An approach to first principles electronic structure calculation by symbolic-numeric computation

Abstract

There is a wide variety of electronic structure calculation cooperating with symbolic computation. The main purpose of the latter is to play an auxiliary role (but not without importance) to the former. In the field of quantum physics [1-9], researchers sometimes have to handle complicated mathematical expressions, whose derivation seems almost beyond human power. Thus one resorts to the intensive use of computers, namely, symbolic computation [10-16]. Examples of this can be seen in various topics: atomic energy levels, molecular dynamics, molecular energy and spectra, collision and scattering, lattice spin models and so on [16]. How to obtain molecular integrals analytically or how to manipulate complex formulas in many body interactions, is one such problem. In the former, when one uses special atomic basis for a specific purpose, to express the integrals by the combination of already known analytic functions, may sometimes be very difficult. In the latter, one must rearrange a number of creation and annihilation operators in a suitable order and calculate the analytical expectation value. It is usual that a quantitative and massive computation follows a symbolic one; for the convenience of the numerical computation, it is necessary to reduce a complicated analytic expression into a tractable and computable form. This is the main motive for the introduction of the symbolic computation as a forerunner of the numerical one and their collaboration has won considerable successes. The present work should be classified as one such trial. Meanwhile, the use of symbolic computation in the present work is not limited to indirect and auxiliary part to the numerical computation. The present work can be applicable to a direct and quantitative estimation of the electronic structure, skipping conventional computational methods

Computer Algebra and Materials Physics

“…[ (1,14,20) (2,15,16) (3,11,17)(4,12,18)(5,13,19) (32,45,46) (33,41,47) (34,42,48) (35,43,49), (1,2,3,4,5)(6,11,16,21,26) (7,12,17,22,27)(8,13,18,23,28)(9,14,19,24,29) (10,15,20,25,30)(31,32,33,34,35)(36,41,46,51,56) (37,42,47,52,57) (38,43,48,53,58) (39,44,49,54,59) (40,45,50,55,60) (9,52,49) (10,51,48) (11,35,28) (12,34,29) (13,…”

“…The newly created group "p" is larger than A5 (the group "G"), and contains A5 as a subgroup. The three generators are given as (34,47,48) (35,43,49) (36,44,50) (37,45,51) gap> p.2; (3,7,6,5,4) (8,28,23,18,13)(9,29,24,19,14)(10,30,25,20,15)(11,31,26,21,16) (12, 32,27,22,17)(33,37,36,35,34) (38,58,53,48,43)(39,59,54,49,44)(40,60,55,50, 45)(41,61,56,51,46) (42,62,57,52,47) gap> p.3;…”

“…[ [ 3,1,4,1 ], [ 16,1,17,1 ], [ 4,1,5,1 ], [ 3,2,61,2 ], [ 22,1,18,1 ], [ 21,1,22,1 ], [ 59,2,44,2 ], [ 17,1,13,1 ], [ 5,1,6,1 ], [ 61,2,11,2 ], [ 16,2,23,2 ], [ 4,2,41,2 ], ...(omitted hereafter)... gap> B1:=Orbit(p, [3,1,8,1],OnTuples);;…”

“…As for the application of combinatorics in quantum physics, a sort of extension of Polya counting theorem can be useful in computing the spin weight in molecules [41][42][43]. In a recent study, it is shown that the framework of the electronic structure could be reconstructed by means of a concept in the algebraic geometry, so-called Groebner basis theory, through the utilization of symbolic computation [44].…”

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

“…As a result, the triangular decomposition yields four subsets of the solutions of equations: the possible electronic configurations are exhausted, as is shown in Table 1.1. the result exhausts the possible four configurations of the ground and the excited states This is one of the featured results in [Kik13]. The author of that work had demonstrated the procedure of the computation in a factual way, but he had not explained the underlying mathematical theory so minutely.…”