Research proposals vs. research excellence

There is generally no reason to assume that the research you are funded to do should exactly coincide with the best possible research that you are capable of doing. Research is unpredictable by its very definition and new ideas, insights, results, etc., that change your optimal course of action, pop up all the time. This makes me wonder what is the higher imperative: Doing what you are funded to do, or doing the most excellent research you are capable of? Let's examine some reasons in favor of the former:

1. You should do what you are funded for simply because you signed a contract to so. End of discussion. But the point is: It is actually not in the best interest of the funding agency to hold you back in case you found a more exciting topic along the way. Are you obliged to comply with the funding agency against its own interest?
2. Your funding agency may not be interested in general research excellence but may have more narrow interests. But even in this case there could be ways to satisfy those more narrow interests outside of your original proposal. Should you grab that opportunity?
3. Your original proposal was certified by peer-review to be productive and worthwhile, switching to a different topic is too risky. But I don't think this makes sense either: It puts the opinion of your "peers" above yours about your own very special research topic.

If I look at my two most cited papers, then I find out that both were not part of a research proposal. The first one, concerned with polyradical character in graphene nanoribbons, was not in a proposal, since we did not expect that graphene nanoribbons would show polyradical character. The second one, about a wavefunction analysis strategy for excitons, was not in the proposal for my scholarship, since I had no idea at the start of my PhD that I would actually be doing that. Do I have to somehow feel bad for writing these papers and working on the related computer codes?

Is a research proposal a "lower bound" to your research and should you modify parts of it as you go along? Or is it cast in stone? Any opinions, experience?

Comparing Wavefunctions by their Overlap

Ever since starting in quantum chemistry I have been trying to avoid looking at orbitals. One reason is laziness. I just do not like sitting there clicking and waiting for all the orbitals to be rendered (even though this can be improved by using the proper scripts and programs). The other reason is a formal one: Orbitals, being one-body functions, can never tell us the whole truth about the many-body wavefunctions. Even worse, the same wavefunction may appear differently depending on the orbital set used to describe it (canonical orbitals, natural orbitals, natural transition orbitals, ...)

Assume that we performed two computations with different computational methods. When we look at the results we find out that, both, the molecular orbitals and wavefunction listings changed between the calculations. Does this mean that the two computations produced different wavefunctions? Not necessarily! The changes in the orbitals might be compensated by changes in the wavefunction expansion, at least in part. If we want to compare such wavefunctions we have to take into account the changes in the MOs and wavefunctions in a consistent fashion. In our newest Communication in J. Chem. Phys. "Unambiguous comparison of many-electron wavefunctions through their overlaps" we suggest to use the many-body wavefunction overlap for this task, i.e. the scalar product in the full many-body Hilbert space.

The outcome looks like this:

What we are doing is computing the 4 lowest excited states with CASSCF(12,9) and with MR-CIS(12,12). And then we compute the overlaps for all pairs of states and summarize them in pie charts. Every chart corresponds to one CASSCF wavefunction and the colors correspond to the MR-CIS wavefunctions. For example the second chart tells us that the Ψ₁' wavefunction at CASSCF has a 67% overlap with the Ψ₁ wavefunction at MR-CIS. But there are also smaller contributions of the Ψ₂ and Ψ₃ wavefunctions (as seen by the green and yellow bits).

The analysis gives us a quick overview of the relations between the wavefunctions computed at the different levels. There are two immediate conclusions: First, the overall state ordering is the same for both methods. Second, the wavefunctions are generally quite different, as seen by the large chunks of the pies missing.

The overlap code described here is actually a side product of a development we did for the nonadiabatic dynamics program SHARC (see this post). It will be released from the SHARC homepage as a standalone module, I hope soon ...

Entanglement Entropy of Electronic Excitations

There is more to excited states than meets the eye. Just looking at the orbitals will not tell you everything there is to know about the many-body wavefunctions. The purpose of my newest paper "Entanglement Entropy of Electronic Excitations," that just appeared in J. Chem. Phys., was to quantify the amount of information that is hidden from view. For this purpse, I used the idea of mutual information from quantum information theory.

The focus of this paper is the eigenvalue spectrum of the natural transition orbital (NTO) decomposition. There is information in the eigenvalue spectrum independent of the orbitals themselves. To illustrate the point, we can look at the first excited singlet state of two interacting ethylene molecules at 6.0 Å

and at 3.5 Å

The orbitals in both cases look similar but the eigenvalues λ₁ and λ₂ are different. For the larger separation both are equal at about 0.45. For the smaller separation, there is one dominant one at 0.86. Clearly, these are different wavefunctions, but what is the significance?

In the paper I am arguing that only the first case is consistent with the idea of a Frenkel exciton, i.e. two coupled local excitations. The second case can be seen as one homogeneous transition. This automatically means that there is admixture of charge transfer, since the orbitals are distributed evenly. And indeed when we apply our charge transfer measures, we find charge resonance character in the second case.

The whole formalism employed is somewhat abstract, unfortunately, and too much for one single blogpost. But the take home message is the following: Quantum and correlation effects appear even for rather simple excited state calculations. In critical cases these may mislead us when interpreting the calculations. Luckily, there is a solution to this conundrum - our wavefunction analysis tools TheoDORE and libwfa :)

Polyradicals

If you are into large aromatic hydrocarbons, here is a new paper for you: "The Polyradical Character of Triangular non-Kekulé Structures, Zethrenes, p-Quinodimethane Linked Bisphenalenyl and the Clar Goblet in Comparison: An Extended Multireference Study," which just appeared in J. Phys. Chem. A. In this work we studied different bonding patterns for polycyclic hydrocarbons and how they lead to the formation of radical character. Some of the systems studied are based on phenalenyl:

In these cases the polyradical character derives from Ovchinnikov's rule. Simply speaking, in all these cases there are more "starred" atoms than "unstarred" atoms. Therefore, some of the "starred" atoms are always left out of the bonds and become radical centers.

For the other systems we studied, Clar's sextet rule comes into play. In an attempt to maximize the number of Clar sextets, the molecule creates empty valences somewhere else, leading to openshell character.

Wavefunction Overlaps

By using wavefunction overlaps you can run nonadiabatic dynamics without computing coupling vectors. This little trick gives you lots of freedom and allows you to run dynamics for any method as long as it gives you gradients. Different people have done that for TDDFT, we have added interfaces for coupled cluster and ADC, and for multireference methods with spin-orbit coupling. However, the one big problem about wavefunction overlaps is that computing them can get really expensive very quickly. Since the orbitals change between the two geometries, the Slater Condon rules do not apply anymore meaning that you have to compute one overlap determinant for every pair of Slater determinants in the expansion.

Fortunately, the situation is not quite as bad, as there are many repetitive terms. If you precompute those and reuse them, things get better. The first step was to identify those repetitive terms, the second one was to think of an algorithm that takes advantage of them, the third one was actually implementing it in a code that works. The result was quite nice: Typical calculations run a thousand times as fast as our previous code and produce exactly the same result. At the same time our formalism is completely general and allows you to do also other things than dynamics. More about that later ...

Read more about our algorithm in our new JCTC paper "Efficient and Flexible Computation of Many-Electron Wavefunction Overlaps". If you want to try it out yourself, wait for the new SHARC release or contact me.