How to interpret orbital energies.

The Fock equation is an eigenvalue equation with solutions that are called orbitals and eigenvalues that are called orbital energies, or, for molecules, also MO energies. Strictly speaking the orbital energies arise from a mathematical derivation that yields the orbitals to form a Slater determinant with the lowest possible electronic energy of an atom, ion, or molecule. The derivation is outside the scope of this course, but, as the objective of it is the orbitals, the orbital energies initially seem just an irrelevant by-product.

However, it turns out that there are various ways to interpret the orbital energy. One interpretation is to regard an orbital energy as the energy of the electron occupying the corresponding orbital. This interpretation is questionable. As the electrons interact with each other, it is rather arbitrary how to assign the energy of this interaction to the different electrons. Moreover, the sum of the energies of the occupied orbitals is not the electronic energy. This sum is on the other hand a reasonable approximation for the total energy of a molecule (see ``The energy of a molecule'').

A better interpretation is based on Koopmans theorem. This theorem states that to remove an electron from an atom, ion, or molecule an energy is required that is equal to minus the energy of the orbital that the electron occupies. This can also be stated as that the energies of occupied orbitals equal minus the ionization potentials. Koopmans theorem also states that when an electron is added to a system an energy is obtained equal to minus the energy of the orbital that the added electron occupies: i.e., the energies of orbitals that are not occupied (so called virtual orbitals) are equal to minus the electron affinities. Koopmans theorem can be proven rigorously provided one assumes that the orbitals do not change when an electron is added or removed. This is in general not the case. Koopmans theorem gives ionization potentials that agree reasonably with experiments, but electron affinities are predicted less well.

Orbital energies are also used to draw MO diagrams. In this context we also use a generalization of orbital energy. This generalization is similar to the definition of an expectation value of the Hamiltonian. An orbital energy is only defined for orbitals that are solutions of the Fock equation. We often also deal with orbitals that are not solutions of this equation. These orbitals appear in MO diagrams as well, and we would like to speak about the energy of such orbitals. We can do that by defining as the energy of an arbitrary orbital j the quantity áj|F|jñ/áj|jñ. We can show that when j is a solution of the Fock equation (i.e., Fj = ej) that áj|F|jñ/áj|jñ = e.

In general, we can write an arbitrary orbital j as a linear combination of solutions of the Fock equation: i.e.,