Valence and extra-valence orbitals in main group and transition metal bonding

We address the issue first raised by Maseras and Morokuma with regard to the questionable treatment of empty p-orbitals in the algorithm for natural atomic/bond orbitals (NAOs, NBOs) and associated natural population analysis. We quantify this issue in terms of the numerical error (root-mean-square density deviation) resulting from the two alternative treatments of empty p-sets, leading to distinct NAOs, atomic charges, and idealized Lewis structural representations. Computational application of this criterion to a broad spectrum of main group and transition group species (employing both single- and multi-structure resonance models) reveals the interesting general pattern of (i) relatively insignificant differences for normal-valent species, where a single resonance structure is usually adequate, but (ii) clear superiority of the standard NAO algorithm for hypervalent species, where multi-resonance character is pronounced. These comparisons show how the divisive issue of "valence shell expansion" in transition metal bonding is deeply linked to competing conceptual models of hypervalency (viz., "p-orbital participation" in skeletal hybridization vs. 3c/4e resonance character). The results provide a quantitative measure of superiority both for the standard NAO evaluation of atomic charges as well as the general 3c/4e (A: B-C<-->A-B :C resonance) picture of main- and transition-group hypervalency.