Preprint Article Version 1 This version is not peer-reviewed

Computational Algebraic Geometry and Quantum Mechanics: An Initiative toward Post-Contemporary Quantum Chemistry

Version 1 : Received: 4 July 2019 / Approved: 8 July 2019 / Online: 8 July 2019 (03:29:04 CEST)

How to cite: Kikuchi, I.; Kikuchi, A. Computational Algebraic Geometry and Quantum Mechanics: An Initiative toward Post-Contemporary Quantum Chemistry. Preprints 2019, 2019070095 (doi: 10.20944/preprints201907.0095.v1). Kikuchi, I.; Kikuchi, A. Computational Algebraic Geometry and Quantum Mechanics: An Initiative toward Post-Contemporary Quantum Chemistry. Preprints 2019, 2019070095 (doi: 10.20944/preprints201907.0095.v1).

Abstract

A new framework in quantum chemistry has been proposed recently (``An approach to first principles electronic structure calculation by symbolic-numeric computation'' by A. Kikuchi). It is based on the modern technique of computational algebraic geometry, viz. the symbolic computation of polynomial systems. Although this framework belongs to molecular orbital theory, it fully adopts the symbolic method. The analytic integrals in the secular equations are approximated by the polynomials. The indeterminate variables of polynomials represent the wave-functions and other parameters for the optimization, such as atomic positions and contraction coefficients of atomic orbitals. Then the symbolic computation digests and decomposes the polynomials into a tame form of the set of equations, to which numerical computations are easily applied. The key technique is Gr\"obner basis theory, by which one can investigate the electronic structure by unraveling the entangled relations of the involved variables. In this article, at first, we demonstrate the featured result of this new theory. Next, we expound the mathematical basics concerning computational algebraic geometry, which are necessitated in our study. We will see how highly abstract ideas of polynomial algebra would be applied to the solution of the definite problems in quantum mechanics. We solve simple problems in ``quantum chemistry in algebraic variety'' by means of algebraic approach. Finally, we review several topics related to polynomial computation, whereby we shall have an outlook for the future direction of the research.

Subject Areas

quantum mechanics; algebraic geometry; commutative algebra; Gr\"onber basis; primary ideal decomposition, eigenvalue problem in quantum mechanics; molecular orbital theory; quantum chemistry; first principles electronic structure calculation; symbolic computation; symbolic-numeric solving; Hartree-Fock theory; molecular integral; Taylor series; polynomial approximation; algebraic molecular orbital equation

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