Molecular Geometry & Bonding
what the shape decides
Figure

Dipole selection rules in Td

For every pair of symmetry species, whether an electric dipole transition between them is allowed and along which polarisation. An entry is allowed exactly when the triple product of representations contains the totally symmetric one.
Dipole selection rules in Td. For every pair of symmetry species, whether an electric dipole transition between them is allowed and along which polarisation. An entry is allowed exactly when the triple product of representations contains the totally symmetric one.

One of the figures on representations: Character tables generated from a molecule's own operations, bases reduced in them, and what a descent in symmetry does to a level.

Three essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

In the essays

Exactly zero

Every electric dipole transition in Td, allowed or forbidden, with the polarisation printed where it is allowed. A dash is not a small number. It is an integrand that cancels in pairs under the twenty-four operations of the group, and the integral is zero for the same reason the overlaps above are.

Selection rules are one theorem

Every electric dipole transition in Td, allowed or forbidden, with the polarisation printed where it is allowed. A dash is not a small number: it is an integrand cancelling in pairs under the group’s twenty-four operations.

C₂ᵥ, small enough to check by hand. Four species, and the allowed entries are exactly the pairs whose product carries x, y or z. A reader who works one entry out with a pencil has done the whole of what group theory contributes to spectroscopy.

Why a d–d band is weak

Every pair of symmetry species in the octahedral group, with the dipole transitions between them marked allowed or forbidden and the polarisation given where one exists. The d orbitals span Eg and T2g; the cell where those two meet is empty.

The same grid for a tetrahedral complex. The E–T₂ cell, which is the d–d cell, is filled: three polarisations, no parity to forbid anything, because a tetrahedron has no centre of symmetry for a function to be even or odd about.

And the case where the restriction is gone entirely, for comparison with the tetrahedron. C3v has no inversion either, and seven of its nine cells carry a polarisation — with three species and no parity to conserve, nearly every pair is connected. Against 24 of a hundred in the octahedral grid, that is what a group without a centre looks like, and it is why the argument in this essay is about one operation rather than about symmetry in general.

Every figure · Every orbital, by what it encloses · All essays