  The well-known spatial integration schemes in molecular electronic structure
theory, immune to cusps and point singularities of some kind at atomic
positions, use a set of weighting functions to split the integrand into a sum
of atom-centered parts, each dealt with in its own spherical coordinate system.
Here, for a given set of integrands in the two-center case, a quality measure
of the weighting functions is defined to compare, design, and optimize them, it
is roughly proportional to the average number of angular quadrature points
needed to reach a given integration accuracy. A study of Becke's fuzzy
Vorono\"i cells has helped to improve their performance by a new modification.
New spherically-symmetric unnormalized weighting functions are found in the
form of a negative power times the negative exponential of the fourth power of
the scaled distance to the atomic center, with the parameters related to the
asymptotic decay of the integrand and the integration accuracy -- these are
much simpler but no less efficient and naturally fit for linear-scaling
calculations. Radial distribution of spherical quadrature orders is studied. A
radial integration scheme of double exponential type is optimized. A symmetric
analog of the pseudospectral approximation is used for the seminumerical
evaluation of two-electron repulsion integrals. Taken together, this allows
efficient calculation of all molecular integrals with well-controlled accuracy,
as shown by tests on a set of molecules.
