Convergence of the multipole expansion for electrostatic potentials of finite topological atoms

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Abstract
The exact atomic electrostatic potential (AEP) and atomic multipole moments are calculated using the topological partitioning of the electron density. High rank (l⩽20) spherical tensor multipole moments are used to examine the convergence properties of the multipole expansion. We vary independently the maximum multipole rank, l max , and the radius of the spherical grid around an atom in a molecule where we measure the discrepancy between the exact AEP and the one obtained via multipole expansion. The root mean square values are between 0.1 and 1.6 kJ/mol for four atoms (C, N, O, S) on a spherical grid with the ρ=0.001 a.u. convergence radius and for l max =4. Our calculations demonstrate that this fast convergence is due to the decay of the electron density. We show that multipole moments generated by finite atoms are adequate for use in the multipole expansion of the electrostatic potential, contrary to some claims made in the literature. Moreover they can be used to model intermolecular and in principle intramolecular interactions as well.