Localization transformations of orthogonal functions

Abstract

A discrete representation of vibrational molecular wavefunctions in coordinate space offers many advantages for the solution of problems in quantum dynamics, in particular if realistic, multidimensional potential energy surfaces are involved. This leads to the problem of how to construct localized representations of operators. While any set of grid points might be defined for a discrete variable representation (DVR) of local operators this is not true for the Hamiltonian operator as a whole. As a consequence of the non-local nature of the kinetic energy operator a correspondence between the localized picture and a delocalized variational basis representation (VBR) or momentum space, respectively, must be established. The arrangement of grid points is therefore determined by the properties of the VBR basis functions. It is shown that linear matrix transformations between localized and delocalized representations and the so-called Fourier method for the evaluation of non-local operations can essentially be reduced to the same idea: localization transformations of orthogonal functions.

Gaussian quadrature is a classical way to obtain localized basis functions derived from polynomials. We investigate the properties of the localized functions related to important polynomials and obtain a new relation for Legendre polynomials. VBR basis functions related to equidistant quadrature points, like particle-in-a-box or Fourier functions, are frequently applied and the associated localized basis functions are shown to possess interesting symmetry features. We give a mathematically consistent theory of matrix transformations between localized and delocalized pictures: a new representation, called nearly discrete variable representation (NDVR), is introduced, that is equivalent to the VBR, in analogy to the correspondence of DVR and finite basis representation (FBR). DVR basis functions are defined in a new way via delta sequences. Finally, a brief analysis of the error made in calculations with a DVR potential matrix is given for two simple one-dimensional models.