15
significantly lower mean absolute error (MAE) and standard deviation ().
It can also be seen that the experimental values are within the uncertainties
predicted by the mBEEF ensemble.
3.3 Calculation of the heats of formation with
the fitting
As mentioned before, the different chemical environment of the multinary compounds
and the reference phases leads to an incomplete error cancellation in
calculating the energy differences i.e. the heats of formation. This behavior
was manifested in the predictions in the previous section which was based on
the DFT reference energies of the elemental phases. Fitted elemental reference
phase energy (FERE) method 25 solves this problem to some extent by
adding corrections to the DFT reference energies. The value of the corrections
is calculated by minimizing the root mean square (RMS) error of the predicted
and the experimental values. The FERE heats of formation can be expressed
as:
HFERE(Ap1Bp2..) = E(Ap1Bp2..)
−pi(μ0i
+ μ0i
), (3.2)
The only difference between the equation above and the equation (3.1) is the
term μ0i
which denotes the correction to reference energy of the elemental
phase.
As mentioned before, a dataset of 257 compounds has been chosen for
experimental heats of formation, 25, 37 on the other hand, the number of
elements relevant for this work is limited to 62. Therefore, the calculation
of the corrections involves solving an overdetermined qP
set of equations which
can be done by minimizing the RMS error
i(HiE
xpt. − HiD
FT )2. Few
points have to be kept in mind while fitting the reference energies, for example,
a reasonable size of the dateset should be taken to avoid over- or under-fitting
and the quality of the fit should be validated on a test dataset which has
compounds not used in the fitting procedure.
The calculated heats of formation with the FERE procedure applied to the
different functionals is shown in the Figure 3.1 (b), (d), (f), (h) and (j). As can
be seen from the figure, different functionals clearly improve the predictions
when augmented with the FERE procedure. After the fitting procedure is
applied all the functionals give similar predictions with almost same MAE
and . It is worth noticing that the mBEEF predictions before the fitting