CHEM01200604005 A. K. Pathak - Homi Bhabha National Institute

where the KS operator
KS
h is given by,
KS 1 z ρ(2)
h =− ∇ − + dr + V
2
2 α
1 ∑ 2 xc
r
∫ (11)
α 1α
r12
And the exchange-correlation potential V
xc
, is found by the functional derivative of
E
xc
V
= δ ρ δρ
(12)
xc
Exc[ ]/
In DFT the most important part is to derive the functional form of the exchangecorrelation
functional. The hybrid density functional, namely B3LYP (Becke’s three
parameter exchange and Lee-Yang-Parr correlation functional), BHHLYP (50% Slater
and 50 % Hartree-Fock exchange and correlation due to Lee-Yang-Parr) etc are the most
popular. They are derived by the adiabatic connection method (ACM), where the
fictitious non interacting system connects to the real system i. e. fully interacting systems.
So these functionals are called hybrid functionals. 37
1.3.4. Basis Set
Basis set is the mathematical functions to represent the wavefunctions of electrons
of an atom. The basis sets having proper ‘cusp’ at the nuclear center are closely resemble
the hydrogenic orbitals. The functional form of such basis set is
ϕ i
Ax y z e ς
2 2 2 1/2
i j k − ( x + y + z )
= (13)
where A is normalization constant, ζ is an exponent and depends on the atomic number,
and i, j, and k are non-negative integers that indicate the nature of the orbital in a
Cartesian sense. They are known as Slater type orbitals (STOs). It is very difficult to
evaluate multicentered four-index integrals with STOs. So, to overcome this problem,
14