## Tuesday, 17 December 2013

### Oscillator strengths

What is the physical significance of the oscillator strength? Following Werner Kuhn's arguments (e.g. in this paper), it marks the number of electrons oscillating per spatial dimension during an electronic transition. The sum over the oscillator strengths of all the excited states amounts to the number electrons, which is the essence of the Thomas-Reiche-Kuhn sum rule. In other words, the oscillator strength counts how much of the total oscillating potential is used for a specific transition.

This interpretation explains for example the linear relationship of the oscillator strength of the lowest excited state with system size in the case of some conjugated organic polymers (see e.g. this paper): If there are more electrons available to oscillate, then the transition strength increases.

The oscillator strength fij between two non-degenerate states i and j is defined (in atomic units) as two thirds of the squared transition dipole moment multiplied by the energy gap



where the vector r contains all 3N spatial coordinates of the N electrons

The Thomas-Reiche-Kuhn sum rule now states that the sum over the oscillator strengths from one state i to all possible other states is equal to the number of electrons in the system, i.e.

In particular, if we consider excitations from the ground state, then all oscillator strengths are positive. Which means that the oscillator strengths can in fact be viewed as a partitioning of the number of electrons.

The derivation of this sum rule starts by realizing that the momentum operator with respect to any spatial coordinate x of any particle (e.g. x=y2) is given as the commutator of the Hamiltonian with this coordinate

This follows whenever H is of the form

where clearly the derivatives with respect to x are the only part, which does not commute with x itself

By applying the product rule twice, the first term of this expression becomes

And in summary

The remaining proof follows what is shown here (sorry that I am switching the notation, but I copy-and-pasted a little bit ...). First one realizes that the commutator of x and px is equal to i

Then one expands the commutators and inserts a resolution of the identity over the eigenstates of the Hamiltonian

Insert the above expression for px

The commutators are evaluated by letting H act either on the bra or the ket, which results in a multiplication with the respective eigenvalue. And after summing together the equivalent terms one obtains

The actual r vector was composed of 3N individual electron coordinates. The above equation holds for each of these coordinates. Thus, in summary:

which is just what we wanted to show.

## Thursday, 5 December 2013

### Pancake bonding

π-system radicals may interact with each other in a special way. An actual recombination of the radicals is not possible due to the delocalization in the π-system. But the binding between two such radicals is still a lot stronger than would be expected from pure van-der-Waals interactions. This phenomenon is called "pancake bonding".

There is a new paper out by us, studying this: "Study of the Diradicaloid Character in a Prototypical Pancake-Bonded Dimer: The Stacked Tetracyanoethylene (TCNE) Anion Dimer and the Neutral K2TCNE2 Complex" in ChemPhysChem.

Pancake bonding is quite demanding from a computational point of view, since you need to describe the static correlation of the radicals as well as dynamic correlation, which determines the dispersion interaction. We used the multireference averaged quadratic coupled cluster (MR-AQCC) method, which should be able to deal with these effects. MR-AQCC is very similar to multireference configuration interaction only that there is an additional correction for size-extensivity.

## Sunday, 24 November 2013

### Ubuntu vs Windows 8

It is quite amazing how much trouble we had with Windows 8. We just got this new Asus Netbook (F201E). The pre-installed Windows 8 was not only completely non-intuitive to use, but so slow that we were close to just throwing the thing out of the window. The task manager showed that there were some processes running all the time on 50% CPU using up hundreds of megabytes RAM. Maybe you can get away with something like that on a workstation if it has multiple cores and lots of RAM. But it was defnitly the wrong thing for our netbook.

What fixed it? Ubuntu. That was my task for yesterday, which went really smooth. Just download the distribution. Then you need to create a bootable USB stick. Tweak the BIOS a little bit to make sure it reads the stick (activate CSM, deactivate SecureBoot). And even the Wifi worked pretty much right away (after I installed the updates).

And now everything runs smooth and as quick as you'd expect a modern computer to work (even if it's a low-end machine).

## Friday, 22 November 2013

### C++

If I wake up screaming tonight, it's because I have been programming C++ all day. I can see that it is a powerful programming language, but if you are not used to it, it just gives you a headache (and nightmares...).

Maybe I should try to understand how pointers work and not just put * and & signs at random until it compiles. That may make things easier ...

## Saturday, 2 November 2013

### Phosphorescent Complexes

If you are into phosphorescent transition metal complexes here is a paper for you: The triplet state of organo-transition metal compounds. Triplet harvesting and singlet harvesting for efficient OLEDs by Yersin et al. It was very interesting to read this. They just give a relly nice picture of spin-orbit coupling effects and how they are affected by the ligand field. As a major effect: if the ligand field allows for quasi-degenerate states (e.g. octahedral complexes), then the zero-field splitting will be much stronger than in other cases (e.g square planar). And later on they show that the zero-field splitting shows a strong inverse correlation with the phosphoresence lifetime. But on the other hand, if the zero field splitting is too strong, then the highest triplet substate will no longer be efficiently thermally populated, which is a problem since this is usually the state with the strongest emission strength. For this reason there is as of yet no triplet harvesting complex with a phosphorescent lifetime below 1 μs.

Of course, I do not know if it is all correct. But it is nice to see that people are actually taking their time to produce realistic models to explain their results.