Chemistry Reference
In-Depth Information
3.4 Methodological Aspects: Density Functional Theory
and Current-Density Functional Theory
In the introduction, it was mentioned that the ground state of a system is fully
determined not only by its ground-stateelectronicwavefunction
C
0
but also by
its ground-state electron density
r
0
(
). Since the wave function depends on 3
N
spatial coordinates (plus
N
additional spin coordinates), and since it is a fairly
complicated object to handle, it would be preferable to have a theory which is
solely based on the electron density, depending only on three spatial coordinates.
For this reason, we give here a brief overview on DFT and its relativistic
extension, namely current DFT, which are in the Kohn-Sham formulation both
single-determinant methods. As discussed in the introduction, all properties of a
system (especially the total energy) can be expressed as functionals of
r
0
(
r
). The
major drawback of DFT is that the analytical expressions for some of the energy
contributions are not known, and thus some parts of the total energy must be
approximated. Contemporary DFT is therefore not suited for highly accurate
calculations, because the achievable accuracy strongly depends on the choice of
the approximate density functionals. On the other hand, DFT is a simple, compu-
tationally not very demanding method which even includes electron correlation
effects via an additional energy functional, and it allows one to perform calcula-
tions on large molecules.
The most widespread implementations of DFT are within the Kohn-Sham (KS)
formalism [
17
], in which an artificial reference system of noninteracting electrons is
introduced that yields exactly the same electron density as the interacting system.
The energy contributions are partitioned in the following way (in Hartree atomic
units):
r
E
tot
½¼
T
s
½þ
V
ext
½þ
J ½þ
E
XC
½rþ
E
NN
ð
d
3
r rðrÞ
X
N=
2
c
KS
i
þ
c
KS
i
2
¼
r
v
ext
ðrÞ
i¼
1
ðð
d
3
r
d
3
r
0
r
ðÞ
r
rðÞ
r r
0
þ
j
þ
E
XC
½þ
E
NN
:
(53)
j
with
T
s
[
r
] denoting the kinetic energy of the noninteracting reference system,
V
ext
[
r
] the external potential energy which is caused by the nuclei,
J
[
r
] the Coulomb
interaction of the electrons,
E
NN
the nuclear repulsion energy,
v
ext
the external
potential, and
E
XC
[
r
] being a sum of the nonclassical part of the electron-electron
interaction and the difference between the kinetic energy of the noninteracting
reference system and that of the interacting system. The exchange-correlation
functional is the only unknown term in this expression, and hence its approximation
determines the accuracy of the whole calculation. The (nonrelativistic) KS orbitals
are calculated from the Kohn-Sham equations:
