Atom–atom partitioning of intramolecular and intermolecular Coulomb energy

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 ${\mathrm{L=l}}_A{\mathrm{+l}}_B\text{+1=5.}$ 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 $\text{L=5}$ 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 $({\mathrm{C}}_{\mathrm{2}}{\mathrm{H}}_{\mathrm{2}}{)}_2;$ $(\mathrm{HF}{)}_2;$ $({\mathrm{H}}_{\mathrm{2}}\mathrm{O}{)}_2;$ 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.