gmolden - exporting densities

For some reason I never realized that there was a program called gmolden that works just as normal molden but looks much cooler (with opengl graphics). You can download the precompiled executable from their homepage just as the normal one. And then, finally, you can have shiny molecules and orbitals in molden.

The reason why I went back to molden is that it lets you produce densities from orbitals and occupation data. In my case I wanted to compute hole/particle densities (which are similar to the attachment/detachment densities but for general wavefunctions not quite the same) as weighted sums over the natural transition orbitals.

And since I like to do my density plotting in VMD, I can also ask molden to export cube files, which I can open in VMD. And then I can plot at the hole density ...

... and the particle density ...

... the way I like to do this.

Orbital relaxation - the natural difference orbitals

As another example for the wavefunction analysis tools from last post, let us look at dimethylaminobenzonitrile (DMABN), a prototype charge transfer molecule and analyze the S₂ state. I will start with the natural transition orbitals (NTOs), the singular vectors of the transition density matrix. The S₂ state of DMABN has only one pair of NTOs with any significant contribution. To the left the hole, to the right the particle NTO are shown. In this representation the state can be clearly identified as a ππ* state with some partial charge transfer character (going from the amino to the nitrile side).

The hole and particle densities, i.e. the weighted sums over all NTOs, closely resemble these primary NTOs:

For comparison, we can look at the attachment/detachment densities as computed from the difference density matrix:

These look notably different from the hole and particle densities! What you can see at first sight, is that they are "bigger" - there is more happening. While the hole and particle densities contain 0.84 electrons each, the integral over the attachment and detachment densities is 1.41 e. To get a more detailed look at this, we can analyze the natural difference orbitals (NDOs), the eigenvectors of the difference density. Here for example the first three detachment and attachment NDOs and their respective eigenvalues:

-0.911	0.908
-0.060	0.059
-0.043	0.043

The first pair of NDOs corresponds to the NTOs as shown above. But aside from that, there are a number of additional contributions. The two most important ones are apparently polarizations of the σ-bonds: while the primary excitation process takes electrons from the π orbital at the amino N-atom, some of the electron density is restored through the σ-system. We can quantify this through a Mulliken analysis and find out that during the primary transition, the N-atom loses 0.33 e and gains 0.04, i.e. there is a primary charge shift of 0.29 e. By contrast the difference density tells us that 0.39 electrons are detached from N-atom and 0.19 are attached leading to a net charge shift of only 0.19 e on this atom. By construction the latter corresponds to the actual change in the Mulliken population. But, I guess also the former has a physical significance.

Anyway, I do not want to go into much more detail now. But I hope I could convince you that there is really a lot of exciting stuff happening with excited states (as the name suggests ...). And just looking at HOMOs and LUMOs is not going to help you with any of that. If you are interested, you can check out the two new papers (Part I, Part II), download the Wave Function Analysis Tools from my homepage, or use Columbus where some of these things are implemented as well. Unfortunately, the whole functionality is not released yet. But it will be made available soon within the ADC module of Q-Chem and as a separate C++ library. Let me know if you have any questions or any suggestions for applications.

Analysis and visualization of excited states

My drive to create pretty orbital pictures lead to two new papers, which I quite like: "New tools for the systematic analysis and visualization of electronic excitations" Part I: Formalism and Part II: Applications. Actually, a number of interesting things happened on the way and people seem to like it so far.

One of the interesting points is the discrepancy between the transition and difference density matrices. Both should give you a compact representation of the transition - but not necessarily the same one ...

For example these are the hole and particle densities (computed from the 1-particle transition density matrix) of the first two singlet excited states of adenine (using Jan's extension of my VMD plotting script) deriving from a ππ* and a nπ* state.

For comparison the attachment/detachment densities (computed from the difference density)

 What you can see is that the attachment/detachment densites are "bigger" than the particle/hole densities. The difference is that many-body effects and orbital relaxation are only included in the latter case giving additional contributions. You can look at this in more detail by analyzing the individual orbitals these are composed of, the "natural difference orbitals". Maybe I'll show that in the next post. Or check out the articles - I think for the first 30 days you can even download them freely.