MOHNISH PANDEY AND KARSTEN W. JACOBSEN PHYSICAL REVIEW B 91, 235201 (2015)
TABLE II. Outliers in the calculations without using the FERE scheme. The compounds exhibiting deviations of the calculated heats of
formation from the experimental values by more than 2σ have been identified as outliers. The values of σ for the different functionals are
shown in Fig. 1. δH denotes the difference between calculated and experimental heats of formation.
PBE δHPBE RPBE δHRPBE PBE+U δHPBE+U TPSS δHTPSS mBEEF δHmBEEF
Al2O3 0.48 Al2O3 0.69 Al2O3 0.48 AlP 0.45 AuF3 −0.30
BaS −0.52 FeF2 0.61 BaS −0.52 BaI2 −0.48 CaF2 −0.35
BaO −0.47 FeO 0.60 BaO −0.47 BiBr3 −0.51 CdF2 −0.34
FeF2 0.61 GaN 0.57 CrS −0.82 CaS 0.48 Cu2Se 0.31
FeO 0.49 HfO2 0.65 CrF3 −0.47 FeF2 0.57 FeSe 0.35
GaS 0.451 NiF2 0.66 Cr2O3 −0.75 GaP 0.43 GaN 0.33
LaN 0.46 GaN 0.42 Ga2S3 0.44 Ga2S3 0.37
MnS 0.60 Ga2S3 0.44 NiF2 0.57 GaS 0.41
NiF2 0.85 GaS 0.45 PbBr2 −0.45 GeSe 0.37
Ge4O8 0.42 SrBr2 −0.49 Ge4O8 0.29
MnS −0.48 SrI2 −0.55 NbF5 −0.38
Mn3O4 −0.42 ZnS 0.43 OsO4 −0.30
V2O3 −0.42 ZrS2 0.47 PbF2 −0.31
SnO2 0.28
TiN −0.30
have none or very few common outliers. For example, the
predictions for barium-containing compounds is a little worse
only in the PBE, PBE+U, and the TPSS whereas the outliers
containing chromium are present in the PBE+U functional
only. On the other hand, even if the gallium is present in all the
functionals it is not the same compound which is an outlier.
Additionally, the outliers present in the mBEEF calculations
do not deviate from the experimental value by more than
0.41 eV per atom whereas the outliers present in the GGA
functionals and the TPSS have deviations as high as 0.85
and 0.57 eV per atom, respectively. The deviations shown
for the mBEEF functional are relative to a common rms error
σ = 0.14 eV and not based on the ensemble estimated errors.
The large variation in outliers with functional seems to indicate
that the appearance of outliers is as might be expected not
due to experimental errors but rather due to limitations of the
different functionals.
Table III shows the outliers after the fitting has been applied.
We see that the outliers are to a large extent different from the
ones before the fitting and again they also vary considerably for
the different functionals. This means that we cannot identify
particular issues with specific systems. The PBE and RPBE
functionals continue to have significant overlap of outliers after
the fitting.
D. Statistical analysis of the mBEEF predictions
The error bars predicted by the mBEEF ensemble are in
reasonable agreement with the actual errors as can be seen
from Fig. 1(i). In order to study the quality of the error bar
prediction in more detail we show in Fig. 2(a) a histogram
of the actual error, i.e., the deviation between the mBEEF
prediction and the experimental value (HmBEEF − HExpt)
divided by the predicted error bar (σBEE). The histogram is a
TABLE III. Outliers in the calculations using the FERE scheme. The compounds exhibiting deviations of the calculated heats of formation
from the experimental values by more than 2σ have been identified as outliers. The values of σ for the different functionals are shown in Fig. 1.
δH denotes the difference between calculated and experimental heats of formation.
PBE RPBE δHFERE
RPBE PBE+U δHFERE
PBE+U TPSS δHFERE
TPSS mBEEF δHFERE
mBEEF
PBE δHFERE
CuF2 0.22 CuF2 0.23 CoS 0.20 BaCl2 0.24 CaF2 −0.18
FeF2 0.33 FeF2 0.27 Co3O4 −0.23 CaS 0.21 CdF2 −0.18
FeSe −0.19 MnO2 −0.21 CrO2 0.17 CsF −0.23 Co3O4 −0.19
MnO2 −0.25 NbF5 −0.32 Fe2O3 −0.17 FeF2 0.29 Fe2O3 −0.17
NbF5 −0.29 Ni3S2 −0.18 GaF3 0.16 KCl 0.32 FeF2 0.19
Ni3S2 −0.21 NiF2 0.38 GeO2 0.20 NbF5 −0.26 GaP −0.16
NiF2 0.65 PbF2 −0.18 MgF2 0.19 NiF2 0.37 KF −0.17
RuO4 −0.19 RuO4 −0.33 NbF5 −0.23 RbI 0.25 Li3Sb 0.18
TaF5 −0.22 TaF5 −0.24 SnO2 0.17 SrS 0.25 MgO −0.15
ZrSi −0.24 ZrSi −0.26 TaF5 −0.18 SrI2 −0.24 MgF2 0.17
ZrS2 0.26 ZrS2 0.24 TiN −0.19 TlI 0.26 MnO2 −0.16
VN 0.26 ZrSi −0.24 NbF5 −0.20
V2O3 −0.36 ZrS2 0.35 TiN −0.17
ZnF2 0.19 ZnF2 0.18
ZrS2 0.16 ZrS2 0.16
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