Chemistry Reference
In-Depth Information
parameters that give rise to the same matrix
O
(R)
:
R(α,n
x
,n
y
,n
z
)
R(
−
α,
−
n
x
,
−
n
y
,
−
n
z
)
R(
−
(7.4)
2
π
+
α,n
x
,n
y
,n
z
)
R(
2
π
−
−
−
−
α,
n
x
,
n
y
,
n
z
)
The transformations of the standard vector form the fundamental irrep of spherical
symmetry. All other irreps can be constructed by taking direct products of this vec-
tor. In particular, the
spherical harmonic functions
can be constructed by taking fully
symmetrized powers of the vector. The symmetrized direct square of the
p
-functions
yields a six-dimensional function space with components:
z
2
,x
2
,y
2
,yz,xz,xy
.
This space is not orthonormal: the components are not normalized, and the first
three components do overlap. In fact, the space is reducible since the sum of the
squares
z
2
{
}
+
y
2
is a radial function, which is totally symmetric under rota-
tions. Taking out this root leaves five components, which are irreducible and cor-
respond to the five
d
orbitals, shown in Table
7.1
. This result parallels the cubic
[
+
x
2
2
T
2
g
coupling [
2
].
When extending these results to the
n
thpowerofthe
p
-irrep, symmetriza-
tion will be governed by the irreducible representations of the corresponding
S
n
permutation group. The
f
-orbitals may be generated by the third power of
the
p
-irrep. Full symmetrization of the three components generates 10 functions,
{
T
1
u
]
=
A
1
g
+
E
g
+
z
3
,x
3
,y
3
,z
2
x,z
2
y,x
2
z,x
2
y,y
2
z,y
2
x,xyz
}
, which in cubic symmetry transform
as
A
2
u
+
T
2
u
. Again, this space is reducible and contains a
p
- and an
f
-subspace. The reduction is based on the removal of the totally symmetric trace.
Indeed, the combination
z(x
2
2
T
1
u
+
+
z
2
)
and its cyclic permutations reduce to the
fundamental
p
-vector. The remainder of the space is irreducible and corresponds
to the seven
f
-functions listed in Table
7.1
. Further explorations of the spherical
symmetry group opens the topic of angular momentum arithmetic and the under-
lying theory of Lie groups.
1
This is outside the present scope. We shall restrict
the treatment to indicating the subduction rules, which describe the decomposition
of the spherical irreps in point-group symmetries. To obtain these rules, we have
to derive the character of function space of the spherical harmonics,
+
y
2
{
Y
m
}
,for
m
=−
1
,
, under the proper and improper rotations of the point
group. We shall start by considering the proper ones first. It is sufficient to limit the
treatment to rotations around the
z
-axis since on a sphere all directions are equiva-
lent. Rotations around
z
will affect only the angular coordinate,
φ
, in the equatorial
plane and leave the azimuthal coordinate,
θ
, unchanged. The
φ
-dependence of the
,
−
+
1
,...,
−
1
See, e.g., [
3
,
4
].
