7.1 The Spherical-Symmetry Group
165
parameters that give rise to the same matrix O(R):
ˆ
R(α,n x ,n y ,n z )
ˆ
R(−α, −n x , −n y , −n z )
ˆ
R(−2π + α, n x ,n y ,n z )
ˆ
R(2π − α, −n x , −n y , −n z )
(7.4)
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 vector. 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 + x 2 + y 2 is a radial function, which is totally symmetric under rotations. Taking out this root leaves five components, which are irreducible and correspond to the five d orbitals, shown in Table 7.1. This result parallels the cubic
[T 1u ] 2 = A 1g + E g + T 2g coupling [2].
When extending these results to the nt hp o w e ro ft h ep-irrep, symmetrization 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,
{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 2u + 2T 1u + T 2u . 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 + y 2 + 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 book of angular momentum arithmetic and the underlying 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 ℓm ℓ },f o r
m ℓ =−ℓ, −ℓ + 1,...,ℓ− 1,ℓ, 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 equivalent. 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 See, e.g., [3, 4].
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parameters that give rise to the same matrix O(R):
ˆ
R(α,n x ,n y ,n z )
ˆ
R(−α, −n x , −n y , −n z )
ˆ
R(−2π + α, n x ,n y ,n z )
ˆ
R(2π − α, −n x , −n y , −n z )
(7.4)
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 vector. 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 + x 2 + y 2 is a radial function, which is totally symmetric under rotations. Taking out this root leaves five components, which are irreducible and correspond to the five d orbitals, shown in Table 7.1. This result parallels the cubic
[T 1u ] 2 = A 1g + E g + T 2g coupling [2].
When extending these results to the nt hp o w e ro ft h ep-irrep, symmetrization 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,
{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 2u + 2T 1u + T 2u . 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 + y 2 + 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 book of angular momentum arithmetic and the underlying 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 ℓm ℓ },f o r
m ℓ =−ℓ, −ℓ + 1,...,ℓ− 1,ℓ, 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 equivalent. 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 See, e.g., [3, 4].