Version 1
: Received: 4 July 2019 / Approved: 8 July 2019 / Online: 8 July 2019 (03:29:04 CEST)
Version 2
: Received: 25 December 2019 / Approved: 26 December 2019 / Online: 26 December 2019 (02:17:44 CET)

How to cite:
Kikuchi, I.; Kikuchi, A. Computational Algebraic Geometry and Quantum Mechanics: An Initiative toward Post-Contemporary Quantum Chemistry. Preprints2019, 2019070095
Kikuchi, I.; Kikuchi, A. Computational Algebraic Geometry and Quantum Mechanics: An Initiative toward Post-Contemporary Quantum Chemistry. Preprints 2019, 2019070095

Cite as:

Kikuchi, I.; Kikuchi, A. Computational Algebraic Geometry and Quantum Mechanics: An Initiative toward Post-Contemporary Quantum Chemistry. Preprints2019, 2019070095
Kikuchi, I.; Kikuchi, A. Computational Algebraic Geometry and Quantum Mechanics: An Initiative toward Post-Contemporary Quantum Chemistry. Preprints 2019, 2019070095

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 Grröbner 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.

Keywords

quantum mechanics; algebraic geometry; commutative algebra; Grönber 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

Subject

PHYSICAL SCIENCES, Mathematical Physics

Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received:
26 December 2019
Commenter:
Ichio Kikuchi
Commenter's Conflict of Interests:
Author

Comment:
In this new version, we have corrected the errors (in the equations, the descriptions, and the reference) which remain in the first version. Besides, we have added two new section (section 2.19 and 2.20), which explain the ideas of "Noether normalization" and "differential Galois theory", respectively. This addition causes the numbering of the sections in Chapter 2. In the first version, the last section of chapter 2 was 2.19. In the new version, this section is renumbered at 2.21.

Commenter: Ichio Kikuchi

Commenter's Conflict of Interests: Author