Sunday 6 April 2014

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 χA and  χB, 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

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.

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