MOHNISH PANDEY AND KARSTEN W. JACOBSEN PHYSICAL REVIEW B 91, 235201 (2015)
antiferromagnetically as reported in the experiments.We use a
real-space description of thewave functions with a grid spacing
of 0.18 A° . A Fermi temperature of 0.05 eV for the solid phases
is used to enhance convergence. Brillouin zone sampling is
done with a k-point mesh of 33a
−1
x
−1
y
× 33a
−1
z with the
× 33a
Monkhorst-Pack 19 sampling scheme. Forces are minimized
down to 0.05 eV/°A
for all the relaxations. Uncertainties in the
heats of formations with the mBEEF functionals are explicitly
calculated using the ensemble of functionals proposed in
Ref. 20. All the experimental heats of formation have been
taken from Refs. 4,21.
III. RESULTS AND DISCUSSIONS
A. Heats of formation with the DFT
The standard heat of formation of a solid calculated with
DFT is
HDFT(Ap1Bp2 . . . ) = E(Ap1Bp2 . . . ) − piμ0
i , (1)
where E(Ap1Bp2 . . . ) indicates the total energy of Ap1Bp2 . . .
calculated withDFTand theμ0
i denotes the chemical potentials
of the elements under standard conditions calculated with
DFT. The entropic and zero-point energy corrections have been
ignored in the above expression. The calculation of the heats
of formation using the above expression with the PBE, RPBE,
TPSS, and PBE+U functionals provide a single number as
the best estimate of the heat of formation. In comparison,
the mBEEF functional provides both a best estimate but also
via the ensemble of functionals an estimation of the error bar
on the calculated heat of formation. The functionals in the
mBEEF ensemble differ from each other by the values of the
parameters defining the functional 15.
The calculated heats of formation versus the experimental
heats of formation (eV/per atom) of a set of 257 binary
compounds with the PBE, RPBE, PBE+U, TPSS, and mBEEF
functionals are shown in Fig. 1, panels (a), (c), (e), (g), and (i).
The set of compounds we use has about 80% overlap with the
set of 252 compounds used by Stevanovi´c et al. 4 and the
full list of compounds is given in Table I along with the heats
of formation calculated with the mBEEF and the mBEEF with
fitting of reference energies. The difference between our data
set and the one of Stevanovi´c et al. gives rise to somewhat
different results in detail but the trends remain the same. In the
figureMAE and σ denote the mean absolute error and standard
deviation with respect to the experimental heats of formation.
The observed trend in Figs. 1(a) and 1(c) is a similar behavior
for the PBE and RPBE functionals with underbinding in most
of the cases with a very few overbinding cases. This behavior
in the GGA functionals arises due to the overbinding of the
reference phases and the underbinding in the multinary compounds
leading to an incomplete cancellation of the errors 22.
In Fig. 1(e) the direction of the deviation in the PBE+U heats
of formation is not very systematic, i.e., underbinding in some
cases and overbinding in others. This behavior has also been
observed in Ref. 23. The predictions do not significantly
improve with the TPSS functional as shown in Fig. 1(g). The
MAE and rms in the TPSS predictions turn out to be similar
to the GGA functionals. An important factor in the calculated
errors is the dissimilar nature of the reactants and the products.
Reactions in which both sides have similar compounds are
FIG. 1. (Color online) (a), (c), (e), (g), and (i) show the heats
of formation calculated with the PBE, RPBE, PBE+U, TPSS, and
mBEEF functionals, respectively, versus the experimental heats of
formation. (b), (d), (f), (h), and (j) show the heats of formation
calculated with PBE, RPBE, PBE+U, TPSS, and mBEEF functionals,
respectively, versus the experimental heats of formation after
correcting the reference phase energies using the experimental heats
of formation as the training set. MAE and σ in (a)–(j) indicate the
mean absolute error and standard deviation of the calculated heats of
formation with respect to the experimental heats of formation.
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