Chemistry Reference
In-Depth Information
(Noga et al., 2001). It is hoped that in this way it will be possible to obtain
'benchmarks' in the calculation of atomization energies, at least for the
small molecules of the first row.
8.6 DENSITY FUNCTIONAL THEORY
DFTwas initially developed by Hohenberg and Kohn (1964) and by Kohn
and Sham (1965), and is largely used today by the quantum chemical
community in calculations on complex molecular systems. It must be
stressed that DFT is a semiempirical theory accounting in part for electron
correlation.
The electronic structure of the ground state of a system is assumed to be
uniquely determined by the ground state electronic density
r
0
(
r
), and
a variational criterion is given for the determination of
r
0
and E
0
from
an arbitrary regular function
r
(
r
). The variational optimization of the
energy functional E[
r
] constrained by the normalization condition:
ð
d
r rðrÞ¼
N
E
½r
E
½r
0
;
ð
8
:
11
Þ
shows that the functional derivative
8
is nothing but the effective one-
KS
electron Kohn-Sham Hamiltonian h
:
d
E
½r
drðrÞ
¼
1
2
r
KS
þ
V
eff
ðrÞ¼
h
2
ðrÞ
ð
8
:
12
Þ
where
V
eff
ðrÞ¼
V
ðrÞþ
J
ðrÞþ
V
xc
ðrÞ
ð
8
:
13
Þ
the effective potential at
r
is the sum of the electron-nuclear attraction
potential V, plus the Coulomb potential J of the electrons of density
,
plus the exchange-correlation potential V
xc
for all the electrons. It is seen
that the effective potential (8.13) differs from the usual HF potential by
the undetermined correlation potential in V
xc
. Since V
xc
cannot be defined
exactly, it can only be given semiempirical evaluations. Most used in
applications is the Becke-Lee-Yang-Parr (B-LYP) correlation potential.
Kohn-Sham orbitals
r
KS
i
f
ðrÞ
, i
¼
1
;
2
; ...;
n, are then obtained from the
8
The Euler-Lagrange parameter
l
of the constrained minimization.
