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.

UV Excited Single- and Double-Stranded DNA

There is a new paper out by us: "Electronic Excitation Processes in Single-Strand and Double-Strand DNA: A Computational Approach" in Topics in Current Chemistry. I made this figure to show all the possible processes happening:

• Monomer-like decay, which is also observed in the gas phase
• Delocalization and excitation energy transfer (with a focus on electron dynamics)
• Proton transfer between the strands, possibly leading to deactivation
• Electron transfer leading to charge transfer states
• Excimers, which may constitute stable trapping sites
• And finally the thing we are trying to avoid: photoproduct formation, which can lead to damage of the DNA

Which ones of these pathways are the important ones is not known yet. The problem is that every research group involved has their own convictions and it is not quite clear who is correct. In this paper we are of course not able to solve the problem, but at least we discuss the different computational methods applied to help make it more clear why the results by different groups are different.

My own contribution to the debate is the hypothesis of an exciplex with strong geometric distortions, small intermolecular separations, and strong orbital interactions that we described in this paper. Some other groups have obtained similar results: In particular Spiridoula Matsika did lots of work (e.g. this paper) where she invokes a bonded exciplex model. And a Chinese group who were the first to obtain these types of results, using semi-empirical calculations.

The calorie counting paradox

Gary Taubes has an interesting purely mathematical argument against calorie counting in his book "Why We Get Fat". I would put the idea as follows:

Assumption: Our weight is primarily affected by our conscious decisions of how much we eat and excercise.
Consequence: The weight of all of us (except for the most rigorous calorie counters) would fluctuate strongly (several kg per year).

The "proof" goes as follows: If our calorie balance was off by only 20-25 kcal a day, this would amount to about 9000 kcal per year, which corresponds to one kg of pure fat. 20-25 kcal is about one percent of our daily intake. It corresponds to about half a slice of bread or 300 m of jogging, respectively. This is way below the precision any of us can excercise on a day-to-day basis. In other words: if it really were up to our concious minds to maintain energy balance, we'd be in a mess. But this is a contradiction to what we actually observe: there are many people whose weight is stable over a long period of time. So, apparently, on a subconscious level there are strong feedback cycles at work, which overrule our conscious actions.

Of course, nothing is definite in medical science, but I think this is kind of an elegant way to think about such a controversial topic. Thanks to the power of maths, we do not have to rely on indirect information from epidemiological or laboraty studies.

Anyway, if we accept that it is hopeless to try adjusting our weight by quantity of food and excercise, can we do anything? Well, we can go for quality! It is well known that hormones such as insuline, human growth hormone, sex hormones, stress hormones etc. strongly affect our body composition. And we also know that we can affect the production of these hormones by our life style and the type of food we eat. Of course, at this point we do have to go into the (secondary) scientific literature to find out what we should do. But at least we know what questions to ask.

Finally, we should mention that the first law of thermodynamics certainly holds (see also this post), i.e.

calories in = calories out + calories stored

But these are things we cannot affect directly. For "calories in" you need perfect willpower. For "calories out" you need complete control over all metabolic pathways including calories leaving you in the form of heat, not completely metabolized molecules etc. And you have to be extremely precise at that ...

One last thought: the concept of calories is actually quite amazing. You can burn food in a calorimeter and measure by how much water is heated up. The number you get, quantifies the ability of this food to support life. This is a fascinating idea showing the power of abstract physical laws. But it is not the whole story.

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 f_ij 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 p_x 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 p_x

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.