8
density. One might conclude that this behavior arises due to the local nature
of the functional. Therefore, a natural way to improve over LDA is to include
the gradients of density in the energy functional. The generalized gradient
approximation provides such a framework to improve over the LDA functional
by an inclusion of the density gradients. The most commonly used functional
under the GGA framework is known as PBE functional named after its developers
9. In the PBE functional the exchange energy density of the LDA
is augmented by an enhancement factor which depends on the density and its
gradient. The PBE exchange energy can be expressed as:
X =
EGGA
Z
d3rX(n(r))LDAFX(s), (2.8)
where FX(s) denotes the exchange enhancement factor with s = |rn(r)/2kF n(r).
One of the crucial property that the enhancement factor should have is that
in the limit of very small s it should behave in a way that the PBE exchange
energy approaches the exchange energy with the LSDA. Keeping this in mind
the following expression for FX(s) has been proposed:
FX(s) = 1 + −
1 + μs2/
. (2.9)
The inclusion of the exchange enhancement factor in the exchange energy
showed significant improvement over the LDA functionals for the systems with
significantly varying charge density. Since then the PBE functional has been
one of the most widely used functional in electronic structure problems.
2.3 Calculation of Bandgaps with DFT
Despite being quite successful in the prediction of ground state properties of
real materials, Kohn-Sham DFT (KS-DFT) has some drawabacks 12. One
of the most commonly known problem with KS-DFT is the systematic underestimation
of bandgaps 13. Over the years, numerous studies have been
performed in order to have an understanding of the bandgap problem and at
the same time finding its solution. A very thorough study to understand the
different sources of the errors in the bandgap prediction has been done in the
Ref. 13. For example, depending on the convexity (concavity) of the functional
between the integer particle number, localization (delocalization) leads
to too high (low) bandgap predictions for the periodic systems. Therefore, it
would be desirable to include an additional localization effect in the concave