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

Atomic multipole moments associated with a spherical volume fully residing within a topological atom (i.e. the β sphere) can be obtained analytically. Such an integration is thus free of quadrature grids. A general formula for an arbitrary rank spherical harmonic multipole moment is derived, for an electron density
comprising Gaussian primitives of arbitrary angular momentum. The closed expressions derived here are also sufficient to calculate the electrostatic potential, the two types of kinetic energy, as well as the potential energy between atoms. Some integrals have not been solved explicitly before but through recursion and substitution are broken down to more elementary listed integrals. The proposed method is based on a central formula that shifts Gaussian primitives from one centre to another, which can be derived from the well-known plane-wave expansion (or Rayleigh equation).