THEORY OF THE EXPANSION OF WAVE-FUNCTIONS IN A GAUSSIAN-BASIS

The convergence properties of the expansions of (a) the function 1/r a nd (b) the function exp(- alpha r) in an even-tempered basis of Gaussi ans are studied analytically. The starting points are the Gaussian int egral representations of 1/r and exp(- alpha r). One arrives at an exp ansion in a finite number of Gaussians in three steps: (1) a restricti on of the integration domain, (2) a variable transformation, and (3) d iscretization of the integral. The cutoff error goes in both cases ess entially as exp(- ah), and the discretization error, as exp(- b/h). Th e minimum overall error is reached for the beta-parameter of an even-t empered basis beta approximately exp(c/ square-root n), where n is the dimension of the basis, and the error itself decreases as epsilon app roximately exp(-d square-root n). Different optimum basis parameters a re obtained depending on which quantity one wants to minimize, e.g., t he error of the energy expectation value, the distance in Hilbert spac e, the variance of the energy, or the density at the nucleus. (C) 1994 John Wiley & Sons, Inc.