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Conference on "Variational and spectral methods in Quantum Field Theory"
Organizers: Volker Bach (Braunschweig), Maria J. Esteban (CNRS), Mathieu Lewin (CNRS), Eric Séré (Paris-Dauphine)


Werner Kutzelnigg (Bochum, Germany): "Rate of Convergence of the Expansion of a Wave Function in a Gaussian basis"
Friday 26 April 2013, 09:30 - 10:15
Amphi Hermite


Traditional expansions in an orthonormal basis of the type of a Fourier series are very sensitive to the singularities of the function to be
expanded. Exponential convergence is only possible, if the basis functions describe the singularities of the expanded functions correctly. Otherwise only an inverse-power-law convergence is realized, which is usually slow. An example is the slow convergence of the CI expansion due to the correlation cusp. An improved convergence can be achieved, if one augments the basis by functions that describe the singularities of the wave function correctly, like in the R12 method.

An alternative approach towards an improved convergence is possible in terms of a discretized integral transformation. This is realized in the conventional Boys-Huzinaga-Ruedenberg expansion of wave functions in a Gaussian basis. Such expansions are surprisingly insensitive to singularities of the wave function to be expanded (e.g. the nuclear cusp). The rate of convergence is of an unconventional type, with an error estimate ~exp(-a sqrt(n)), if n is the basis size. We compare a local and a global criterion for convergence. We consider especially the convergence of the wave function at the position of a nucleus, as well as the energy expectation value. The two criteria lead to different sets of basis parameters. We further compare an even-tempered (geometric sequence) basis with a non-even-tempered basis.

keywords: Gaussian basis, even-tempered basis, rate of convergence, discretized integral transformation, relativistic quantum chemistry