What dominates the error in the CaO diatomic bond energy predicted by various approximate exchange-correlation functionals?

In order to understand what governs the accuracy of approximate exchange-correlation functionals for intrinsically multiconfigurational systems containing metal atoms, the properties of the ground electronic state of CaO have been studied in detail. We first applied the T1, TAE(T), B1, and M diagnostics to CaO and confirmed that CaO is an intrinsically multiconfigurational system. Then, we compared the bond dissociation energies (BDEs) of CaO as calculated by 49 exchange-correlation functionals, three exchange-only functionals, and the HF method. To analyze the error in the BDEs for the various functionals, we decomposed each calculated BDE into four components, in particular the ionization potential, the electron affinity, the atomic excitation energy of the metal cation to prepare the valence state, and the interaction energy between prepared states. We found that the dominant error occurs in the calculated atomic excitation energy of the cation. Third, we compared dipole moments of CaO as calculated by the 53 methods, and we analyzed the dipole moments in terms of partial atomic charges to understand the contribution of ionic bonding and how it is affected by errors in the calculated ionization potential of the metal atom. We then analyzed the dipole moment in terms of the charge distribution among orbitals, and we found that the orbital charge distribution does not correlate well with the difference between the calculated ionization potential and electron affinity. Fourth, we examined the potential curves and internuclear distance dependence of the orbital energies of the lowest-energy CaO singlet and triplet states to analyze the near-degeneracy aspect of the correlation energy. The most important conclusion is that the error tends to be dominated by the error in the relative energies of s and d orbitals in Ca⁺, and the most popular density functionals predict this excitation energy poorly. Thus, even if they were to predict the BDE reasonably well, it would be due to cancellation of errors. The effect of the cation excitation energy can be understood in terms of an orbital picture, as follows. For most functionals the predicted cation excitation energy is too small, so it is too easy to delocalize charge from the oxygen 2p orbital to the Ca⁺ d orbital; this overestimates the covalency and explains why most functionals overestimate the bond energy.