Keywords on Dnh MOs

The inorganic professor explained something to me today (twice) and I still don't really know what he's talking about. So I am just putting down what I remember so I can look them up after the real analysis final. Or possibly before, as I am easily distracted.

Two disparate parts, really.

First: for that hexavalent allene-like carbon compound? Deriving the MO for that uses the exact same procedure as for say, SF₆, except the energy separations are different (the C-O "bond" is very weak). D_4h symmetry, by the way, after ignoring the aromatics. The MOs should correspond well with the Lewis structure for the molecule.

Second: the LGOs for the pi system of benzene has the representation a_2u + b_2g + e_1g + e_2u. Both b_2g and e_2u lack a corresponding basis function, so it's not as easy to sketch the MOs for the LGOs following these two symmetries. Of course, everyone kind of knows what the pi system for benzene (and other aromatic/antiaromatic compounds) look like from organic anyway, but still.
There are three ways to go:

1. Trial and error. Fine for small molecules.
2. Do it algebraically via a "projection operator". Which is out in the literature somewhere, though not the textbook. Time to borrow Cotton again.
3. "Symmetry reduction", where one notes that any D_nh point group is a subgroup of D_infinity-h and the irreducible representations in the latter consists of cross-sections of atomic orbitals (s, p, d, f, ...). Mapping the LGOs using this (using the corresponding angle instead of an arbitrary &phi ) should give information as to the phasing of each contributing AO in the overall LGO of interest.