78
4 Representations
four-site structure is the square, but, from the tables in Appendix D, the induction
from the C 2v sites in a square-planar structure yields
Γ(a 1 C 2v ↑ D 4h ) = A 1g + B 2g + E u
(4.90)
This matches the symmetry of sp 2 d hybrids. It is thus not suitable for carbon, but
indeed describes the valence structure of square-planar transition-metal complexes
where d-orbitals are involved in the bonding.
4.8 Application: The Vibrations of UF 6
As we have already mentioned, representations not only apply to orbitals, but
equally well to vibrational coordinates. The function space in such a case consists
of a set of distortions. When applying symmetry operations we do not move the
atoms, but the distortions. Let us consider the vibrations of an octahedral complex,
such as UF 6 , which can be brought into the gas phase and which has been studied
in great detail since it is the carrier of uranium in the gas diffusion process for enrichment of nuclear fuel. The atoms are labeled as in Fig. 4.5, and on each atom
we define a local coordinate system that parallels the central system. In the notation adopted, Y 2 is a variable for a displacement of atom 2 over a distance
in the positive y-direction. A symmetry operation such as ˆ
C
z
4 transforms this Y 2
into −X 3 . The seven atoms give rise to 21 distortions, which include six spurious modes, corresponding to three translations and three rotations. The seven atoms
form two different orbits: the orbit containing the six fluoride ligands and the oneatom orbit of the central uranium atom. The displacements of one ligand can further
be separated into a radial or σ -mode and two tangential or π -modes, which, in the
C 4v site group, transform as a 1 and e, respectively. Altogether, the distortion space
thus contains three different basis sets: the central atom, the ligand σ -modes, and
the ligand π -modes. For each of these, the symmetry content may be determined by
directly applying the character theorem, or—for the case of the ligands—by using
induction. The three displacements of the uranium atom transform as the T 1u irrep
of the central translation mode. The ligand inductions are as follows:
Fσ : Γ(a 1 C 4v ↑ O h ) = A 1g + E g + T 1u
Fπ : Γ(eC 4v ↑ O h ) = T 1g + T 2g + T 1u + T 2u
(4.91)
We can now determine the symmetry-adapted coordinates by applying the projection operators, but the results can be written down almost immediately by again
using the criterion of overlap with central symmetry functions. The A 1g , T 1u , and
E g + T 2g SALCs reflect the nodal patterns of central s, p, and d functions, respectively. The T 1g mode corresponds to the rotation and evidently consists of tangential
displacements of ligands in the equator perpendicular to the rotation axis. Finally,
the T 2u is a buckling mode, which has the symmetry of central f orbitals, viz.
4 Representations
four-site structure is the square, but, from the tables in Appendix D, the induction
from the C 2v sites in a square-planar structure yields
Γ(a 1 C 2v ↑ D 4h ) = A 1g + B 2g + E u
(4.90)
This matches the symmetry of sp 2 d hybrids. It is thus not suitable for carbon, but
indeed describes the valence structure of square-planar transition-metal complexes
where d-orbitals are involved in the bonding.
4.8 Application: The Vibrations of UF 6
As we have already mentioned, representations not only apply to orbitals, but
equally well to vibrational coordinates. The function space in such a case consists
of a set of distortions. When applying symmetry operations we do not move the
atoms, but the distortions. Let us consider the vibrations of an octahedral complex,
such as UF 6 , which can be brought into the gas phase and which has been studied
in great detail since it is the carrier of uranium in the gas diffusion process for enrichment of nuclear fuel. The atoms are labeled as in Fig. 4.5, and on each atom
we define a local coordinate system that parallels the central system. In the notation adopted, Y 2 is a variable for a displacement of atom 2 over a distance
in the positive y-direction. A symmetry operation such as ˆ
C
z
4 transforms this Y 2
into −X 3 . The seven atoms give rise to 21 distortions, which include six spurious modes, corresponding to three translations and three rotations. The seven atoms
form two different orbits: the orbit containing the six fluoride ligands and the oneatom orbit of the central uranium atom. The displacements of one ligand can further
be separated into a radial or σ -mode and two tangential or π -modes, which, in the
C 4v site group, transform as a 1 and e, respectively. Altogether, the distortion space
thus contains three different basis sets: the central atom, the ligand σ -modes, and
the ligand π -modes. For each of these, the symmetry content may be determined by
directly applying the character theorem, or—for the case of the ligands—by using
induction. The three displacements of the uranium atom transform as the T 1u irrep
of the central translation mode. The ligand inductions are as follows:
Fσ : Γ(a 1 C 4v ↑ O h ) = A 1g + E g + T 1u
Fπ : Γ(eC 4v ↑ O h ) = T 1g + T 2g + T 1u + T 2u
(4.91)
We can now determine the symmetry-adapted coordinates by applying the projection operators, but the results can be written down almost immediately by again
using the criterion of overlap with central symmetry functions. The A 1g , T 1u , and
E g + T 2g SALCs reflect the nodal patterns of central s, p, and d functions, respectively. The T 1g mode corresponds to the rotation and evidently consists of tangential
displacements of ligands in the equator perpendicular to the rotation axis. Finally,
the T 2u is a buckling mode, which has the symmetry of central f orbitals, viz.