Band gaps

Traditionally, as a quantum chemist, you leave the talking about orbitals to physicists, organic chemists and all the others denying the many-particle nature of reality. And also the periodic analogues of orbitals, the "bands," are seen more as artifacts of Kohn-Sham theory than as actual physical entities. But these days there are several reasons to jump off this high horse. Firstly, experiment: with orbtial imaging by attosecond spectroscopy and scanning tunneling microscopy it becomes harder and harder to argue that orbitals are just meaningless mathematical quantities. The second one is the increasing size of the molecules considered. Large polymeric molecules start to resemble "small solids" in the sense that many of their excited state properties can be understood in a one-particle (or at least one-quasi-particle) picture. That is why I spent some time trying to understand what bands actually are and what the analogues of HOMO and LUMO are in many particle theory (starting with a chapter in this review by Truhlar).

In a Hartree-Fock calculation it loooks like it is shown below. There are a number of doubly occupied MOs, including the highest occupied MO (HOMO) and a number of unoccupied ones including the LUMO. According to Koopmans' theorem the energy of the HOMO is related to the ionization potential (IP). Precisely: if the orbitals do not relax after the ionization process, then the Hartree-Fock energy of the ion will differ from the neutral system exactly by the HOMO energy. The Koopmans IPs are in many cases quite accurate because of some favorable error compensation between electron correlation and orbital relaxation. In a less rigorous way one may relate the LUMO energy to the electron affinity (EA). This derives from applying Koopmans' theorem to the anion. However, this only works if the LUMO of the neutral system is in fact a good approximation to the singly occupied MO of the anion, which it is not in many cases.

Finally, one considers the HOMO-LUMO gap as a central quantity in molecular orbital theory and calls it "band gap" in the periodic equivalent.

In many-particle calculations there is no direct way to obtain a similar picture. There are some ways to recover orbitals out of a many particle theory. But they are not unique. For example you may diagonalize the 1-particle density matrix to obtain the natural orbitals. These will have well defined occupations but it is not possible to assign energies to them. On the other hand we can diagonalize effective Fock matrices to obtain orbitals with energies but no well defined occupations.

 However, one can try to recover a band picture by talking in terms of physical observables. The IP is the energy difference between the ground state of the cation D₀ (M⁺) and the neutral molecule  S₀ (M) . The EA is obtained in the same way considering the anion  D₀ (M^-). For completeness I am also showing the (singlet) excitation energy E_exc derived from the energy of the first excited state  S₁ (M). These are physical properties of the molecule, which are well defined, no matter what method you use to describe them.

As a side remark: these quantities may in general either be computed by energy differences from separate calculations as shown here or alternatively by direct propagator approaches.

Now, if someone asks us to tell them the band gap of our molecule, we can simply subtract the EA from the IP and have a well defined quantity. Alternatively, we may tell them the excitation energy, which is sometimes referred to as "optical band gap", where the difference between those two is called the "exciton binding energy".

Finally, if you really want to talk to a physicist, you should also understand the concept of the Fermi energy. For a metal the Fermi energy is simply coinciding with the highest occupied level. In an insulator it is somewhere in the "forbidden region" between the HOMO and the LUMO, but it is not so clear to me where.