Fully analytical integration over the 3D volume bounded by the β sphere in topological atoms.

Atomic properties of a topological atom are obtained by 3D integration over the volume of its atomic basin. Algorithms that compute atomic properties typically integrate over two subspaces: the volume bounded by the so-called β sphere, which is centered at the nucleus and completely contained within the atomic basin, and the volume of the remaining part of the basin. Here we show how the usual quadrature over the β sphere volume can be replaced by a fully analytical 3D integration leading to the atomic charge (monopole moment) for s, p, and d functions. Spherical tensor multipole moments have also been implemented and tested up to hexadecupole for s functions only, and up to quadrupole for s and p functions. The new algorithm is illustrated by operating on capped glycine (HF/6-31G, 35 molecular orbitals (MOs), 322 Gaussian primitives, 19 nuclei), the protein crambin (HF/3-21G, 1260 MOs, 5922 primitives and 642 nuclei), and tin (Z = 50) in Sn(2)(CH(3))(2) (B3LYP/cc-pVTZ and LANL2DZ, 59 MOs, 1352 primitives).