Saturday, July 11, 2009

The effect of the environment

I attach some rough notes on Hush-Marcus electron transfer theory, which I have referred to in a few previous posts. I think contains very important concepts and equations that are very relevant to many biomolecular processes and issues in organic electronics and photonics.

A few points I stress:

This theoretical formalism does not just apply to electron transfer but also many other processes involving transitions between two weakly coupled quantum states which are strongly coupled to an environment which can be treated classically. (The result can be derived from a spin-boson model)

A key physical quantity is the reorganisation energy.

The Marcus inverted parabola shows that the process occurs at the greatest rate, not when the two states are at the same energy, but rather when their energy separation equals the reorganisation energy. (i.e, how much the energy of the environment changes as a result of the process).

The matrix element coupling the two states (e.g, donor and acceptor molecules) falls off exponentially with increasing spatial separation. As a result, I think charge transfer won't be fast enough for many desired processes (e.g, charge separation in organic bulk heterojunction solar cells) unless there is pi-stacking of molecules. Sulfur atoms (as in thiophenes) are also desirable for this reason.

3 comments:

  1. It's becoming apparent to me that what the CMT/COPE group really needs to see at this point is a clear discussion of what the 'reorganization energy' actually IS. This readily apparent from listening to comments at weekly meetings, which go like this:

    Speaker 1: "by comparing the energy of the excited state with the S1 minimized geometry, we can get the reorganization energy..."

    Speaker 2: "well, wait, no... really the reorganization energy is referring to the effects of solvent..."

    Spearker 1: ".. oh no, but it's not in solvent, its in a thin film..."

    et cetera, et cetera.

    The point that all of these speakers is missing is that the theory never DEFINES the reorganization energy AT ALL, any more than it invokes a particular physical meaning of the 'reaction coordinate'.

    This latter point - the physical identity of the reaction coordinate - really made me turn into knots when I read Marcus' paper, because the whole theory is based on it, but it is not explicitly based on anything at all! It took me a long time to come to terms with the fact that it did not really matter, as long as the particular coordinate COULD exist, subject to certain constraints. I am still up in the air about what these constraints might be, mind you, but I am now sure that a reasonable go could be had at formulating them.

    Clearly, for a complicated physico-chemical situation like a real device or beaker full of stuff to be subject to such a simple law implies a MASSIVE reduction in complexity. Therefore, it is only reasonable to assume that the model refers to an ensemble of systems, and not a particular one. In this light, the question "what is the reorganization energy?" is very dependent on the question "what is the ensemble?".

    Conceptually, there is a way forward (see my comment on the July 12 post) in the principle of maximum entropy. The least biased ensemble would be the one that maximizes the information entropy subject to constraint, but this just moves the question "what is the ensemble?" to "what are the contraints?". The most logical constraints that I can think of are just the ones that say simply "the model can be applied here". Namely, that a reaction coordinate such as postulated should exist, along which two states differing by electron transfer simultaneously have definite energies (or at least a dispersion small enough to be ignored).

    How would you write these constraints? I haven't worked that out yet, but I can point out that they intrinsically constrain not just the electronic manifold, because we have already supposed that an external coordinate exists along which the electronic energies can be well-defined. That is, the constraints are on the combined electronic/non-electronic state. I am still thinking about this one...

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  2. If you start with the spin-boson model the reaction co-ordinate is the expectation value of Q = sum_alpha g_alpha (a_alpha + a^\dagger_alpha).

    In a dielectric continuum model the reaction co-ordinate is the value of the electric polarisation at the location of the molecule.

    The re-organisation energy does depend on the environment but for a specific system (thin film, solvent) it is a well-defined measurable quantity.

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