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.

Conjugated Organic Polymers

There is another recent paper by us: "Electronically Excited States in Poly(p-phenylenevinylene): Vertical Excitations and Torsional Potentials from High-Level Ab Initio Calculations." We looked at Poly(p-phenylenevinylene) (PPV), which is a prototypical conjugated organic polymer to understand the behaviour of excitation energies with respect to different chain lengths, to analyze the structure of the exciton, and to evaluate the effect of torsions.

In the following picture you can see our excited state analysis methods (as described here) when applied to PPV. The model to understand this structure is to think of a hydrogen atom (or a "quasi-particle") in a one-dimensional box. First there are different translation states for the ground state of the hydrogen atom. These should be differentiated by a different number of nodal planes (shown in the first three cases). Then a different state of the "hydrogen atom" comes into play and there is a somewhat different shape. Unfortunately this method does not show any phases or signs but I still think one can recognize the nodal structure.

If there is a torsion in the center of the molecule, the situation changes. Now there are two equivalent, effectively decoupled, units. Therefore the excited states come in pairs, corresponding to the "+" and "-" linear combinations of the fragment states.

Referee reports

What is the better kind of referee report: a five line acceptance or a two page rejection?

If I get the first type, I tend to feel kind of disappointed. If the referee had only five lines to write, it means that you probably did not have anything interesting to say anyway and may as well not have done it at all.

But if someone takes the time to actually write two pages trying to dump everything on you that they can think of, then you know you are doing something interesting...

Photoluminescent Thiophene Derivatives

The next paper with me as a co-author just came out, actually: "Synthesis, Spectroscopy, and Computational Analysis of Photoluminescent Bis(aminophenyl)-Substituted Thiophene Derivatives" in collaboration with colleagues from the TU Vienna.

The idea was to simulate luminescence spectra of different thiophene derivatives and compare them to experimental results. The main methodological outcomes were that M06-2X proved to be a great functional for those substances and that state-specific PCM was superior to standard linear response PCM.

The next step will be to apply this methodology to a larger number of molecules.

Graphene Nanoribbons

Our Angewandte paper just came out: "The Multiradical Character of One- and Two-Dimensional Graphene Nanoribbons."

What we looked at is how open-shell character develops in polyacenes and bigger graphene nanoribbons. Some time ago I learned in basic organic chemistry that phenantren is more stable then anthracene. And interestingly this difference becomes enhanced when you move to longer chains. If you just keep adding rings in a linear fashion (forming polyacenes) you can go until five rings, reaching pentacene, and then things become really unstable. By contrast you can create longer stable phenacenes without any problem. The reason is that unpaired electrons accumulate at the zigzag edges, as shown in this TOC graphic.

Similar things also hold for the 2-dimensional systems and the precise shape of the edge decides whether or not the system obtains open-shell character. Apparently people have been discussing these things a lot. And since the paper is about graphene and contains the word "nano" we decided to try submitting it to Angewandte ...

There is also a German version: "Der Multiradikalcharakter ein- und zweidimensionaler Graphen-Nanobänder." Apparently the Gesellschaft Deutscher Chemiker likes to keep German as an active science language. It was kind of fun to translate it actually. Because the only German papers I ever read were from the very early times, like Hückel, Förster, Kuhn that write like quantum theory is something really new and special. So I felt a little bit like one of them. And actually I also got a free book from Wiley for my translation work: "Relativistic Quantum Chemistry".