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Designer polynomials, discrete variable representations, and the Schrödinger equation
Author(s) -
Weatherford Charles A.,
Red Eddie,
Wynn Albert
Publication year - 2002
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.10356
Subject(s) - orthogonal polynomials , polynomial , recursion (computer science) , mathematics , jacobi polynomials , exponential function , variable (mathematics) , term (time) , function (biology) , classical orthogonal polynomials , pure mathematics , mathematical analysis , physics , quantum mechanics , algorithm , evolutionary biology , biology
The general procedure for constructing a set of orthonormal polynomials is given for an arbitrary positive definite weight function, w ( x ), in the interval [ a , b ]. The Lanczos method is used to generate the three‐term recursion relation, which is then used to produce the polynomial coefficients. A discrete variable representation (DVR) is constructed from Gaussian nodes and weights that result from the three‐term recursion relation. These are termed “designer polynomials” and the associated “designer DVRs.” It will be shown by construction that every such set of “synthetic polynomials” carries an associated DVR. The term “designer” derives from the fact that the interval [ a , b ] and the weight function w ( x ) are arbitrary (except that w ( x ) must be positive definite on [ a , b ] and must have continuous derivatives except at a finite number of isolated discontinuities) and may be adapted to the physical problem of interest. The difficulties of applying a DVR to a “bare” Coulomb problem will be illustrated on a “toy” model in one dimension (1‐D hydrogen atom). A solution for the 1‐D Coulomb problem will be given, thereby motivating the need for designer DVRs. In doing so, a new set of polynomials is defined with a weight function w ( x ) = | x | k exp(−α|x|), (such that k = −1, 0, +1, +2, …) between the symmetrical limits [−∞, +∞]. These are called “synthetic Cartesian exponential polynomials (SCEP).” These polynomials are then used in a spectral and pseudospectral (DVR) representation to solve the 1‐D hydrogen atom problem. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002