Of course the next step is to extend the approach to more interesting systems. But the nice thing was that most of the physcially important parameters could be adjusted easily. So it was a good way to learn the basic phenomena of charge transfer.

I showed in a previous post what such a dynamics looks like. Here I want to talk a little bit about some of the math behind it. Before we did this work, I never really understood what a non-adiabatic coupling vector was. I will briefly describe that now - for a longer explanation you can of course download and read our paper.

The components of the non-adiabatic coupling vector

are obtained as the derivative of the electronic wave function of one state projected onto the wave function of another state. R

_{j}means the displacement of a nuclear coordinate. Note that this is a derivative in the Hilbert space of electronic wave functions, parametrically dependent on nuclear coordinates.[1]

The interpretation comes when using the mixing angle η between diabatic and adiabatic functions. Because it turns out that

What I am doing now, is to start with a delocalized charge (η=π/4) and localize the charge (η=0) by changing the bond length alternation.

At an intermolecular distance of 5 Angstrom, the interactionbetween the fragments is quite large. The wavefunctions change slowly over a wide geometric range and you have a wide peak of the coupling vector accordingly.

At an intermolecular distance of 7 Angstrom, the interaction is almost vanishing. The wave function localizes after only a small geometric displacement. Accordingly there is a highly peaked coupling vector.

According to the second equation above, the area under both curves is π/4. If you look at the graphs you can see that this could be right, i.e. the area should be about unity.

[1] Similar difficulties occur in the derivation of the Hellman-Feynman theorem - in that post there is even the third component λ which represents the possibility of having a not converged wave function...

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