Orthogonal Functions, Discrete Variable Representation, and Generalized Gauss Quadratures

Abstract

The numerical solution of most problems in theoretical chemistry involve either the use of a basis set expansion (spectral method) or a numerical grid. For many basis sets, there is an intimate connection between the spectral form and numerical quadrature. When this connection exists, the distinction between spectral and grid approaches becomes blurred. In fact, the two approaches can be related by a similarity transformation. By the exploitation of this idea, calculations can be considerably simplified by removing the need to compute difficult matrix elements of the Hamiltonian in the original representation. This has been exploited in bound-state, scattering, and time-dependent problems using the so-called, discrete variable representation (DVR). At the core of this approach is the mathematical three-term recursion relationship satisfied by the classical orthogonal functions. This three-term recursion can be used to generate the orthogonal functions as well as to generate the points and weights of Gauss quadratures on the basis of these functions. For the classical orthogonal functions, the terms in the three-term recursion are known analytically. For more general weight functions, this is not the case. However, they may be computed in a stable numerical fashion, via the recursion. In essence, this is an application of the well-known Lanczos recursion approach. Once the recursion coefficients are known, it is possible to compute the points and weights of quadratures on the basis of the generalized weight functions. We review these ideas below and apply then to the generation of the points and weights of the Rys polynomials which have proven useful in the evaluation of multicenter integrals, using Gaussian basis sets in quantum chemistry. In contrast to some approaches, the method advocated is general, numerically stable, and trivial to program.