Chemistry Reference
In-Depth Information
/(2
m
e
c
2
Truncation of the
X
-operator after the zeroth-order term
X
0
¼
c
s
·
p
V
)
leads then to the two-component ZORA Hamiltonian:
c
2
2
m
e
c
2
h
ZORA
¼ s p
V
s p þ
V
:
(32)
The order of the regular approximation is thus determined by the order of the
expansion of the energy-dependent
X
-operator in terms of
e
i
/(
V
2
m
e
c
2
). The
derivation of the ZORA Hamiltonian is related to a Foldy-Wouthuysen transfor-
mation [
60
,
61
]:
0
1
1
1
1
1
X
0
q
q
@
A
X
0
X
0
X
0
X
0
þ
þ
U
0
¼
;
(33)
1
1
1
1
q
X
0
q
X
0
X
0
X
0
X
0
þ
þ
yielding
1
1
1
1
U
0
h
D
U
1
q
q
¼
h
ZORA
:
(34)
0
X
0
X
0
X
0
X
0
þ
þ
The ZORA Hamiltonian is then obtained by truncating the expansion of the
q
1
X
0
X
0
2
X
0
X
0
þ
reciprocal square root 1
after the zeroth order
term which is simply unity. Since only the four-component wave function is
normalized and not
=
þ
¼
1
1
=
the large component,
it has to be normalized such that
q
1
X
0
X
0
c
L
describes the normalized ZORA wave function.
The DKH and the ZORA Hamiltonian reduce the number of components in the
wave function from four to two. So-called scalar-relativistic one-component meth-
ods are obtained by neglecting all spin-dependent terms in the DKH and the ZORA
Hamiltonian. A further approximation, which also reduces the dimension to one, is
given by the nonrelativistic limit of the Dirac equation, where the speed of light
approaches infinity. The result is the one-component many-electron Schr
c
ZORA
¼
þ
€
odinger
equation with the corresponding Hamiltonian:
X
X
X
X
M
N
M
M
I
2
m
I
þ
i
2
m
e
þ
Z
I
Z
J
e
2
R
I
R
J
p
p
H
non
rel
:
¼
j
j
I
¼
1
i
¼
1
I
¼
1
J
¼
I
þ
1
X
X
X
X
N
N
N
M
e
2
r
i
r
j
Z
I
e
2
r
i
R
I
þ
j
;
(35)
j
i
¼
1
j
¼
i
þ
1
i
¼
1
I
¼
1
being the standard reference for nonrelativistic calculations.
