An atom-atom partitioning of the (super)molecular Coulomb energy is proposed on the basis of the topological partitioning of the electron density. Atom-atom contributions to the molecular intra- and intermolecular Coulomb energy are computed exactly, i.e., via a double integration over atomic basins, and by means of the spherical tensor multipole expansion, up to rank L=l(A)+l(B)+1=5. The convergence of the multipole expansion is able to reproduce the exact interaction energy with an accuracy of 0.1-2.3 kJ/mol at L=5 for atom pairs, each atom belonging to a different molecule constituting a van der Waals complex, and for nonbonded atom-atom interactions in single molecules. The atom-atom contributions do not show a significant basis set dependence (3%) provided electron correlation and polarization basis functions are included. The proposed atom-atom Coulomb interaction energy can be used both with post-Hartree-Fock wave functions and experimental charge densities in principle. The Coulomb interaction energy between two molecules in a van der Waals complex can be computed by summing the additive atom-atom contributions between the molecules. Our method is able to extract from the supermolecule wave function an estimate of the molecular interaction energy in a complex, without invoking the reference state of free noninteracting molecules. We provide computational details of this method and apply it to (C2H2)(2); (HF)(2); (H2O)(2); butane; 1,3,5-hexatriene; acrolein and urocanic acid, thereby covering a cross section of hydrogen bonds, and covalent bonds with and without charge transfer.