76
4 Representations
Fig. 4.4 Tetrahedral SALCs
in methane. The open circles
are positive and the filled
circles are negative. The T 2
orbitals match the sign
pattern of central p-orbitals
Using reciprocity, we can thus immediately infer the SALC symmetries of the hydrogen basis functions by selecting the C 3v irreps that subduce a in the case of the
1s orbitals and b in the case of the bending coordinates:
Hydrogen 1s : aC s ↑ C 3v = A 1 + E
Hydrogen φ : bC s ↑ C 3v = A 2 + E
(4.84)
Some induction schemes for σ, π, and δ orbital basis sets on C nv sites of polyhedral complexes are to be found in Appendix D. In addition to the Frobenius theorem, there is also a stronger result for induction theory based on the concept of a
fiber bundle. This requires the coupling of representations and will be considered in
Sect. 6.9.
4.7 Application: The sp 3 Hybridization of Carbon
Methane is the prototype of the saturated aliphatic hydrocarbons. The four hydrogen
atoms occupy the corners of a regular tetrahedron, as in Fig. 4.4. Their site symmetry
is C 3v .The1s atomic orbital on hydrogen is totally symmetric in the site group. The
symmetries of the corresponding SALCs can of course be obtained by the standard
character procedure, as for the case of ammonia, but we might as well directly obtain
them by induction:
Γ(a 1 C 3v ↑ T d ) = A 1 + T 2
(4.85)
The corresponding SALCs can easily be projected:
|A 1 | T 2x | T 2y | T 2z
=
|1s A | 1s B | 1s C | 1s D
× T
(4.86)
4 Representations
Fig. 4.4 Tetrahedral SALCs
in methane. The open circles
are positive and the filled
circles are negative. The T 2
orbitals match the sign
pattern of central p-orbitals
Using reciprocity, we can thus immediately infer the SALC symmetries of the hydrogen basis functions by selecting the C 3v irreps that subduce a in the case of the
1s orbitals and b in the case of the bending coordinates:
Hydrogen 1s : aC s ↑ C 3v = A 1 + E
Hydrogen φ : bC s ↑ C 3v = A 2 + E
(4.84)
Some induction schemes for σ, π, and δ orbital basis sets on C nv sites of polyhedral complexes are to be found in Appendix D. In addition to the Frobenius theorem, there is also a stronger result for induction theory based on the concept of a
fiber bundle. This requires the coupling of representations and will be considered in
Sect. 6.9.
4.7 Application: The sp 3 Hybridization of Carbon
Methane is the prototype of the saturated aliphatic hydrocarbons. The four hydrogen
atoms occupy the corners of a regular tetrahedron, as in Fig. 4.4. Their site symmetry
is C 3v .The1s atomic orbital on hydrogen is totally symmetric in the site group. The
symmetries of the corresponding SALCs can of course be obtained by the standard
character procedure, as for the case of ammonia, but we might as well directly obtain
them by induction:
Γ(a 1 C 3v ↑ T d ) = A 1 + T 2
(4.85)
The corresponding SALCs can easily be projected:
|A 1 | T 2x | T 2y | T 2z
=
|1s A | 1s B | 1s C | 1s D
× T
(4.86)