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The well-known spatial integration schemes in molecular electronic structuretheory, immune to cusps and point singularities of some kind at atomicpositions, use a set of weighting functions to split the integrand into a sumof 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 measureof the weighting functions is defined to compare, design, and optimize them, itis roughly proportional to the average number of angular quadrature pointsneeded to reach a given integration accuracy. A study of Becke's fuzzyVorono\"i cells has helped to improve their performance by a new modification.New spherically-symmetric unnormalized weighting functions are found in theform of a negative power times the negative exponential of the fourth power ofthe scaled distance to the atomic center, with the parameters related to theasymptotic decay of the integrand and the integration accuracy -- these aremuch simpler but no less efficient and naturally fit for linear-scalingcalculations. Radial distribution of spherical quadrature orders is studied. Aradial integration scheme of double exponential type is optimized. A symmetricanalog of the pseudospectral approximation is used for the seminumericalevaluation of two-electron repulsion integrals. Taken together, this allowsefficient calculation of all molecular integrals with well-controlled accuracy,as shown by tests on a set of molecules.

Spherical-harmonic-analysis-based optimization of atomic weighting functions for multicenter numerical integration in molecules