Chemistry Reference
In-Depth Information
To obtain this result, the following sum-rule was used, which is obtained by carrying
out a straightforward division:
N
−
r
N
1
1
r
n
=
(7.8)
−
r
n
=
0
As an example, for
=
1, the rotation matrix corresponds to the matrix
O
(R)
in Eq.
(
7.3
). Its trace is given by
sin
(
3
α/
2
)
sin
(α/
2
)
χ
p
(C
α
)
=
4sin
2
(α/
2
)
=
3
−
(7.9)
Improper rotations can always be written as the products of a proper rotation and
the inversion operation. The resulting character is then given by the product of the
rotational character in Eq. (
7.7
), times the parity of the spherical harmonics, which
is given by
1
)
Y
m
ıY
m
=
ˆ
(
−
(7.10)
Finally, we also remind that the definition of the spherical harmonics in the standard
phase convention implies complex conjugation, as
Y
m
=
(
−
1
)
m
Y
−
m
(7.11)
In Sect.
A.2
the character tables for the groups
O(
3
)
and
SO(
3
)
are given. The
subduction relations are listed in Sect.
C.1
.
7.2 Application: Crystal-Field Potentials
Model treatments of transition-metal and lanthanide complexes are essentially based
on the
d
n
or
f
n
open-shell states of the central metal atom, which are perturbed by
the electrostatic field of the surrounding ligating groups:
e
2
Z
L
4
π
0
|
V
CF
=
(7.12)
R
L
−
r
|
L
This term describes the electrostatic repulsion between an electron residing in the
metal orbital at a position
r
and a negatively charged ligand, with charge
e
Z
L
at
a position
R
L
. In crystal-field theory the electrostatic field of the surroundings is
written as an expansion in spherical harmonic operators, as these elements will be
evaluated in the
d
or
f
function space of the central metal atom. Series expansion
for distances
r<R
L
yields
−
∞
m
=+
e
2
Z
L
4
π
0
r
4
π
2
R
+
1
Y
m
(θ,φ) Y
m
(θ
L
,φ
L
)
V
CF
=
(7.13)
+
1
L
=
0
m
=−
