynomial equations whose solutions are eigenstate, which is designated as algebraic molecular orbital equation. Symbolic computation, especially, Gr\"obner bases theory, enables us to rewrite the polynomial equations into more trimmed and tractable forms with identical roots, from which we can unravel the relationship between physical parameters (wave function, atomic coordinates, and others) and numerically evaluate them one by one in order. Furthermore, this method is a unified way to solve the electronic structure calculation, the optimization of physical parameters, and the inverse problem as a forward problem.