Computational Algebraic Geometry and Quantum Mechanics: an Initiative Toward Post Contemporary Quantum Chemistry
Total Page：16
File Type：pdf, Size：1020Kb
J Multidis Res Rev 2019 Inno Journal of Multidisciplinary Volume 1: 2 Research and Reviews Computational Algebraic Geometry and Quantum Mechanics: An Initiative toward Post Contemporary Quantum Chemistry Ichio Kikuchi International Research Center for Quantum Technology, Tokyo, Japan Akihito Kikuchi* Abstract Article Information A new framework in quantum chemistry has been proposed recently Article Type: Review Article Number: JMRR118 symbolic-numeric computation” by A. Kikuchi). It is based on the Received Date: 30 September, 2019 modern(“An approach technique to first of principlescomputational electronic algebraic structure geometry, calculation viz. theby Accepted Date: 18 October, 2019 symbolic computation of polynomial systems. Although this framework Published Date: 25 October, 2019 belongs to molecular orbital theory, it fully adopts the symbolic method. The analytic integrals in the secular equations are approximated by the *Corresponding Author: Akihito Kikuchi, Independent polynomials. The indeterminate variables of polynomials represent Researcher, Member of International Research Center for the wave-functions and other parameters for the optimization, such as Quantum Technology, 2-17-7 Yaguchi, Ota-ku, Tokyo, Japan. Tel: +81-3-3759-6810; Email: akihito_kikuchi(at)gakushikai.jp the symbolic computation digests and decomposes the polynomials into aatomic tame positionsform of the and set contraction of equations, coefficients to which ofnumerical atomic orbitals.computations Then Citation: Kikuchi A, Kikuchi I (2019) Computational are easily applied. The key technique is Gröbner basis theory, by Algebraic Geometry and Quantum Mechanics: An Initiative which one can investigate the electronic structure by unraveling the toward Post Contemporary Quantum Chemistry. J Multidis Res Rev Vol: 1, Issu: 2 (47-79). demonstrate the featured result of this new theory. Next, we expound theentangled mathematical relations basics of the concerning involved variables. computational In this algebraic article, at geometry, first, we Copyright: © 2019 Kikuchi A. This is an open-access which are necessitated in our study. We will see how highly abstract ideas article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the problems in quantum mechanics. We solve simple problems in “quantum original author and source are credited. chemistryof polynomial in algebraic algebra variety”would be by applied means toof algebraicthe solution approach. of the definiteFinally, 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, Quantum chemistry in algebraic variety, First principles electronic structure calculation, Symbolic computation, symbolic-numeric solving, Hartree-Fock theory, Taylor series, Polynomial approximation, Algebraic molecular orbital theory. Introduction Dear Readers, If you are researchers or students with the expertise of physics or chemistry, you might have heard of “algebraic geometry” or “commutative algebra”. Maybe you might have heard only of these words, and you might the department of mathematics, not in those of physics and chemistry. Younot havemight definite have heard ideas of about advanced them, regions because of theoretical these topics physics, are taught such asin super-string theory, matrix model, etc., where the researchers are seeking the secret of the universe by means of esoteric theories of mathematics with the motto algebraic geometry and quantum mechanics. And you might be desperate in imagining the required endurance to arrive at the foremost front of the study... However, algebraic geometry is originated from rather a primitive region of mathematics. In fact, it is an extension www.innovationinfo.org of analytic geometry which you must have learned in high- the Gröbner basis theory and triangulation of polynomial set. Let us review the exemplary computation of hydrogen of the word goes as follows: molecule in [2]. schools. According to Encyclopedia Britannica, the definition algebraic geometry, study of the geometric properties Let φ , , be the wavefunctions, the atomic charges, i Za Ra of solutions of polynomial equations, including solutions in and the atomic positions. The total energy functional of three dimensions beyond three... Hartree-Fock theory is given by It simply asserts that algebraic geometry is the study of ∆ 2 Z polynomial systems. And polynomial is ubiquitous in every Ω= φφ− + a ∑∑∫ dr ii() r ()r branch of physics. If you attend the lecture of elementary ia2 ||rR− a quantum mechanics, or you study quantum chemistry, you always encounter secular equations in order to compute ′′ 1 ′ φφφii(rr ) ( ) j (r ) φ j (r ) the energy spectrum. Such equations are actually given by + ∑ drdr 2∫ ||r − r′ the polynomial systems, although you solve them through ij, linear algebra. Indeed, linear algebra is so powerful that you ′′ 1 ′ φφφijji(rr ) ( ) (r ) φ (r ) have almost forgotten that you are laboring with polynomial − ∑∫ drdr algebraic equations. 2ij, ||r − r′ Be courageous! Let us have a small tour in the sea of ZZ + ab QUANTUM MECHANICS with the chart of ALGEBRAIC ∑ a,b ||RRa − b GEOMETRY. Your voyage shall never be in stormy and misty mare incognitum. Having chartered a cruise ship, − λ(dr φφ ()r (r) − δ) the “COMMUTATIVE ALGEBRA”, we sail from a celebrated ∑ ij,,∫ i j ij ij, seaport named “MOLECULAR ORBITAL THEORY”. The secular equation is derived from the stationary The molecular orbital theory [1] in the electronic structure condition with respect to the wave-function computation of quantum chemistry is computed by the solution of the secular equation, if one adopts the localized δΩ atomic orbital basis {φi} =0. positions with other optimizable variables ζ : δφi {Ri} { i} defined on the set of atoms at the The energy minimum with respect to the atomic coordinate gives the stable structure: H,({Rij} {ζζ})⋅Ψ = ER S,({ ij} { })⋅Ψ , Let us consider the simple hydrogen, as in Figure 1. in which the matrix elements are defined by H,ij = drφφi j ∫ Assume that the two hydrogen atoms are placed at RA and RB . By the simplest model, we can choose the trial and 3 wave-functions for up-and down-spin at the point z ∈ as = φφ follows: S.ij ∫dr i j 1 In these expressions, is the Hamiltonian; the vector φ ( z) =( aexp( −+ zb) exp( − z) ) Ψ up π AB Ψ= χφ ∑ ii; E is the energy spectrum. The Fock matrix H and represents the coefficients in LCAO wave-function 1 the overlap matrix S are, in theory, computed symbolically φdown ( z) =( cexp( −+ zdAB) exp( − z) ) and represented by the analytic formulas with respect to π the atomic coordinates and other parameters included in Where the Gaussian- or Slater- type localized atomic basis; they are, in practice, given numerically and the equation is solved z= zR − by means of linear algebra. In contrast to this practice, it is AB//AB demonstrated by Kikuchi [2] that there is a possibility of and abcd,,, are the real variables to be determined. Let φ φ symbolic-numeric computation of molecular orbital theory, ev and ew to be Lagrange multipliers λij for up and down . Since which can go without linear algebra: the secular equation these two wave-functions are orthogonal in nature, due to is given by the analytic form and approximated by the the difference of spin, we assume that the multipliers are polynomial system, which is processed by the computational algebraic geometry. The symbolic computation reconstructs and decomposes the polynomial equations into a more tractable and simpler form, by which the numerical solution H H of the polynomial equation is applied for the purpose of obtaining the quantum eigenstates. The key technique is Figure 1: The image of a hydrogen molecule, composed of two atoms. J Multidis Res Rev 2019 48 www.innovationinfo.org 12470*t=0 diagonal: λij = 0 for ij≠ . From these expressions, all of the integrals involved S[2]156*s*u^2*r^4-1068*s*u^2*r^3+2248*s*u^2*r^2- in the energy functional are analytically computed. (As 80*s*u^2*r for the analytic forms of the integrals, see the supplement ....................................................... of [2].) Then, with respect to inter-atomic distance , RAB Taylor expansion of the energy functional is computed up +992*t*u*v*r^2-160*t*u*v*r+40*t*u*v=0 to degree four, at the center of RAB = 7/5. The polynomial S[3]156*s^2*u*r^4-1068*s^2*u*r^3+2248*s^2*u*r^2- approximation is given by 80*s^2*u*r OMEGA = (3571 - 1580*a^2 - 3075*a*b - 1580*b^2 - ....................................................... 1580*c^2 +126*u*r^4-882*u*r^3+1748*u*r^2+1896*u*r- + 625*a^2*c^2 + 1243*a*b*c^2 + 620*b^2*c^2 - 12470*u=0 3075*c*d S[4]-52*s^2*v*r^4+236*s^2*v*r^3-32*s^2*v*r^2- + 1243*a^2*c*d + 2506*a*b*c*d + 1243*b^2*c*d - 104*s^2*v*r 1580*d^2 ....................................................... .......................................................................................................... -30*v*r^4+330*v*r^3-1148*v*r^2+760*v*r-170*v=0 .......................................................................................................... δΩ The stationary conditions δΩ , , with respect to , δ ev δ ew .......................................................................................................... , are the normalization condition for the wave-functions. - 86*a*b*c*d*r^4 - 17*b^2*c*d*r^4 + 12*d^2*r^4 - They yield two equations: 4*a^2*d^2*r^4 S[5]-13*s^2*r^4+139*s^2*r^3-458*s^2*r^2+63*s^2*r - 17*a*b*d^2*r^4 + 13*a*b*ev*r^4 + 13*c*d*ew*r^4)/1000 ....................................................... where the inter-atomic distance RAB is represented by r -63*t^2*r-3986*t^2+1000=0 (for the convenience of polynomial processing).