Tests of Exchange-Correlation Functional Approximations Against Reliable Experimental Data for Average Bond Energies of 3d Transition Metal Compounds

One of the greatest challenges for the theoretical study of transition-metal-containing compounds is the treatment of intrinsically multiconfigurational atoms and molecules, which require a multireference (MR) treatment in wave function theory. The accuracy of density functional theory for such systems is still being explored. Here, we continue that exploration by presenting the predictions of 42 exchange-correlation (xc) functionals of 11 types [local spin density approximation (LSDA), generalized gradient approximation (GGA), nonseparable gradient approximation (NGA), global-hybrid GGA, meta-GGA, meta-NGA, global-hybrid meta-GGA, range-separated hybrid GGA, range-separated hybrid meta-GGA, range-separated hybrid meta-NGA, and DFT augmented with molecular mechanics damped dispersion (DFT-D)]. DFT-D is tested both for Grimme's DFT-D3(BJ) model with Becke-Johnson damping and for ωB97X-D, which has the empirical atom-atom dispersion parametrized by Chai and Head-Gordon. The Hartree-Fock (HF) method has also been included because it can be viewed as a functional with 100% HF exchange and no correlation. These methods are tested against a database including 70 first-transition-row (3d) transition-metal-containing molecules (19 single-reference molecules and 51 MR molecules), all of which have estimated experimental uncertainties equal to or less than 2.0 kcal/mol in the heat of formation. We analyze the accuracy in terms of the atomization energy per bond instead of the enthalpy of formation of the molecule because it allows us to test electronic energies without the possibility of cancellation of errors in electronic energies with errors in vibrational energies. All the density functional and HF wave functions have been optimized to a stable solution, in which the spatial symmetry is allowed to be broken to minimize the energy to a stable solution. We find that τ-HCTHhyb has the smallest mean unsigned error (MUE) in average bond energy, in particular 2.5 kcal/mol, for the full set of 70 molecules, and it also gives the smallest MUE for MR systems. For single-reference systems, MPW1B95 has the best performance, with an MUE of 1.6 kcal/mol. Among local functionals, which are the least expensive, the best performance (MUE = 3.4 kcal/mol) for the total database is achieved by OreLYP. It is observed that adding HF exchange does not guarantee better accuracy for GGAs or for the NGA, but inclusion of the kinetic energy densities can benefit the GGAs and NGA calculations. The metal hydrides and metal oxides are demonstrated to be the most difficult bond types to predict, and CrO3, FeH, CrO, VH, and MnS are found to be the most difficult molecules to predict. The middle transition metals (V, Cr, and Mn) lead to larger errors on average than either the early or late transition metals.