CHEM01200604005 A. K. Pathak - Homi Bhabha National Institute

terms, the energy can be written in terms of only six spatial coordinates. This has
prompted the search for a function which has fewer variables than the wave function and
also can be used to calculate the energy of the systems. Unfortunately, no convenient
method like the variational principle is developed to determine the density matrix without
first constructing the wave function. Hohenberg and Kohn (HK) shows that the ground
state wave function, energy and other molecular electronic properties are uniquely
calculated from the electron density ρ(x,y,z), which is a function of only three variables.
The ground state energy E 0 is a function of ρ and can be writen as, E 0 = [ρ(x,y,z)], where
the square bracket denotes the functional relation. But HK principle does not show how
to calculate E 0 from ρ or hoe to find ρ without first calculating the wave function. In
order to solve this problem Kohn and Sham (KS) proposed a self consistent method.
They showed that the ground state electronic energy would be given by, 37
1 z ρ(1) 1 ρ(1) ρ(2)
E χ (1) | | χ (1) dν dν dν E [ ρ]
N
2
α
0
=− 〈
i
∇
1 i
〉−
1+ 1 2
+
xc
2 i=
1 α r1 α
2 r12
∑ ∑∫ ∫∫ (8)
Where, the KS orbital χ
i
are found by the procedure discussed below and the exchange –
correlation energy, Exc[ ρ ] is a function of ρ. The symbols χ
i
(1) and ρ(1) indicates that
χ
i
and ρ are taken as function of the spatial coordinates of electron 1. Kohn and Sham
also showed that the exact ground state ρ could be found from the expression given
below,
ρ
N
2
= ∑ | χi
|
(9)
i=
1
KS orbitals are found by solving the one electron equations,
KS
h (1) χ (1) = ε (1) χ (1)
(10)
i i i i
13