MOHNISH PANDEY AND KARSTEN W. JACOBSEN PHYSICAL REVIEW B 91, 235201 (2015)
TABLE IV. Heats of formation of the solid compounds calculated with the different functionals with and without employing the fitting
procedure. The set below was not used in the training set for fitting the reference phase energies. All the energies are in eV/atom.
Compound HExpt HPBE HRPBE HPBE+U HTPSS HmBEEF HFERE
PBE HFERE
RPBE HFERE
PBE+U HFERE
TPSS HFERE
mBEEF
AgNO3
−0.26 −0.40 −0.31 −0.53 −0.47 −0.60±0.22 −0.58 −0.67 −0.68 −0.45 −0.63±0.16
AlPO4 −2.99 −2.71 −2.58 −2.71 −2.86 −2.97±0.19 −2.95 −2.97 −2.94 −3.02 −3.03±0.07
BeSO4 −2.16 −1.99 −1.83 −1.99 −2.09 −2.19±0.16 −2.22 −2.23 −2.21 −2.19 −2.25±0.11
BiOCl −1.27 −1.26 −1.11 −1.26 −1.62 −1.26±0.16 −1.25 −1.20 −1.23 −1.32 −1.23±0.09
CdSO4 −1.61 −1.42 −1.27 −1.42 −1.53 −1.59±0.17 −1.61 −1.62 −1.60 −1.60 −1.63±0.12
CuCl2 −0.76 −0.51 −0.32 −0.70 −0.80 −0.74±0.21 −0.75 −0.60 −0.84 −0.79 −0.81±0.07
TiBr3 −1.42 −1.24 −1.23 −1.52 −1.71 −1.37±0.08 −1.38 −1.39 −1.61 −1.38 −1.43±0.05
NaClO4 −0.66 −0.54 −0.41 −0.54 −0.67 −0.63±0.15 −0.76 −0.77 −0.73 −0.68 −0.65±0.16
CaSO4 −2.48 −2.24 −2.06 −2.24 −2.37 −2.40±0.17 −2.41 −2.41 −2.41 −2.46 −2.43±0.12
Cs2S −1.24 −1.01 −0.92 −1.01 −1.47 −1.16±0.18 −1.27 −1.24 −1.33 −1.97 −1.23±0.06
CuWO4 −1.91 −1.59 −1.41 −1.76 −1.72 −1.68±0.21 −1.62 −1.60 −1.75 −1.65 −1.71±0.07
PbF4 −1.95 −2.13 −2.05 −2.13 −2.26 −2.32±0.23 −2.19 −2.19 −2.11 −2.24 −2.13±0.08
MgSO4
−2.22 −1.97 −1.79 −1.97 −2.09 −2.16±0.16 −2.22 −2.21 −2.21 −2.20 −2.24±0.10
SrSe −2.00 −2.04 −1.98 −2.04 −2.76 −2.29±0.16 −2.25 −2.26 −2.29 −2.66 −2.29±0.05
NiSO4 −1.51 −1.11 −0.96 −1.35 −1.23 −1.42±0.23 −1.35 −1.36 −1.54 −1.33 −1.50±0.11
FeWO4 −1.99 −1.73 −1.58 −2.01 −1.87 −1.84±0.21 −1.81 −1.81 −1.94 −1.86 −1.89±0.06
GeP −0.11 +0.04 +0.09 +0.04 −0.19 +0.14±0.08 −0.01 +0.03 −0.05 −0.28 −0.02±0.07
VOCl −2.10 −1.79 −1.68 −2.45 −2.07 −2.11±0.24 −1.97 −1.98 −2.41 −2.05 −2.12±0.07
LiBO2 −2.67 −2.42 −2.30 −2.42 −2.57 −2.58±0.17 −2.61 −2.61 −2.58 −2.64 −2.61±0.05
NaBrO3 −0.69 −0.52 −0.41 −0.52 −0.71 −0.60±0.13 −0.74 −0.76 −0.72 −0.69 −0.66±0.11
CoSO4 −1.53 −1.09 −0.95 −1.43 −1.24 −1.40±0.23 −1.30 −1.34 −1.56 −1.31 −1.43±0.11
PbSeO4 −1.05 −0.94 −0.81 −0.94 −1.13 −1.04±0.16 −1.07 −1.09 −1.06 −1.07 −1.08±0.09
Mn2SiO4 −2.56 −1.83 −1.77 −2.58 −2.01 −2.29±0.23 −2.17 −2.19 −2.38 −2.10 −2.25±0.08
ZnSO4 −1.70 −1.37 −1.20 −1.37 −1.47 −1.53±0.16 −1.61 −1.61 −1.61 −1.60 −1.62±0.11
MAE 0.24 0.35 0.16 0.20 0.12 0.12 0.13 0.11 0.15 0.09
σ 0.28 0.39 0.19 0.26 0.16 0.16 0.17 0.15 0.25 0.14
IV. CONCLUSION
The need for accurate predictions of material stabilities
has led to the development of schemes combining DFT total
energy calculations with experimental information. We have
analyzed one such scheme for calculation of heats of formation
which fit the reference energies for elemental systems. The
scheme was developed with the PBE+U functional, but we
show that comparable predictive power is obtained using other
GGAs such as PBE or RPBE or the meta-GGA TPSS.We have
furthermore seen that the mBEEF functional, which is a meta-
GGA and which has been extensively optimized to a variety of
experimental data, leads to much improved estimation of heats
of formation even without applying the fitting procedure. The
mBEEF functional furthermore includes realistic ensemble
estimates of the calculated formation energies. Applying
the fitting scheme to mBEEF leads to a further reduction
of the error and narrows the ensemble error estimation
accordingly.
The FERE scheme clearly has its limitations. The correction
of only the binding energies of the reference systems makes
most sense if the character of the bonding differs significantly
between the material at hand and the reference systems. This
is for example the case for a metal oxide, in which the bonding
may be quite different from the one in an oxygen molecule
and in the pure metal. However, oxygen can enter in many
different ways in different materials and only improving on
themolecular energy cannot be a solution to improved heats of
formation in the long run.Moving to more accurate functionals
is therefore a must, and the current work shows that applying
a meta-GGA such as mBEEF already provides a significant
improvement in the prediction of solid heats of formation.
ACKNOWLEDGMENTS
The authors acknowledge CASE (Catalysis for Sustainable
Energy) initiative. CASE is funded by the Danish Ministry of
Science, Technology and Innovation.
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