HEATS OF FORMATION OF SOLIDS WITH ERROR . . . PHYSICAL REVIEW B 91, 235201 (2015)
TABLE I. (Continued.)
Compound HExpt HmBEEF HFERE
mBEEF Compound HExpt HmBEEF HFERE
mBEEF
TiO2 −3.26 −3.37±0.21 −3.32±0.05 TlI −0.64 −0.70±0.10 −0.67±0.04
TlBr −0.90 −0.95±0.11 −0.97±0.05 TlCl −1.06 −1.16±0.20 −1.08±0.07
Tl2O −0.62 −0.68±0.15 −0.61±0.08 TlF −1.68 −1.80±0.21 −1.62±0.03
Tl2S −0.34 −0.37±0.12 −0.32±0.05 VN −1.13 −0.99±0.13 −1.00±0.08
V2O3 −2.53 −2.58±0.29 −2.62±0.11 V2O5 −2.29 −2.31±0.20 −2.35±0.06
VO2 −2.47 −2.48±0.23 −2.52±0.04 WBr6 −0.52 −0.41±0.07 −0.51±0.03
WO3 −2.04 −2.08±0.19 −2.05±0.02 YAs −1.68 −1.75±0.08 −1.68±0.02
YCl3 −2.59 −2.63±0.22 −2.67±0.05 YF3 −4.45 −4.45±0.27 −4.36±0.04
ZnO −1.81 −1.57±0.21 −1.75±0.04 ZnSe −0.85 −0.76±0.12 −0.97±0.02
ZnTe −0.61 −0.56±0.08 −0.64±0.01 ZnS −1.07 −0.89±0.14 −1.10±0.03
ZnCl2 −1.43 −1.36±0.19 −1.44±0.05 ZnF2 −2.64 −2.50±0.25 −2.46±0.05
ZrO2 −3.80 −3.75±0.23 −3.79±0.05 ZrCl4 −2.03 −2.11±0.22 −2.08±0.03
ZrSi −0.80 −0.93±0.11 −0.94±0.05 ZrN −1.89 −1.84±0.13 −1.85±0.06
ZrS2 −1.96 −1.73±0.14 −1.79±0.06
shown to give smaller errors when compared to experiments
22. Figure 1(i) shows the calculated heats of formation
with themBEEFfunctional with calculated error bars indicated
with green bars. The calculated values are significantly closer
to the experimental values compared to the values obtained
from the PBE, RPBE, PBE+U, and TPSS functionals. As can
be seen from the figure the experimental values are within the
error bars predicted by the mBEEF functional.
The mBEEF functional thus seems to be more accurate
than both the GGA functionals and the TPSS which is also
a meta-GGA functional. However, it should also be noted
that in the construction of the mBEEF functional considerable
optimization to experimental databases was performed. In
the following we investigate how the scheme suggested by
Stevanovi´c et al. 4 helps in improving the predictions for the
different functionals.
B. Heats of formation with the FERE
In the previous section we noticed that the limited predictability
of the TPSS and the GGA functionals mainly arises
from the different nature of the bonding in themultinary phases
and the reference phases. The FERE scheme 4 circumvents
this problem by adding corrections to the reference phase
energies. The heats of formation calculated with the FERE
can be written as
HFERE(Ap1Bp2 . . . )
= E(Ap1Bp2 . . . ) − pi
μ0
i
+ δμ0
i
, (2)
where the δμ0
i ’s are the corrections added to the reference
phase energies to improve the heats of formation. The values
of the δμ0
i ’s can be calculated by a linear regression fit by
minimizing the root mean square (rms) error between the
calculated (HDFT) and the experimental (HExpt) heats of
formation. The size of the training set has to be sufficiently
large to avoid any overfitting and the quality of the fit must be
validated on a test set. The linear regression requires that the
number of parameters in the linear model which need to be
fitted to the observations be smaller than the number of data
points; i.e., the system of the equations has to be overdetermined.
We calculate 62 parameters which correspond to the
corrections to the reference phase energies of 62 elements by
using a training set of 257 compounds with the experimental
heats of formation available. The parameters can be calculated
using singular value decomposition (SVD) 24 byminimizing
the rms error |HExpt − HDFT|2. The calculated reference
energies are tabulated in the Supplemental Material 25.
Figure 1, panels (b), (d), (f), (h), and (j), shows the heats
of formation calculated after adding the corrections to the
reference phase energies. The comparison with panels (a),
(c), (e), (g), and (i) of the figure clearly shows that the MAE
and σ are significantly reduced after applying the corrections
to the reference phase energies. Interestingly, all the GGA
functionals give similar heats of formation after employing
the corrections even though they differ before applying the
corrections. The TPSS functional does not perform any better
than the GGA functionals after applying the corrections.
As noted the performance of mBEEF before fitting is
somewhat better than the GGAs and the TPSS functional and
in fact, as we shall see later, it is comparable to the fitted GGAs
and the TPSS on a test set. However, for comparison we also
apply the FERE fitting procedure to the mBEEF functional
and this does naturally lead to an improvement on the training
set. As mentioned before, we furthermore employ the fitting
procedure on all the functionals in the mBEEF ensemble
anticipating a reduction of the error and the fluctuations within
the ensemble. This is indeed the case. In Fig. 1(j) we can see
that the uncertainties are significantly reduced as compared to
Fig. 1(i). The reduction in the size of the uncertainties is in
agreement with the fact that the fitted mBEEF predictions are
more accurate.
C. Analysis of outliers
The appearance of outliers with and without the fitting
for the PBE, RPBE, and PBE+U functionals may occur
for two reasons: (1) error in the experimental data, and (2)
some systems are poorly described with the given functional.
The compounds having the deviation of the calculated heat
of formation (δH) from the experimental value by twice of
the standard deviation (2σ) are shown in Table II. The table
clearly shows that all the functionals except for PBE and RPBE
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