Rate of Convergence of Basis Expansions in Quantum Chemistry

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 similar to exp(-a root n), if n is the basis size.