Effect of a solvent on excited state dynamics

Last two days I have read through a nice paper Modeling the Nonradiative Decay Rate of Electronically Excited Thioflavin T
by Yuval Erez, Yu-Hui Liu, Nadav Amdursky, and Dan Huppert

The relevant molecule [chromophore] is shown below. It is of particular interest because its fluorescence intensity increases significantly when bound to amyloid fibrils which are associated with Parkinson's, type II diabetes, and Alzheimer's disease.
The key photophysics is associated with twisting about the central carbon-carbon bond.

The experimental results that need to be explained are
-the non-radiative life time increased linearly with the solvent viscosity over 3 orders of magnitude
-the fluorescence intensity decreases with time on the scales of tens of picoseconds

The calculated [via TDFT = Time-Dependent Density Functional Theory] dependence of the ground and excited state energies as a function of the twist angle is shown below.

As the twist angle increases from zero to 90 degrees the transition dipole moment between the ground and excited states decreases by 2 orders of magnitude.

Much the above can be qualitatively understood in terms of a two-site Hubbard model where the two sites correspond to two orbitals localised on the two opposite sides of the molecule.

Excitation from S0 to S1 at 30 degrees then leads to downward movement on the S1 potential energy surface towards the minimum at 90 degrees [referred to as the TICT =Twisted Intra-molecular Charge Transfer] state.
Hence, with increasing time the fluorescence intensity decreases due to decreasing oscillator strength for the S1-S0 transition. However, the twisting motion of the large rings is opposed by friction (viscosity) arising from the solvent.

The paper applies a theory due to van der Meer, Zhang, and Glasbeek that I discussed in an earlier post, to give a quantitative description of the experiment. It is assumed that the viscosity is sufficiently large that the twisting motion is classical and over-damped and so can be described by a Smoluchowski diffusion equation. This is solved to given emission line shapes as a function of time. For times larger than 3 picosecond a diffusion constant of D=0.1/psec gives results consistent with experiment. This value is nicely consistent with that predicted by combining the Einstein relation [fluctuation-dissipation relation]
D= k_B T/friction
with a Stokes formula relating the viscosity to the friction for a rotating disc with the size of a phenyl ring.

A few open issues

• it would be nice to see a plot showing the calculated relationship between the non-radiative lifetime and the solvent viscosity with a comparison of the correlations seen experimentally.

• describing the short time dynamics (greater than 3 psec) requires a larger diffusion constant (smaller friction) consistent with the idea of a frequency dependent friction due to the finite relaxation time of the solvent.

• I assume the classical dissipative dynamics is justified because the timescales of interest are much larger than the relevant thermal time ~ hbar/k_B T ~ 20 fsec.

• A very broad "line shape function characteristic of the Frank-Condon factor" is used. The width is 3300 cm-1. It is not clear what the physical origin of this large width is.

• The line shape function is taken to be a log normal distribution. This form is often used to empirically describe emission lines, going back to a 1969 paper by Siano and Metzler. However, as far as I am aware, there is still no clear theoretical justification for this form.