Evaluation of group theoretical characteristics using the symbolic manipulation language MAPLE

Abstract

Relying on theoretical developments exploiting quasispin and the pseudo-orthogonal group in the Hubbard model of cyclic polyenes, the general expressions for generating polynomials, providing the dimensional information for relevant irreducible representations, were derived (M. D. Could, J. Paldus, and J. Ciiek, Int. J. Quantum Chem., in press). These generating polynomials result from qdimensional formulas through rather tedious algebraic manipulations involving ratios of polynomials with fractional powers. It is shown that these expressions may be efficiently handled using the symbolic manipulation language MAPLE and the dimensional information for an arbitrary spin, isospin, and quasimomentum obtained. Exploitation of symbolic computation for other group theoretical problems that are relevant in quantum chemical calculations and their relationship with Gaussian polynomial based combinatorial approaches is also briefly addressed and various possible applications outlined. 0 1994 John Wiley & Sons, Inc.

Symbolic calculation in chemistry: Selected examples

“…Taneri and Paldus constructed dimensional information concerning the irreducible representations needed in a Hubbard model of cyclic polyenes using MAPLE 286.…”

“…Further work on the Hubbard model was reported by Angelescu and Bhatt 16, Bartkowiak et al 32, and Steeb et al 132. Taneri and Paldus 286 and Bishop et al 41 considered Heisenberg and Ising models.…”

Reconstruction of secular polynomials for Hubbard model from energy perturbation series for weak and strong coupling cases

“…For systems that contain more than 6 atoms, the full CI matrix is extremely large. 4 Any reconstruction of secular polynomials in these cases will be only approximate. However, the accuracy of the results for practical purposes is found to be excellent.…”

“…Even with this quasi-diagonalization, when calculations are done on systems with more than six sites, one obtains matrices which become extremely large. 4 It should be noted that to construct the matrix would lead to a nonlinear problem, while constructing a secular polynomial is linear. However, in the latter case, the equations are ill conditioned, while in integer arithmetic, one has no trouble.…”

“…At the present time, it has not been possible to obtain secular polynomials for Nϭ10 particles because of the size of the problem. 4 One other possible disadvantage of this procedure is that it can be rather unstable. It should be carried out with a large number of digits, or in terms of rational arithmetic.…”