This week in Telluride I had some really helpful discussions with Dmitry Matyushov. I finally understood a few papers that he wrote a decade ago with Greg Voth and Marshall Newton. I had looked at least one of these paper before when Joel Gilmore and I were modelling decoherence of excited states of biological chromophores. But I did not understand them or fully appreciate their relevance or significance. I can also see the relevance to current work on methine dyes (with Seth Olsen) and on photoactive organometallic complexes (with Anthony Jacko and Ben Powell).
In my papers with Joel we made the simplifying assumption (which is a standard one following Hush's treatment of intervalence charge transfer transitions):
Assumption*:for the ground to excited state transition in the chromophore the transition dipole moment is much smaller than the difference of the dipole moments between the ground and excited states.
This allows a direct mapping of the problem to the independent boson model and exact evaluation of the full time dependence of the reduced density matrix (and thus the optical absorption lineshape) for any coupling to the environment.
Some consequences of the analysis are that:
- the absorption and emission lines obey the mirror image rule
- in the "high" temperature limit (which is reasonable for most solvents at room temperature) the line width is proportional to the Stokes shift and to the temperature
However, the chromophores of interest (those that occur naturally or are used to tag biomolecules) are of interest because they are bright (i.e. they have a large transition dipole moment) and so Assumption * is actually not true.
This large transition dipole moment is a natural consequence of the fact that the transitions with large oscillator strengths are coherent charge transfer transitions.
This turns out the reason why often the mirror image rule is not satisfied. In particular, the emission line can be substantially narrower than the absorption line, and they have different shapes. It also explains why one often sees vibronic structure in emission spectra but not absorption spectra. This paper discusses the basic effects of the electronic delocalisation associated with the transition dipole moment.
I think one must then modify the independent boson model to include off-diagonal terms coupling the boson bath to the two-level system. That Hamiltonian can then be transformed to a spin-boson model (at a non-zero bias) with diagonal coupling to the bath. That is then a highly non-trivial problem to solve.
Dmitry did not consider that problem but the physically relevant and analytically tractable one with the limit that:
- the reorganisation energy is much less than the transition energy[this just means one can clearly resolve the absorption and emission bands]
- the solvent degrees of freedom are treated classically.
There turns out that there is another effect which needs to be taken into account to get a quantitative description of line shapes: that the polarisability of the ground and excited states is not the same. This effect actually acts in the opposite direction to that of the transition dipole moment.
With this analysis Dmitry and Marshall Newton were able to obtain a quantitative description of the lineshapes of both emission and absorption for the dye Coumarin-153 in a wide range of solvents.