14
the other hand, most of the higher level methods are computationally quite
expensive and cannot be used for large scale computations.
Recently a method has been proposed by Stevanovic et al. which uses the
experimental heats of formation and DFT total energies to fit the elemental
reference energies in order get better prediction for the standard heats of formation
28, 25. In the work by Stevanovic et al. DFT+U 29 has been used
with non-zero U for the transition metals. However, in our work we find that
the other functionals like PBE 9, RPBE 30 and TPSS 31, 32, 33, 34, 35
give similar prediction as PBE+U after fitting the reference energies. Surprisingly
TPSS being a meta-GGA does not improve the prediction and has
similar error as the standard GGA functionals. But, the recently developed
Bayesian error estimation meta-GGA functional known as mBEEF improves
the predictions significantly. Additionally, it provides the uncertainties in the
formation energies as well thus giving the information of the trust radius of
the results. The details of the mBEEF functional can be found in the Ref.
36.
3.2 Calculation of the heats of formation without
the fitting
The heat of formation of a solid calculated with DFT can be written as:
HDFT (Ap1Bp2..) = E(Ap1Bp2..) − piμ0i
, (3.1)
0i
where E(Ap1Bp2..) indicates the total energy of Ap1Bp2.. calculated with DFT
and the μdenotes the chemical potentials of the elements under standard
conditions calculated with DFT. The entropic and zero point corrections have
been ignored in the expression above.
For the current work, a set of 257 compounds has been selected to compare
different functionals for the calculation of heats of formation. Compounds
have been selected to ensure that the space of relevant elements is spanned.
Figure 3.1 (a), (c), (e), (g) and (i) show the calculated heats of formation versus
the experimental values for the different functionals. The figure indicates
that the RPBE functional deviates the most from the experimental values,
which is also expected since the functional parameters have been fitted to give
accurate adsorption energies which makes it a bit worse for the prediction of
the bulk properties. Additionally, PBE, PBE+U and TPSS give similar predictions
thus TPSS despite being meta-GGA does not perform better than
the functionals at the GGA level. Therefore, before any fitting of the experimental
values, mBEEF outperforms other functionals in the predictions with