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

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.

Explanation of Nikiforov-Uvarov method in quantum mechanics – with a view toward algebraic geometry

“…Note that the decomposition of the ideal is the decomposition of the eigenspace; namely, the primary ideal decomposition serves as the solution of the eigenvalue problem (which quantum mechanics always requires in the Heisenberg picture). As for the details of this sort of argument, see [1].…”

“…LIB "primdec.lib"; ring r=0,(A,B,s,alpha,beta,lambda,L),lp; ideal I=A^2-alpha^2,2*A*B+2*A+beta-lambda,B^2+B-L*(L+1); I; list pr=primdecGTZ(std(I)); pr; print("Primary ideal decomposition"); for (int i=1;i<=4;i++) { print("Branch:"); ideal j=pr[i] [1]; std(j); }…”

“…In the above list, the primary ideals and the corresponding prime ideals are given respectively. We obtain four primary ideals pi (indexed by [1]... [4]) as the result of the decompo-…”

“…In [1], the authors (I. K. and A. K.) have pointed out the relation between algebraic geometry and quantum mechanics. It is summarized as follows.…”

Galois and Class Field Theory for Quantum Chemists

“…As for the mathematical concepts concerning this sort of algebraic method, see [KK19] and the references therein. In [Kik13], the secular equation, derived from the first-principles molecular orbital theory, is transformed into the set of polynomial equations, which are chosen to be the generators of an ideal.…”