Correcting density functional theory for accurate predictions of compound enthalpies of formation: Fitted elemental-phase reference energies

Despite the great success that theoretical approaches based on density functional theory have in describing properties of solid compounds, accurate predictions of the enthalpies of formation ($\Delta H_f$) of insulating and semiconducting solids still remain a challenge. This is mainly due to incomplete error cancellation when computing the total energy differences between the compound total energy and the total energies of its elemental constituents. In this paper we present an approach based on GGA $+$ $U$ calculations, including the spin-orbit coupling, which involves fitted elemental-phase reference energies (FERE) and which significantly improves the error cancellation resulting in accurate values for the compound enthalpies of formation. We use an extensive set of 252 binary compounds with measured $\Delta H_f$ values (pnictides, chalcogenides, and halides) to obtain FERE energies and show that after the fitting, the 252 enthalpies of formation are reproduced with the mean absolute error $\text{MAE}=0.054$ eV/atom instead of $\text{MAE}\approx 0.250$ eV/atom resulting from pure GGA calculations. When applied to a set of 55 ternary compounds that were not part of the fitting set the FERE method reproduces their enthalpies of formation with $\text{MAE}=0.048$ eV/atom. Furthermore, we find that contributions to the total energy differences coming from the spin-orbit coupling can be, to a good approximation, separated into purely atomic contributions which do not affect $\Delta H_f$. The FERE method, hence, represents a simple and general approach, as it is computationally equivalent to the cost of pure GGA calculations and applies to virtually all insulating and semiconducting compounds, for predicting compound $\Delta H_f$ values with chemical accuracy. We also show that by providing accurate $\Delta H_f$ the FERE approach can be applied for accurate predictions of the compound thermodynamic stability or for predictions of Li-ion battery voltages.

Figure 1

Histogram showing absolute errors of the GGA (upper part) and of the FERE approach (lower part) in reproducing measured enthalpies of formation[10, 11] for 45 binary pnictides and chalcogenides containing 3$d$ transition metals. Dashed lines represent the mean absolute error (MAE) of the two methods corresponding to the full set of 252 binary compounds listed in Table . Numerical values for the compounds shown in this figure are provided in Table .

Figure 2

The relation of the chemical potentials $\Delta \mu$ of Ni and O for equilibrium with the NiO and Ni${}_2$O${}_3$ compound phases, as calculated in GGA and in the present FERE method based on GGA $+$ $U$ compound energies. In GGA, NiO is unstable against Ni${}_2$O${}_3$ for the full range of allowed chemical potentials $\Delta \mu ⩽0$. The correct phase stability of NiO is recovered in the present method.

Figure 3

Part of the periodic table listing the $\delta {\mu }_i^{\mathrm{FERE}}$ values (in eV) of Eq. (4) for 50 chemical elements. Colors denote the values of the Hubbard $U$ parameter used in the calculations: $U=3$ (light blue), 5 (orange), and 0 eV (light grey). Absolute $\left\{{\mu }_i^{\mathrm{FERE}}\right\}$ values are given in Table of the Appendix. It is important to note that $\left\{{\mu }^{\mathrm{FERE}}\right\}$ are obtained by fitting using a set of 252 binary compounds with measured $\Delta H_f$ within which elements from groups I-IV of the periodic table appear formally as cations and from groups V, VI, and VII as anions (see text for details).

Figure 4

Histogram showing deviations of $\Delta H_f^{\mathrm{FERE}}$ [see Eq. (5)] relative to $\Delta H_f^{\exp }$ for 55 ternary compounds. Dashed line represent the mean absolute error (MAE) of the method. Numerical values are provided in Table .

Figure 5

(a) Projection of the allowed ranges of chemical potentials onto $(\Delta {\mu }_{\text{Mn}},\Delta {\mu }_{\text{Si}})$ plane with the green polygon defined by Eq. (8) representing the region of thermodynamic stability for Mn${}_2$SiO${}_4$. (b) Stability region from (a) displayed on a oxygen partial pressure versus temperature plot. Experimental growth conditions reported in Ref. 25 are also shown.