The theory of atoms in molecules (AIM) defines bounded atomic fragments in real space that generate transferable atomic properties. As part of a program that investigates the topological partitioning of electromagnetic properties based on the electron density, we have calculated the exact atomic electrostatic potential (AEP) of an AIM atom in a molecule. Second we expand this atomic electrostatic potential in terms of AIM electrostatic multipole moments based on spherical tensors. We prove that the convergence of this expansion is faster than previously assumed, even for complicated atomic shapes. © 2000 American Chemical Society.