Chemistry Reference
In-Depth Information
3.3 Electron Density Distribution for Different Choices of
Hamiltonian and Wave Function
The next step in the calculation of the electron density and the current density is to
choose and insert a Hamiltonian operator and an ansatz for the wave function into
the continuity equation (
5
), which contains the commutator of the Hamiltonian and
the density operator. First, we discuss the case in which the many-electron wave
function is approximated by a single Slater determinant. The electron density
r
(
)is
then calculated by inserting the definition of the Slater determinant given by (
8
). By
applying the Slater-Condon rules [
62
,
63
] for the calculation of matrix elements,
the integrals containing the Slater determinants collapse to a sum of one-electron
integrals:
r
*
+
X
N
d
ð
3
Þ
r ðÞ¼Y
ð
r
fg
Þ r
j
Y
ð
r
fg
Þ
i
¼
Y
ð
r
fg
Þ
ð
r r
i
Þ
Y
ð
r
fg
Þ
h
i¼
1
c
i
ðsÞ
D
E
X
N
ðÞd
ð
3
Þ
¼
c
i
ð
r s
Þ
;
(36)
i
¼
1
for which we choose the arbitrary integration variable
. The sum of these one-
electron integrals yields the trace of the density matrix, whose diagonal elements
define orbital densities:
s
D
c
i
E
X
N
ðÞd
ð
3
Þ
r ðÞ¼
c
i
ð
r s
Þ
ðÞ
i¼
1
X
X
N
N
c
i
2
¼
ðÞc
i
ðÞ¼
j
c
i
ðÞ
j
;
(37)
i¼
1
i¼
1
where
c
i
denotes the transposed and complex conjugate of the four-component
orbital
c
i
(if
c
i
denotes a one-component orbital, then the dagger
is replaced by a
star * for complex conjugation as transposing is inapplicable). Proceeding now with
the insertion of the DCB Hamiltonian in the right-hand side of the continuity
equation in (
5
), all multiplicative operators cancel in the commutator so that it
reads:
{
*
+
h
i
C
X
N
@ Crj C
h
i
i
p
i
; d
ð
3
Þ
¼
c
a
i
ð
r r
i
Þ
C
@
t
h
i¼
1
D
C
E
cN C a
1
d
ð
3
Þ
¼r
ð
r r
1
Þ
:
(38)
!
cN
hCja
1
d
ð
3
Þ
From this equation, we can define the current density
j
DCB
¼
ðr r
1
ÞjCi
for any type of four-component wave function. Hence, in the case of
