European PhD positions open

If anyone is interested: there are three PhD positions opening up in our group (back in Vienna). These are EU projects where half of the time will be spent in Vienna and the other half in either Pisa, Toulouse, or Madrid.

To learn more, check out the Jobs page of the Gonzalez group. And if you need more information you can of course contact me or write directly to Leticia Gonzalez.

Iridium complexes and the failure of ADC(2) to describe them

When I started working on iridium complexes two and a half years ago, there was one simple thing I could not explain: The results at the ADC(2) level were more than one eV off from the ADC(3) reference. And that is not good: you would not expect that a correlated ab-initio method would fail so badly. What followed was a two year detour through the world of excited state analysis (seen in some previous posts: [1], [2], [3], [4], [5]). And finally, I feel like I am in a position to discuss the above question properly in our new paper "High-Level Ab Initio Computations of the Absorption Spectra of Organic Iridium Complexes", which just appeared in JPCA.

The first step was to properly quantify the charge transferred using population analysis schemes. This revealed that ADC(2) had a strong bias toward overstabilizing charge transfer states. Usually you would think that this is a problem of TDDFT. But interestingly even TDDFT/B3LYP performed much better with errors only on the order of 0.5 eV.

To quantify the problem of ADC(2) we needed a new tool. For this purpose we took a closer look at the attachment/detachment analysis of Head-Gordon. While it is common to visualize the attachment and detachment densities, there is also an important meaning to the integral over them, the promotion number p. This quantity counts the total number of rearranged electrons and proved as a useful measure for orbital relaxation effects, which are difficult to understand otherwise.

So what happened when we made this analysis? At the ADC(1) level (whose energies and state densities are identical to CIS) p has to be equal to 1 per construction. Then ADC(2) apparently overestimates orbital relaxation effects with p values above 1.5. And only the ADC(3) level is balanced enough to cover some but not too much orbital relaxation.

Exciton sizes

When you do not understand the wikipedia article about the very topic you are supposed to be working on, what do you do? Do you ignore this fact, do you give up, or do you see it as a chance?

When I started working on excitons five years ago, I first checked out the wikipedia article, which talked about weird things like exciton binding energies and exciton sizes. An excited state is a transition between orbitals, right? So why would you talk about quasi-particles? Is a "hole" an actual thing and why is it paired with a specific "electron" when you already have tons of electrons floating around?

In a sense my work since then has been occupied with understanding that wikepedia article. It started with our work on excitonic and charge resonance interactions, continued with a series in JCP about analysis and visualization of excited states, went on with the release of the TheoDORE analysis package, and is now finally at a point where we are computing exciton sizes (as demanded by the wikipedia article). The new paper is called "Exciton analysis of many-body wave functions: Bridging the gap between the quasiparticle and molecular orbital pictures" which it appeared yesteday in Phys. Rev. A.

Our idea is to interpret the transition density matrix as an effective exciton wavefunction. This creates a clear rule for computing operator expectation values and reduces the problem to linear algebra. We started with exciton sizes since it is simple to deal with the required multipole integrals. But remembering the wikipedia article, I would also be very interested in the binding energies. All we need for this, is the electron repulsion integrals, which are easily accessible as well. But the physical interpretation is a little bit more difficult because of "screening" (another one of those terms which quantum chemists don't like).

The main area of application will probably be the case of large quasi-periodic systems like conjugated organic polymers where the molecular and the solid-state viewpoints meet. But it is also interesting to look at simple dimers of molecules. In such a case, the exciton size of a charge separated state is equivalent to the distance between the two molecules while it is independent of the distance for locally excited states. Below we are showing this in the case of the pyrdine dimer, considering (a) the excitation energies, (b) the exciton size, and (c) the charge transfer measure that we defined earlier.

The added complication in the case of this system is the symmetry which makes all the states and orbitals delocalized over the whole system and takes away the possibility for net transfer of charge or a dipole moment along the separation direction. As shown above, the states are still divided into excitonic and charge resonance states. The difference between them is the correlation between the electron and hole.

TheoDORE 1.0 release

The first version of the Theoretical Density, Orbital Relaxation, and Exciton analysis package TheoDORE is released. You can find the project's homepage here, download the package from here, or check out the documentation wiki.

The central feature of this analysis are the electron-hole correlation plots of the charge transfer numbers (as shown in the top part of the figure). These allow you to get a two-dimensional representation of correlations between the electron and hole quasiparticles involved in the excitation. In other words: this analysis allows you to detect dynamic charge separation effects even in the absense of any net charge transfer. In the lower panel, a natural transition orbital is shown, which is convenient and compact way to represent an excited state.

Formally, the above quantities are defined with respect to the transition density matrix (1TDM). But, if you are satisfied with approximate results, then you can simply use the response vector of the quantum chemical method and regard it as the 1TDM. This way we extended TheoDORE to work with the Columbus, Turbomole, and Q-Chem packages and support for ADC, CC, TDDFT, and multi-reference methods is available (more information).

There is also support for the analysis of state and difference density matrices: Analysis of effectively unpaired electrons, attachment/detachment analysis, and a population analysis of the resulting densities.

HOMO-LUMO gaps and spin eigenfunctions

Statistically speaking, writing sequels is not a good idea,[1] but there is something I withheld from you in my previous post about HOMO-LUMO gaps and excitation energies: the spin of the electron. Spin makes everything a bit more complicated but also more interesting. What are the excitation energies of the singlet and triplet eigenfunctions? And what is exchange splitting?

Compared to last time, we have to construct spin-adapted eigenfunctions. Slater determinants  (constructed from restricted orbitals) are always eigenfunctions of S_z but not necessarily of S². One simple possibility of creating an S² eigenfunction is the construction of a high-spin determinant. And this is how I will start here. For example, we can create a spin-eigenfunction by exciting from spin-down occupied orbital k into the spin-up unoccopied orbital a. As shown in the previous post, the energy of the resulting determinant is given according to (here the bar marks the spin-down quantities)

The second (exchange type) integral vanishes and the expression can be rewritten as

In other words: In the triplet case the Coulomb interaction between electron and hole is the only relevant term.

If we are interested in the singlet then we have to consider linear combinations of excited determinants. The standard construction uses the spin-up and spin-down excited determinants

Here "plus" yields the singlet and "minus" the triplet. The energy of this wavefunction is expanded as:

The first two terms are the energies of the individual Slater determinants (as discussed in the last post). The third term is the coupling element, which is also readily calculated.

The spin-up and spin-down energies are equivalent and of the coupling terms only the first (exchange-like) one remains:

If we use the energy expression of the excited Slater determinant from the previous post we obtain for the singlet energy

When compared to the simple band picture, the energy is lowered by the Coulomb interaction between the electron and hole and raised by twice the exchange interaction. For the triplet we obtain

which is (luckily for me trying to write this down) the same as the high-spin triplet discussed above. There is only a Coulomb but no exchange interaction. The splitting between the singlet and triplet amounts to twice the exchange integral, hence it is called "exchange splitting".

The Coulomb term corresponds to the attraction between the hole density (computed as the square of orbital k) and the electron density (the square of orbital a). It is a long range interaction decaying with the reciprocal distance of the two orbitals. The exchange term is computed by multiplying orbitals k and a with each other (yielding the transition density) and computing the electrostatic repulsion of this with itself. It is a short range interaction, which requires that a and k occupy the same space. These considerations show that the triplet state will always be lesser or equal in energy than the corresponding triplet singlet state and that this is particularly pronounced for localized states.

---

[1] The problem is that there is a very low chance that the sequel to your best post will be your new very best post. Even if it is a good post, it will look bad in comparison to the original one. But this is only a question of the reference point and part of a general phenomenon called Regression to the Mean.