Transforming the molecular orbitals.

The theory of the chemical bond uses MO's so extensively, and MO's are such an integral part of this theory, that one might easily forget that MO's are a purely theoretical construct and that they do not correspond to anything real. The fundamental quantity in quantum chemistry is the all-electron wave function. We use MO's to construct these wave function via Slater determinants, but that is something that we do, not because the laws of physics request it, but because that makes working with wave functions easier. In fact, even if we construct a Slater determinant from MO's, we can do this in many ways. There is an infinite number of sets of MO's, each of which give exactly the same Slater determinant. It is up to us to choose which set to use, and we make this choice based on what is most convenient to us.

To see how different sets of MO's can give the same Slater determinant, we have to look at the properties of a determinant. A n×n matrix A consists of n² numbers

A Slater determinant has the form

If we approximate an all-electron wave function by a Slater determinant and use the variation principle, then we get the Fock equation. There are many ways to choose the orbitals that give the Slater determinant with the lowest energy as we have just seen. Therefore it may be a bit of a surprise that the Fock equation gives unique and well-defined orbitals. The reason for this can be found in the derivation of the Fock equation. At a certain point in that derivation a choice is made for a particular set of orbitals. This set simplifies the rest of the derivation. Although how this is done is beyond the scope of this course, it is important to realize that this has been done. It is also important to know that we can only interpret the energies of the orbitals that are solutions of the Fock equation as ionization potentials and electron affinities by using Koopmans's theorem.

The orbitals that the Fock equation gives are not always convenient. In general, it is not possible to assign an orbital to a single bond. Orbitals that solve the Fock equation are often spread out over more than one bond. For many molecules it is possible to make a unitary transformation of the orbitals that form the Slater determinant to get new orbitals that are each localized at bonds. (Note that you should only use the orbitals that form the Slater determinant, and not the unoccupied orbitals that you also get from the Fock equation.) This turns out to be very useful to explain the structure of a molecule. For some molecules (e.g., benzene) it is not possible to make such a transformation. Such molecule have special properties, because of this. One one talks about delocalization.