Theory of the expansion of wave functions in a gaussian basis

The convergence properties of the expansions of (a) the function 1/ r and (b) the function exp(‐α r ) in an even‐tempered basis of Gaussians are studied analytically. The starting points are the Gaussian integral representations of 1/ r and exp(−α r ). One arrives at an expansion in a finite number of Gaussians in three steps: (1) a restriction of the integration domain, (2) a variable transformation, and (3) discretization of the integral. The cutoff error goes in both cases essentially as exp(− ah ), and the discretization error, as exp(− b / h ). The minimum overall error is reached for the β‐parameter of an even‐tempered basis β ∽ exp( c /√ n ), where n is the dimension of the basis, and the error itself decreases as ϵ ∽ exp(− d √ n ). Different optimum basis parameters are obtained depending on which quantity one wants to minimize, e.g., the error of the energy expectation value, the distance in Hilbert space, the variance of the energy, or the density at the nucleus.