Chemistry Reference
In-Depth Information
ð
d
3
r
c
KS
i
h
2
2
m
e
r
r ðÞ
r r
0
dE
XC
½
dr ðÞ
2
i
ðÞ¼e
i
c
KS
þ
v
ext
ðÞþ
j
þ
ðÞ;
(54)
i
j
which are obtained using the Lagrange method of undetermined multipliers to the
variational procedure applied to the energy expression. The KS equations are one-
electron equations that exhibit many similarities to the Hartree-Fock equations
which are the basis of almost all wave function-based methods.
There exists a relativistic generalization of DFT called current DFT, which is
based on fundamental relativistic concepts. The first requirement for this extension
is a generalization of the Hohenberg-Kohn theorems which has been given by
Rajagopal and Callaway [
67
]. The functionals then depend not only on the electron
density
r
(
r
) but also on the current density
j
(
r
). Both combine to the relativistic
4-current
j
(
cr
,
j
). The total electronic energy in a four-component Kohn-Sham
model then reads:
E
tot
r; j
½
¼
T
s
r; j
½
þ
V
ext
r; j
½
þ
J r; r
½
þ
E
XC
r; j
½
:
(55)
Following Rajagopal and Callaway [
67
], the most general form of the relativistic
KS-DFT equations reads:
h
i
c
i
e
c
A
eff
m
e
c
2
c
a p þ
ðÞ
þ b
ef
eff
ðÞ
ðÞ¼e
i
c
i
ðÞ;
(56)
where the Dirac Hamiltonian is clearly visible and supplemented by effective
electromagnetic potentials defined as:
ð
d
3
r
0
r rðÞ
r r
0
e
c
c
dE
XC
r
; ½
f
eff
ðÞ¼f
ext
ðÞ
j
þ
dr ðÞ
;
j
ð
d
3
r
0
(57)
e
c
jðr
0
Þ
r r
0
c
dE
XC
r
; ½
A
eff
ðÞ¼A
ext
ðÞ
j
þ
djðÞ
;
j
where the external potentials
f
ext
and
A
ext
represent the interaction between the
electron and the nuclei in the case of isolated molecules.
4 Analysis of Approximate Electron Densities
Approximate wave functions and the specific choice of the Hamiltonian operator
can cause errors in the calculation of the electron density, which have to be
investigated systematically. The first part of this section focuses on the effect of
approximate Hamiltonian operators, whereas the second part deals with the effects
of the correlated motion of the electrons on the electron density.
