  The electronic Schr\"odinger equation describes the motion of N electrons
under Coulomb interaction forces in a field of clamped nuclei. The solutions of
this equation, the electronic wavefunctions, depend on 3N variables, three
spatial dimensions for each electron. We study the approximability of these
wavefunctions by linear combinations of anisotropic Gauss functions, or more
precisely Gauss-Hermite functions, products of polynomials and anisotropic
Gauss functions in the narrow sense. We show that the original, singular
wavefunctions can up to given accuracy and a negligibly small residual error be
approximated with only insignificantly more such terms than their convolution
with a Gaussian kernel of sufficiently small width and that basically arbitrary
orders of convergence can be reached. This is a fairly surprising result, since
it essentially means that by this type of approximation, the intricate
hierarchies of non-smooth cusps in electronic wavefunctions have almost no
impact on the convergence, once the global structure is resolved.
