Formation energies of rutile metal dioxides using density functional theory

We apply standard density functional theory at the generalized gradient approximation (GGA) level to study the stability of rutile metal oxides. It is well known that standard GGA exchange and correlation in some cases is not sufficient to address reduction and oxidation reactions. Especially the formation energy of the oxygen molecule and the electron self-interaction for localized $d$ and $f$ electrons are known shortcomings. In this paper we show that despite the known problems, it is possible to calculate the stability of a wide range of rutile oxides $M{\text{O}}_2$, with $M$ being Pt, Ru, Ir, Os, Pb, Re, Mn, Se, Ge, Ti, Cr, Nb, W, Mo, and V, using the electrochemical series as reference. The mean absolute error of the formation energy is 0.29 eV using the revised Perdew-Burke-Ernzerhof (PBE) GGA functional. We believe that the reason for the success is due to the reference level being ${\text{H}}_2$ and ${\text{H}}_2\text{O}$ and not ${\text{O}}_2$ and due to a more accurate description of exchange for this particular GGA functional compared to PBE. Furthermore, we would expect the self-interaction problem to be largest for the most localized $d$ orbitals; that means the late $3d$ metals and since Co, Fe, Ni, and Cu do not form rutile oxides they are not included in this study. We show that the variations in formation energy can be understood in terms of a previously suggested model separating the formation energy into a metal deformation contribution and an oxygen binding contribution. The latter is found to scale with the filling of the $d$ band.

Figure 1

(Color online) Calculated formation energies of various representative rutile metal dioxides as a function of the experimental values extracted from the electrochemical series (Refs. 38, 39, 40). The zero free-energy reference has been taken as the total energy of the metal oxide bulks.

Figure 2

(Color online) The formation energy, the metal bulk deformation energy, and the bonding energy in the expanded metal lattice as functions of the filling of the metal $t_{2g}$ band for some representative rutile $4d$ metal dioxides (Nb, Mo, and Ru). The variation in formation energy is clearly dominated by the oxygen binding energy.