7.2 Application: Crystal-Field Potentials
167
To obtain this result, the following sum-rule was used, which is obtained by carrying
out a straightforward division:
N
n=0
r
n =
1 − r N
1 − r
(7.8)
As an example, for ℓ = 1, the rotation matrix corresponds to the matrix O(R) in Eq.
(7.3). Its trace is given by
χ
p (C α ) = 3 − 4sin
2 (α/2) =
sin(3α/2)
sin(α/2)
(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
ˆ
ıY ℓm ℓ = (−1)
ℓ Y ℓm ℓ
(7.10)
Finally, we also remind that the definition of the spherical harmonics in the standard
phase convention implies complex conjugation, as
¯
Y ℓm ℓ = (−1)
m ℓ Y ℓ−m ℓ
(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:
V CF =
L
e 2 Z L
4πǫ 0 |R L − r|
(7.12)
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 −eZ 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 V CF =
L
e 2 Z L
4πǫ 0
∞
ℓ=0
m ℓ =+ℓ
m ℓ =−ℓ
4π
2ℓ + 1
r ℓ
R ℓ+1 Y ℓm ℓ (θ, φ) ¯
Y ℓm ℓ (θ L ,φ L )
(7.13)