Molecular Algebraic Geometry: Electronic Structure of H3+ as Algebraic Variety

Abstract

In this article, we demonstrate the restricted Hartree-Fock electronic structure computation of the molecule H3+ 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 Groebner basis and primary ideal decomposition. This treatment enables us to describe the electronic structures as algebraic varieties in terms of polynomials.

Feasibility of first principles molecular dynamics in fault-tolerant quantum computer by quantum phase estimation

“…This sort of advanced theory is equally represented by polynomials, and our algorithm has the potential to treat it. In [3], we discussed the computation of virtual states that would be located above the ground state molecular orbitals. Those virtual orbitals could be encoded in quantum states lying on the qubits.…”

“…The shape of the molecule is an equilateral triangle with edge length R. Note that this molecule has attracted the interests of researchers in various fields since this molecule is the simplest triatomic molecule and abundant in the universe; this molecule is regarded as a benchmark problem of quantum chemistry [14,15,16,17,18,19,20,21,22,23,24]. The electronic structure of this molecule as an algebraic variety is analyzed by the authors of the present works [3].…”

“…The authors of the present article have developed a method of quantum simulation of materials, wherein numeric, symbolic, and quantum algorithms are employed [1,2,3,4]. This approach uses the following steps.…”