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.

Suitcases with wheels

More than 5000 years passed between the invention of the wheel and that of the wheeled suitcase, as Nassim N. Taleb points out in his book "Antifragility". By the time humanity sent a man to the moon we still had three decades left of straining our shoulders lugging around bags whenever we traveled. Quantum theory and relativity had been discovered for more than fifty years and the scientists uncovering the subtleties of these theories with ever more detail had to pick up and carry their suitcases when they wanted to go and discuss their findings at a conference. And then someone started putting wheels on a suitcase. This completely changed luggage carrying. Wherever you look, you'll find people dragging around their trolleys.

I think this is a big lesson for people like us who are supposed to be in the "invention business." The paradigm changing innovation may be right in front of our eyes, and it might have been there for a long time. It is just that we are too domain dependent to see it. "Interdisciplinarity" (in this case between the fields of luggage carrying and transportation) is an overused buzzword but it is a key element of efficient research. Let's build suitcases with wheels rather than reinvent the wheel.

Correlation and entanglement in H2

Why does correlation increase when you dissociate an H₂ molecule? The quick answer is that for the single determinant wavefunction at the equilibrium geometry the two electrons move independently: they could be located at the same nucleus (ionic configuration) or at different nuclei (covalent configuration) with equal probability. At infinite separation the two electrons are "statically" correlated: if the first electron is located at the first nucleus, the second electron has to be at the other nucleus and vice versa. More importantly, if you measured spin up for the electron at one of the nuclei, you would know with certainty that the electron at the other nucleus would be spin down. This leads us to the fancy answer: if you were somehow able to dissociate H₂ in a fully coherent way, you would obtain two entangled hydrogen atoms, see also this paper by Garnet Chan.

Let's look at the math...

The closed shell wavefunction of H₂  is given as the Slater determinant

where ψ is the bonding MO

constructed from the AOs χ_A and  χ_B, situated on atoms A and B,which are assumed to be orthogonalized. (And we do not worry about normalization.) Here, the Slater determinant can be factorized into a spatial and a spin part

The spatial part is a simple product, and it is clear that there is no kind of spatial correlation between the two electrons. The next step in this discussion is to insert the AOs:

And it can be seen that the ionic configurations (both electrons are on either A or B) have the same statistical weight as the covalent configurations (one electron is on A and the other one on B). The position of electron 1 does not affect the position of electron 2 - they are statistically independent (uncorrelated).

At infinite separation the doubly excited determinant mixes into the wavefunction with equal weight to the closed shell

where

And with respect to AOs the wavefunction reads

Let's look at this equation in detail: There is an a priori probability of 50% that electron 1 is on atom A and 50% for atom B. But when you specify the position of electron 2, then there is a 100% chance that electron 1 will be on the other atom. In other words their positions are stastically dependent of each other (correlated).

From the point of view of the atoms, there is another subtlety. A priori there is an equal probability that either one will have spin α or spin β. But if you do measure the spin on one of these atoms, you immediately know that the other one has the opposite spin, even if it is infinitely far away. This phenomenon is called entanglement.

These types of counterintuitive behaviour (entanglement and correlation at infinite separation) only come into play because of the special form of the wavefunction as given above. The more intuitive form

where the electrons are simply located on a spin orbital on either atom would not show any of this. However, this wavefunction is not admissable because it does not fulfill the Pauli principle.

This leads us to the final point of the discussion. It makes sense to use the pure "coherent" wavefunction of the isolated  H₂ system as given above only if there were no interactions with the "bath". Otherwise we would have to describe the wavefunction of a larger part of the world (or use a density matrix formalism). In such a case this strange type of correlation and the entanglement would disappear. In summary: from a mathematical point of view it is clearly possible to find "static correlation" in the dissociated  H₂ system. But from an experimental point of view it is probably extremely difficult to create this situation. Should we discuss this at all then? Probably yes because we need a consistent theoretical framework. But we should remember that the situation is actually quite artificial.

Non-adiabatic dynamics with coupled-cluster and ADC

For non-adiabatic dynamics you could either be using CASSCF, which is a pain with respect to choosing the active space, or multireference CI, which is more reliable but also much more expensive. You could be computing approximate non-adiabatic interactions at the TDDFT level, which is much faster but carries all the problems of TDDFT. All these methods have their justifications (and are in fact available in Newton-X), but we wanted to open another route: coupled cluster (CC) and algebraic diagrammatic construction (ADC) dynamics, which we introduce in our new JCTC paper: "Surface Hopping Dynamics with Correlated Single-Reference Methods: 9H-Adenine as a Case Study".

CC2 and ADC(2) are available in a very efficient implementation in Turbomole. All we had to do is produce some approximate non-adiabatic coupling terms. Our "trick" was the following (in a similar spirit to what they do for TDDFT): take only the single excitation part of the response vector, construct a renormalized CIS vector out of this, and compute the wavefunction overlap of these vectors at two different timesteps. The first numerical tests were very promising and it looks like this can really be a viable method.

The limitation is that CC and ADC (just as TDDFT) are not able to describe intersections with the closed shell ground state correctly. But, if you are interested in non-adiabatic excited state processes ADC(2) may be the method of choice for you. According to some of our previous experience, potential applications include:

• nπ* and ππ* crossings during excited state proton transfer
• energy and electron transfer
• excimer formation
• and in fact any application where you are not interested in the precise details of the activation to S₀

So, if you do have a Turbomole license, it definitly makes sense to check out the new methods.