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functionals (like LDA) whereas employing a delocalization effect in the convex
functionals would improve the bandgap predictions.
As explained in Ref. 13 the energy in LDA like functional behave linearly
between integer points in periodic systems. Therefore one would expect it to
give correct bandgaps. But, the linear behavior has wrong slopes due to which
it systematically underestimates the bandgap. To account for the incorrect
slopes the correction in the derivative discontinuity can be applied leading
to improved prediction of the bandgap. One such functional is the GLLB-SC
functional which includes an explicit calculation of the derivative discontinuity.
The details of the functional can be found in Ref. 14, 15.
The other method to improve over the LDA/GGA functionals is to incorporate
a fraction of Hartree-Fock exchange (or exact exchange) which has a
convex behavior. Thus, the Hartree-Fock exchange when added in an appropriate
fraction in the LDA exchange gives a reasonable behavior between the
integer points of the particle number. Generally, the LDA/GGA functionals
having a fraction of exact exchange are called hybrid functionals. Most commonly
used hybrid functionals in condensed matter systems are PBE0 and
HSE03/HSE06 10, 11, 16, 17. A brief introduction to the HSE functional
will be provided here since its implementation in GPAW was carried out as a
part of this thesis.
2.3.1 A Brief Introduction to the Hybrid Functionals
The PBE0 or HSE functionals have 25 % of exact exchange (at least that is
how it started) mixed with 75 % of GGA exchange. The exchange correlation
energy in the PBE0 functional can be written as:
EPBE0
xc = 0.25EHF
x + 0.75EPBE
x + EPBE
c . (2.10)
The (1 /|r − r0|) dependence of HF exchange gives rise to a singularity at r =
r’( or q = q’ in reciprocal space). Therefore, it is essential to get rid of the
singularity to prevent divergence. Additionally, a very high density of k-points
is required to resolve the interaction near the singularity.
The singularity problem has been remedied in the HSE functionals by having
an additional term which prevents the exchange term from diverging. The
HSE functional has many commonalities with the PBE0 functionals. However,
in the HSE functional the exchange is screened by a screening parameter
as opposed to the PBE0 functional which has a bare (or unscreened) exact
exchange. The exchange interaction in the HSE is divided into a short range