3. Quantitative modeling of molecular dynamics
3.3 From absorption changes to state populations:
The interplay of solvation and intramolecular charge transfer is also the key aspect in the TA study of crystal violet lactone (CVL) that is summarized in the following publication (ap- pendix A5):
The key role of solvation dynamics in intramolecular electron transfer: time-resolved photophysics of crystal violet lactone
U. Schmidhammer, U. Megerle, S. Lochbrunner, E. Riedle, J. Karpiuk Journal of Physical Chemistry A 112, 8487-8496 (2008)
In analogy to the TABs, the evaluation of the solvatochromic behavior together with ab ini- tio calculations could identify the nature of the CT states involved in the absorption and emission process [59]. As shown in Fig. 3.5a, the interesting extension in this case is that the emission can occur from two different CT states depending on the solvent polarity. Both CT states are characterized by a negative charge on the lactone ring and a positive charge on one of the differently situated dimethylaniline groups (Fig. 3.5b)
Fig. 3.5 (a) Steady-state absorption and emission spectra of CVL in nonpolar (n-hexane, dotted lines) and polar (acetonitrile, solid lines) solvent. In the latter, CVL shows a dual fluorescence (A and B) from the two charge transfer states shown in (b). In nonpolar media, the CTB state is not
accessible and emission is only seen in the A-band.
We showed that in this system the intramolecular ET between the initially excited CTA
state and the highly polar CTB state proceeds with a time-dependent rate that is controlled by
the solvent. Within the timescale of the solvation process, the relative energetics of the two states changes from a higher lying CTB state to the opposite situation in the quasi-
equilibrium. The intrinsic ET reaction is probably as fast as in the structurally analogous malachite green lactone (on the 100 fs time scale). Nevertheless, the slower solvation dy- namics is decisive for the actual ET rate and the final population distribution.
From the steady-state spectra, one could expect qualitatively similar TA signals for CVL as for the TABs. In particular, the strong CT character of the emission would suggest signifi- cant spectral shifts in polar solvents. However, the stimulated emission is completely out- weighed by the excited state absorption at all delay times and over the whole spectral range (see Fig. 3.6a). In contrast to the S1 → S0 transition, the transitions contributing to the ESA
obviously comprise only minor changes of the static dipole moment. The most striking evi- dence for this is that instead of spectral shifts a clear isosbestic point is observed around 500 nm. By comparison of the CVL spectra with data from related molecules, we can iden- tify the initial double-peaked signature with the CTA state whereas the final single-peaked
spectrum with a maximum at 470 nm corresponds to the CTB state. The isosbestic point then
indicates the direct interconversion between these two states by intramolecular ET. In con- trast to the situation in the core-substituted NDIs (see section 3.1), the back-ET in CVL pro- ceeds on the nanosecond timescale [59] and is practically not observed in the ultrafast ex- periments. From this one can infer a larger energy gap between the S1 state and the ground
state, even after the stabilization of the excited CT states by the solvation process. The larger energy gap is in line with the UV absorption of CVL being shifted by roughly 10000 cm-1 or 1.2 eV compared to the visible CT absorption band of the NDIs.
Fig. 3.6 From kinetic traces to population dynamics: (a) TA spectra of CVL in propylene car- bonate (PC) after 370 nm excitation. The isosbestic point corresponding to the CTA → CTB con-
version is marked by a red circle. (b) Relative population of the CTA state in PC (open circles) and
acetonitrile (closed circles). The solid lines show the simulation of the population dynamics using the time-dependent rate determined from Marcus theory.
The kinetics of the intramolecular ET from CTA to CTB can best be monitored in the rise
of the 470 nm maximum or in the decay of the long wavelength band around 550 nm. As expected from the observation of the isosbestic point, the rise and decay dynamics are corre- lated in each of the investigated solvents. Within the experimental accuracy we always find the same exponential time constants in both bands (see appendix A5). However, the ob- served biexponential behavior has no straightforward explanation, even if the multiexponen- tial characteristics of solvation processes are taken into account. Most evidently, the second
time constant is always significantly larger than the slowest solvation process [49]. There- fore, the exponential fit merely serves as a parameterization of the kinetic traces.
The reason for the failure of a kinetic model based on exponential fits is again the fact that the underlying process is only poorly described with a static rate model. When the rela- tive energetics between CTA and CTB change, the barrier between them is also subject to
changes. This in turn affects the rate for the population transfer from CTA to CTB and back.
The first step towards a quantitative kinetic modeling is therefore to extract the relative population of the involved states from the TA data.
Since effectively only the two CT states contribute to the observed signal at a given wave- length and we know that the CTB state is not populated directly by the optical excitation,
ΔOD(t) is given by (3.2)
[
]
A A B B A A A A B OD(t) c (t) d c (t) d c (t) d c (0) c (t) d , Δ = ⋅ε ⋅ + ⋅ε ⋅ = ⋅ε ⋅ + − ⋅ε ⋅where d is the sample thickness, cX(t) is the concentration of molecules in state CTX and εX
is the extinction coefficient of this state at the investigated wavelength. The normalized population c (t)A of the CTA state can then be expressed as
A A A A A B c (t) OD(t) c (0) d c (t) . c (0) d( ) B Δ − ε ⋅ = = ε − ε (3.3)
The unknown extinction coefficients can be eliminated using the amplitudes A0 at t = 0 and
A∞ at t > 100 ps together with the calculable* fraction x = cA(∞)/cA(0) of the CTA population
after the solvation, i.e. in quasi-equilibrium:
A A 0 A A A B OD(0) c (0) d A ; OD( ) x c (0) d (1 x) c (0) d A .∞ Δ = ⋅ε ⋅ = Δ ∞ = ⋅ ⋅ε ⋅ + − ⋅ ⋅ε ⋅ = (3.4)
Substituting these expressions into the equation for the normalized CTA population yields:
0 A 0 OD(t) (1 x) A A x c (t) . A A ∞ ∞ Δ ⋅ − − + ⋅ = − (3.5)
Thus, we can use the fit amplitudes from the above mentioned exponential parameterization to convert the kinetic traces at individual wavelengths to the physically more relevant popu- lation dynamics (see Fig. 3.6b). This paves the way for a quantitative description of the sol- vent-dependent ET.
The model includes the determination of a number of microscopic and thermodynamic quantities such as the initial separation between the two CT states, the total amount of ener- getic lowering caused by the solvent reorganization and an estimate for the solvation and local heat dissipation timescales. The proper combination of these considerations leads to a knowledge of the energetics at each point in time and therefore allows the application of the
* The final population in the CT
A and CTB state corresponds to a Boltzmann distribution and depends on the
energy difference between the states after solvation. This can be inferred from the analysis of the solvato- chromism, see equation 1 and 2 in appendix A5.
Marcus theory [60, 61] to determine a time-dependent ET rate kET(t). With this the popula-
tion dynamics can be found by numerical integration of the rate equation
(
A ET A ET A dc k (t) c k (t) 1 c , dt + − = − ⋅ + −)
(3.6)where the plus sign refers to the forward and the minus sign to the backward ET. The results for the depopulation of the CTA state in propylene carbonate and acetonitrile are shown in
Fig. 3.6b as solid lines. The agreement with the experimental data is remarkable given that all kinetic information used in the model is taken from independent sources. The only pa- rameters that are adjusted are the reorganization energy λ and the initial increase in local temperature ΔkT. The significantly higher value of the latter in PC is due to the high polarity of this solvent. Thus, the absorption band of CVL in PC is redshifted compared to MeCN and the same excitation wavelength leads to a higher excess energy.
The analysis of the intramolecular ET of CVL demonstrates that even clear spectral signa- tures might sometimes require non-standard modeling to reach a deeper understanding. In this regard, CVL is the counter-example to the TABs described above, especially CB, where rather complicated spectral evolutions could be explained by a fairly simple model. In both cases, the continuous characteristics of the solvation process impede the application of a classic rate model implying discrete states with time-independent energies and constant rates for the transitions between them. For CVL, the application of the time-dependent rate model not directly to kinetic TA traces but to the derived state population was decisive to prove the control of the ET by the solvent, both in terms of energetics and dynamics. The considera- tion of both these aspects explains why simple exponential fits to kinetic traces fail to repro- duce the known solvation times. The obtained time-constants are therefore not particularly relevant for the final description of the process.
3.4 Band integrals, diffusion model and species associated spectra: