Chemistry Reference
In-Depth Information
summing up weighted squares of the DKH orbitals. The PCE for any operator
O
is
given by:
C
D
E
D
E
C
L
O
LL
L
L
L
ðÞ¼C
C
O
LL
PCE O
;
(27)
where
O
LL
denotes the upper left block of
O
[see (
25
)], while
O
LL
denotes the upper
left block of the
transformed operator O
. The notation reads then as DKH(
n
,
m
),
with
n
being the order of the DKH transformation of the wave function and
m
the
order of the DKH transformation of the property operator. If no property operator
is used, the notation is simply given by DKH
n
, with
n
being the order of the
transformation of the wave function.
A second way to reduce the four-component Dirac Hamiltonian to a two-
component Hamiltonian is given by the regular approximation approach, which
was introduced in 1986 by Heully et al. [
53
], Durand [
54
], and Chang et al. [
55
], and
rediscovered by van Lenthe, van Leeuwen, Baerends, and Snijders [
34
,
56
-
59
]. It
relates the small component of the four-component wave function via the energy-
dependent
X
-operator to the large component. The starting point for the regular
approximation is the Dirac equation in split notation after applying an energy shift
of
m
e
c
2
:
h
@
@
s pc
i
þ
Vc
i
¼
m
e
c
2
c
i
c
i
t
;
(28)
|
{z
}
!e
i
h
@
@
s pc
i
2
m
e
c
2
c
i
þ
Vc
i
¼
m
e
c
2
c
i
:
c
i
t
(29)
|
{z
}
!e
i
The energy-dependent
X
(
e
i
) operator is obtained from the lower part of the Dirac
equation as:
1
c
s p
s p
2
m
e
c
2
c
e
i
c
i
¼
c
i
¼
c
i
1
þ
2
m
e
c
2
2
m
e
c
2
e
i
V
V
V
|
{z
}
X eðÞ
(30)
k
V
1
k
s p
2
m
e
c
2
c
e
i
c
i
¼
V
2
m
e
c
2
¼
0
and can then in the next step be expanded in a geometric series with the expansion
parameter
e
i
/(
V
2
m
e
c
2
). The expanded form of the
X
-operator is then inserted in
the upper part of the Dirac equation and yields:
"
#
!
k
V
1
k
c
2
ð
s p
Þ
e
i
c
i
ðÞ¼e
i
c
i
V
þ
ð
s p
Þ
ðÞ:
(31)
2
m
e
c
2
V
2
m
e
c
2
¼
0
