Accuracy of density functional theory in predicting formation energies of ternary oxides from binary oxides and its implication on phase stability

The evaluation of reaction energies between solids using density functional theory (DFT) is of practical importance in many technological fields and paramount in the study of the phase stability of known and predicted compounds. In this work, we present a comparison between reaction energies provided by experiments and computed by DFT in the generalized gradient approximation (GGA), using a Hubbard $U$ parameter for some transition metal elements (GGA+$U$). We use a data set of 135 reactions involving the formation of ternary oxides from binary oxides in a broad range of chemistries and crystal structures. We find that the computational errors can be modeled by a normal distribution with a mean close to zero and a standard deviation of 24 meV/atom. The significantly smaller error compared to the more commonly reported errors in the formation energies from the elements is related to the larger cancellation of errors in energies when reactions involve chemically similar compounds. This result is of importance for phase diagram computations for which the relevant reaction energies are often not from the elements but from chemically close phases (e.g., ternary oxides versus binary oxides). In addition, we discuss the distribution of computational errors among chemistries and show that the use of a Hubbard $U$ parameter is critical to the accuracy of reaction energies involving transition metals even when no major change in formal oxidation state is occurring.

Figure 1

Illustration of the procedure to determine the reaction critical to phase stability for an ABC${}_3$ compound. By building the phase diagram using the convex-hull construction on all phases in the A-B-C system with the exception of ABC${}_3$, we find that the equilibrium triangle consists in A${}_{\text{2}}$C, BC${}_{\text{3}}$, and BC${}_{\text{2}}$. The critical reaction energy to phase stability is therefore $\frac{1}{2}$A${}_{\text{2}}$C+$\frac{1}{2}$BC${}_{\text{3}}$+$\frac{1}{2}$BC${}_{\text{2}}$ $\to$ ABC${}_3$.

Figure 2

Experimental reaction energy as function of the computed reaction energy (in meV/atom). The error bar indicates the experimental error. As the reaction energies are typically negative, the graph actually plots the negative of the reaction energy.

Figure 3

Histogram of the difference between computed ($\Delta E_{0\text{K}}^{\mathrm{comp}}$) and experimental ($\Delta E_{0\text{K}}^{\mathrm{expt}}$) energies of reaction (in meV/atom).

Figure 4

Matrix of the difference between computed and experimental energies of reaction ($|\Delta E_{0\text{K}}^{\mathrm{comp}}-\Delta E_{0\text{K}}^{\mathrm{expt}}|$). The $x$ ($y$) axis represents the oxides of element A (B). Each element in the matrix corresponds to a chemical system A-B-O. When several reaction energies are available in a chemical systems (i.e., several ternary compounds are present), we plot the maximum absolute error energy in this system. The matrix is symmetric as A-B-O is equivalent to B-A-O. The elements are sorted by Mendeleev number and the first row indicates the mean of the difference across the different chemistries.

Figure 5

Two examples of ternary oxides 0-K phase diagrams: Li-Al-O (a) and Li-Cr-O (b). All phases are from the ICSD. We excluded phases with partial occupancies. The Li-Al-O phase diagram illustrates the case of elements with only one oxidation state. On the other hand, Cr in the Li-Cr-O phase diagram is common in oxidation states from $+$3 to $+$6. The black (dark gray) arrows indicate the critical reaction energies determining the stability of ternary phases. The light blue (gray) dashed arrow indicates the Li${}_{\text{3}}$CrO${}_{\text{4}}$ phase.

Figure 6

Heat capacity vs temperature for several compounds (MgO, BaO, ZnO, Cs${}_2$O, FeO, and CoFe${}_2$O${}_4$). The black dots are experimental values and the red line is the fitted Debye-type model. All heat capacities are given in J/mol-fu K.