6.9 Induction Revisited: The Fibre Bundle
149
This is precisely the set of fluorine displacements that we constructed in Sect. 4.8 in
order to describe the vibrational modes of UF 6 . One remarkable result of induction
theory is that the mechanical representation can also be obtained as the direct product of the positional representation and the translational representation, T 1u ;thisis
the representation of the three displacements of the centre of the cluster.
Γ mech = T 1u × (A 1g + E g + T 1u )
= T 1u + (T 1u + T 2u ) + (A 1g + E g + T 1g + T 2g )
(6.125)
It is as if the displacements of the central point of the octahedron were relocated to
every ligand site. The elementary function space of the displacements of the central
atom, which transforms as the translational irrep, T 1u , is called the standard fibre.
This fibre is attached to every site of the cluster, and the set of these fibres is the
fibre bundle. The action of the group permutes fibres of the bundle. The following
induction theorem holds:
Theorem 14 Consider a standard fibre, consisting of a function space that is invariant under the action of the group. In a cluster of equivalent sites, we can form
a fibre bundle by associating this standard fibre with every site position. The induced representation of the fibre bundle is then the direct product of the irrep of the
standard fibre with the positional representation.
For V being the representation of the standard fibre, T 1u in our example, and P
the positional representation of the set of equivalent sites in the molecule, one has
for the induced representation:
Γ
{V}H ↑ G
= V × P(H ↑ G)
(6.126)
For a proof of this theorem, we refer to the literature [19, 20]. The theorem is not
only applicable to molecular vibrations but is also directly in line with the LCAO
method in molecular quantum chemistry. In this method the molecular orbitals
(MOs) are constructed from atomic basis sets that are defined on the constituent
atoms. An atomic basis set, such as 3d or 4f , corresponds to a fibre, emanating, as
it were, from the atomic centre. Usually, such basis sets obey spherical symmetry,
since they are defined for the isolated atoms. As such, they are also invariant under
the molecular point group [21]. As an example, a set of 4f polarisation functions on
a chlorine ligand in a RhCl
3−
6 complex is itself adapted to octahedral symmetry as
a 2u + t 1u + t 2u . This representation thus corresponds to V.IntheC 4v site symmetry
these irreps subduce: a 1 + b 1 + b 2 + 2e. According to the theorem, the LCAOs based
on the 4f orbitals thus will transform as:
Γ
{a 1 + b 1 + b 2 + 2e}C 4v ↑ O h
= (a 2u + t 1u + t 2u ) × (a 1g + e g + t 1u )
= a 1g + a 2g + 2e g + 2t 1g + 3t 2g + a 2u + e u + 3t 1u + 3t 2u (6.127)
149
This is precisely the set of fluorine displacements that we constructed in Sect. 4.8 in
order to describe the vibrational modes of UF 6 . One remarkable result of induction
theory is that the mechanical representation can also be obtained as the direct product of the positional representation and the translational representation, T 1u ;thisis
the representation of the three displacements of the centre of the cluster.
Γ mech = T 1u × (A 1g + E g + T 1u )
= T 1u + (T 1u + T 2u ) + (A 1g + E g + T 1g + T 2g )
(6.125)
It is as if the displacements of the central point of the octahedron were relocated to
every ligand site. The elementary function space of the displacements of the central
atom, which transforms as the translational irrep, T 1u , is called the standard fibre.
This fibre is attached to every site of the cluster, and the set of these fibres is the
fibre bundle. The action of the group permutes fibres of the bundle. The following
induction theorem holds:
Theorem 14 Consider a standard fibre, consisting of a function space that is invariant under the action of the group. In a cluster of equivalent sites, we can form
a fibre bundle by associating this standard fibre with every site position. The induced representation of the fibre bundle is then the direct product of the irrep of the
standard fibre with the positional representation.
For V being the representation of the standard fibre, T 1u in our example, and P
the positional representation of the set of equivalent sites in the molecule, one has
for the induced representation:
Γ
{V}H ↑ G
= V × P(H ↑ G)
(6.126)
For a proof of this theorem, we refer to the literature [19, 20]. The theorem is not
only applicable to molecular vibrations but is also directly in line with the LCAO
method in molecular quantum chemistry. In this method the molecular orbitals
(MOs) are constructed from atomic basis sets that are defined on the constituent
atoms. An atomic basis set, such as 3d or 4f , corresponds to a fibre, emanating, as
it were, from the atomic centre. Usually, such basis sets obey spherical symmetry,
since they are defined for the isolated atoms. As such, they are also invariant under
the molecular point group [21]. As an example, a set of 4f polarisation functions on
a chlorine ligand in a RhCl
3−
6 complex is itself adapted to octahedral symmetry as
a 2u + t 1u + t 2u . This representation thus corresponds to V.IntheC 4v site symmetry
these irreps subduce: a 1 + b 1 + b 2 + 2e. According to the theorem, the LCAOs based
on the 4f orbitals thus will transform as:
Γ
{a 1 + b 1 + b 2 + 2e}C 4v ↑ O h
= (a 2u + t 1u + t 2u ) × (a 1g + e g + t 1u )
= a 1g + a 2g + 2e g + 2t 1g + 3t 2g + a 2u + e u + 3t 1u + 3t 2u (6.127)