Explanation of the Source of Very Large Errors in Many Exchange–Correlation Functionals for Vanadium Dimer

Vanadium dimer is a notoriously difficult case for Kohn-Sham (KS) density functional theory with currently available approximations to the exchange-correlation (xc) functionals, and many approximate xc functionals yield an exceedingly large error in the calculated bond energy. In this paper, we first test the bond energies estimated by 43 xc functionals and the Hartree-Fock (HF) method. The results further confirm the large errors and show that, with the experimental bond energy being 64.2 kcal/mol, the KS calculations give predictions all over the map with errors ranging from -61.5 to +60.5 kcal/mol, and the HF method performs much worse with an error of -124.4 kcal/mol! The reason for these very large errors is examined in this article by analyzing the atomic and molecular orbital energies calculated by various xc functionals. The results show that the errors in estimates of the bond energy of vanadium dimer can primarily be related to the calculated energy gap between the 4s and 3dz2 atomic orbitals of the vanadium atom and especially to the 3dz2 orbital energy. This interesting relation between the errors in the calculated bond energy and the magnitudes of the single-particle orbital energies provides a constructive alternative to the common but more sterile explanation that it is the static correlation energy due to multicenter left-right correlation that makes the vanadium dimer and many other transition metal compounds so difficult for Kohn-Sham calculations. One of the most important factors in determining the critical atomic orbital energy is the amount of nonlocal HF exchange that is included in the xc functional, but it is still difficult to explain why different local functionals (functionals with no HF exchange) yield quite different results. We conclude that improving calculations of orbital energies of atoms may provide a route to improving the accuracy of theoretical predictions of molecular bond energies for systems containing metal atoms.