Density Functional Theory in Transition-Metal Chemistry:  Relative Energies of Low-Lying States of Iron Compounds and the Effect of Spatial Symmetry Breaking

The ground and lower excited states of Fe₂, Fe₂^-, and FeO⁺ were studied using a number of density functional theory (DFT) methods. Specific attention was paid to the relative state energies, the internuclear distances (r_e), and the harmonic vibrational frequencies (ω_e). A number of factors influencing the calculated values of these properties were examined. These include basis sets, the nature of the density functional chosen, the percentage of Hartree−Fock exchange in the density functional, and constraints on orbital symmetry. A number of different types of generalized gradient approximation (GGA) density functionals (straight GGA, hybrid GGA, meta-GGA, and hybrid meta-GGA) were examined, and it was found that the best results were obtained with hybrid GGA or hybrid meta-GGA functionals that contain nonzero fractions of HF exchange; specifically, the best overall results were obtained with B3LYP, M05, and M06, closely followed by B1LYP. One significant observation was the effect of enforcing symmetry on the orbitals. When a degenerate orbital (π or δ) is partially occupied in the ⁴Φ excited state of FeO⁺, reducing the enforced symmetry (from C_6v to C_4v to C_2v) results in a lower energy since these degenerate orbitals are split in the lower symmetries. The results obtained were compared to higher level ab initio results from the literature and to recent PBE+U plane wave results by Kulik et al. (Phys. Rev. Lett. 2006, 97, 103001). It was found that some of the improvements that were afforded by the semiempirical +U correction can also be accomplished by improving the form of the DFT functional and, in one case, by not enforcing high symmetry on the orbitals.