7
particles can be written as:
−
12
r2 + vext(r) +
Z
d3r
n(r)
|r − r0|
+ vxcn(r)
i(r) = ii(r). (2.5)
The first term denotes the kinetic energy operator, second term is the external
potential which typically comes from nuclei, third term is the Hartree potential
and vxcn(r) represents the exchange-correlation potential which arises from
the antisymmetric and many body nature of the wavefunction. The exchagecorrelation
potential vxcn(r) entering the Kohn-Sham equation can be written
as:
vxcn(r) = Exc
n(r) . (2.6)
Up to this point no approximations in the Kohn-Sham system has been made,
therefore, the formalism in principle is exact. But our ignorance about the exact
form of Exc demands approximations to calculate the ground state properties
of the system hence deviating us from exactness. Fortunately, the approximations
for the exchange-correlation energy make the quantum mechanical
treatement of complex materials tractable with a reasonable accuracy. Few of
the well know approximations are the local density approximation (LDA) 8,
the generalized gradient approximation (GGA) 9, and hybrid functionals e.g.
HSE06 10, 11. A brief overview of the different approximations is given in
the following subsection.
2.2.1 Local (Spin) Density Approximation (L(S)DA) and
Generalized Gradient Approximation (GGA)
The local density approximation is the first approximation employed in the
density functional theory. It is built using the free electron gas as a model
system, and is therefore expected to perform well for systems with reasonably
homogeneous charge density. Since its inception it has been widely used and
has produced remarkable results. The exchange energy density under the
framework of the LDA can be written as:
X(n(r))LDA = −
3
4
3
1/3
n(r)1/3. (2.7)
The correlation part has been derived from quantum Monte Carlo calculations
and can be found in Ref. 6. Despite being quite succesful LDA occasionally
performs badly especially for the systems having very inhomogeneous charge