Distributed Gaussian discrete variable representation

Abstract

A discrete variable representation (DVR) made from distributed Gaussians gn(x) = e-c2(x/d-n)2, (n = - ?,..., ?) and its infinite grid limit is described. The infinite grid limit of the distributed Gaussian DVR (DGDVR) reduces to the sine function DVR of Colbert and Miller in the limit c › 0. The numerical performance of both finite and infinite grid DGDVRs and the sine function DVR is compared. If a small number of quadrature points are taken, the finite grid DGDVR performs much better than both infinite grid DGDVR and sine function DVR. The infinite grid DVRs lose accuracy due to the truncation error. In contrast, the sine function DVR is found to be superior to both finite and infinite grid DGDVRs if enough grid points are taken to eliminate the truncation error. In particular, the accuracy of DGDVRs does not get better than some limit when the distance between Gaussians d goes to zero with fixed c, whereas the accuracy of the sine function DVR improves very quickly as d becomes smaller, and the results are exact in the limit d › 0. An analysis of the performance of distributed basis functions to represent a given function is presented in a recent publication. With this analysis, we explain why the sine function DVR performs better than the infinite grid DGDVR. The analysis also traces the inability of Gaussians to yield exact results in the limit d › 0 to the incompleteness of this basis in this limit. © 2005 Wilev Periodicals. Inc.