2. Ultrafast, quantitative transient absorption spectroscopy
2.7 Peak shifts as sensitive probe on a solute’s geometry and environment
Spectral Shifts in Time-Resolved Measurements. The electronic transitions of molecules are
not static and unalterable but they can be changed by a diversity of effects. A prominent ex- ample is the red or blue shift of a molecular absorption band when changing the solvent po- larity (solvatochromism). The shift is caused by differential solvation of the ground and first excited state of the molecule [51]. However, solvation shifts are not only visible in steady- state absorption or fluorescence spectroscopy when the solvent is changed. They can also be observed on ultrafast timescales when the solvent adapts to a new charge distribution.
A frequently used technique is the photo-excitation of a chromophore with a pronounced change in the dipole moment upon excitation and a time-resolved measurement of the fluo- rescence. The evaluation of the fluorescence peak shift then allows for a determination of characteristic solvation times [52-55]. The same information can also be obtained by tran- sient absorption spectroscopy by following the temporal evolution of the spectral position of absorption bands or the stimulated emission [56,57]. However, the interpretation of the ob- served spectral shift can be complicated due the presence of further contributions, particu- larly vibrational relaxation [58] and dynamics of the excited state absorption.
The impact of the solvent and dynamic solvation on the transition energies highlights the influence of the dielectric environment on a chromophore. Further changes in this environ- ment – apart from solvent effects – should therefore also have an effect on the peak position. A first insight into the sensitivity of absorption bands of carbocations towards changes in their vicinity was given by Schneider et al. [43] who showed that the absorption band (measured by steady state UV/Vis absorption spectroscopy) of persistent benzhydryl cations in the vicinity of an anion is slightly blue shifted by about 2 nm compared to unpaired cations. A closer inspection of Figure 2.2 reveals that the benzhydryl cation absorption band observed after UV excitation of (tol)2CH-PPh2Ph-SO3¯ in dichloromethane (see) also exhib- its a temporal change of the position. Knowledge on the vicinity of a reactant or the temporal evolution of the distance between two reaction partners – by diffusion or a distance depend- ent reaction – is an important aspect in the understanding of ultrafast chemical reactions. Therefore, a consistent measure for the position of the absorption band for every measured delay time is needed.
Parabola Fitting Algorithm. The determination of the center of gravity of the benzhydryl
cation or radical absorption band is biased by the significant spectral overlap of the two product bands. An iterative algorithm has therefore been developed which fits a parabola to the maximum of an absorption band. This ansatz relies on the Taylor expansion of a function f(x) around an extremum at x0 which is given by:
( )
(
)
(
0) (
)
2(
0) (
)
3 0 f x 0 f x 0 f x f x x x x x ... 2 6 ′′ ′′′ = + − + − + (2.21)Hereby, the linear term vanishes since the first derivative is zero at x0. Therefore, every
maximum of an observed absorption band can be – within a sufficiently small interval – well approximated by a parabola. Any other definition of the band position leads hereby at most to a systematic deviation which is constant for all times. The dynamics of spectral shifts is thus not influenced by the choice of the measure for the band position and even the relative shift amplitudes remain unchanged.
Figure 2.10: Peak position determination for the benzhydryl cation band observed ~15 ps after UV excitation of (tol)2CH-PPh2Ph-SO3¯ in dichloromethane (see Figure 2.2). (a) Iterative fit of parabolas within a narrowing interval. The first two iterations are shown in blue and green to- gether with the borders of the interval. The last fit and interval is shown in red. The complete transient absorption spectrum is shown in the inset. (b) Temporal evolution of the cation peak po- sition determined by the algorithm.
The algorithm performs an iterative fitting of a parabola to the absorption band maximum within a dynamically adapted data interval around the maximum. In a first step a parabola is fitted to the transient data of an sufficiently large interval comprehending the absorption peak (see Figure 2.10a). Around the maximum of this parabola a new, smaller interval is chosen and a second parabola is fitted to the data. This procedure is repeated with smaller intervals until the identified peak position is independent of the size of the interval. The al-
gorithm allows for the accurate determination of the time-dependent shift of an absorption band maximum with an accuracy of ~ 0.1 nm
The temporal evolution of the band position can be evaluated by applying the algorithm to the entire transient absorption data set, thus allowing to track the peak position for time
delays from below 200 fs to nanoseconds or longer (see Figure 2.10b).4 It allows to track
changes in the observed solute (e.g., rearrangements) or its environment (e.g., solvation or diffusional separation) and to indicate the time scale on which these changes occur.
Figure 2.11: Influence of the noise amplitude on a) the deviation Δλ of the fitted to the nominal peak position and b) the standard deviation for a series of 100 spectra with a Gaussian, a Lor- entzian and a log-normal peak shape. A Gaussian-shaped spectrum (grey dots) for a signal-to- noise ratio of 1 is shown in the inset, together with the noise-free Gaussian (black) and the parab- ola fit (red).
Performance of the algorithm. The influence of noise on the precision of the fitted peak
position was simulated with three peak shapes often found in spectroscopy: a Gaussian, a
Lorentzian and a log-normal5 function. All three were set to have their maximum at 400 nm
and a FWHM (full width at half maximum) of 20 nm. The spacing of the wavelength axis (~0.6 nm) was taken from the experiment. A series of 100 different spectra for each of these three peak shapes was generated by adding white noise (see inset in Figure 2.11 for a show- case). The ratio between the noise and the peak amplitude was varied between 0.01 and 2.
4 The upper temporal limit is given by the ability to measure absorption spectra at such long delays.
5 The log-normal function is defined as
( )
max 2 ln exp 2 f A w λ λ λ = ⋅ − ⎡ ⎤ ⎢ ⎥ ⎣ ⎦.The generated spectra were fitted with the algorithm with a final data interval of 20 nm. As a
measure for the accuracy of the peak position determination the deviation Δλ = λ −400 nm
of the mean peak position λ from the nominal value of 400 nm and the standard deviation
λ
σ were evaluated.
In Figure 2.11a the deviation Δλ of the fitted mean peak position from the nominal value
of 400 nm is shown. It is most notable that no significant systematic deviation is found for all three peak shapes although a rather large fit interval was chosen. This insensitivity allows to fit the parabola to a large part of the spectrum around the peak without biasing the result. The standard deviation of the fitted peak position for the generated 100 spectra is shown in Figure 2.11b. While at small noise levels the standard deviation is negligible small it rises monotonously with the amplitude of the noise. However, even when the noise amplitude is twice as large as the amplitude of the peak and the bare eye has already problems to recog- nize the presence of a peak (see inset in Figure 2.11), the algorithm is able to find the maxi- mum with a standard deviation of only a few nanometers.
Technical and Spectral Prerequisites. In order to obtain a reliable band position with sub-
nanometer precision several prerequisites (both, technical, i.e., related to the transient ab- sorption setup, and spectral) have to be fulfilled:
• The studied absorption band has to be strong enough. The position determination for
bands with absorption changes á 1 mOD is difficult and special care has to be taken in
order to obtain sufficiently low-noise transient spectra.
• The absorption band should not be overlapped by a further evolving band. However, an
underlying broad and unstructured absorption background (e.g., many excited state ab- sorption bands) will not change the result significantly.
• The achievable precision of the band position determination is directly linked to the
width of the absorption band.6 The broader a band becomes the less defined is the fit of a parabola on it.
• The absorption band has to be covered by a sufficiently high number of detection chan-
nels, thus increasing the stability of the fit against statistical deviations. For the benzhy- dryl radical and cation absorption bands typically 20 to 40 points are used for the fit.
• Over one transient spectrum the noise from detection channel to detection channel
should be as small as possible. On the contrary, the variations between several spectra are only of minor importance. Interestingly, when using CaF2 white-light the signal fluc-
6 It suffices if a part of the entire absorption band exhibits a sharp feature which can be fitted by a parabola. The vibronic progression of a transition leads to such a behavior.
tuations are highly correlated leading to a systematic increase or decrease over a large part of the spectrum. The origin of this behavior is not yet fully understood. However, it seems to originate from the concerted generation of different parts of the whitelight spec- trum.
The high accuracy allows for the observation of even tiny effects on time scales ranging from 100 fs to microseconds. In the sub-picosecond range solvation and intramolecular rear- rangements (e.g., planarization) are dominant. Both can lead to a distinct shift of the peak wavelength. After their completion further processes can lead to a shift of the absorption bands. As pointed out above, the presence of an anion in the vicinity of a benzhydryl cation blue shifts the cation absorption band by about 2 nm. Therefore, the change in the mean cation-anion distance can be followed by the dynamic peak shift.
By analyzing the benzhydryl cation and radical peak shift after UV excitation of Ph2CHCl
in CH3CN we were able to corroborate the microscopic reaction scheme proposed by the
theoretical Marcus-Smoluchowski model. While the distance increase in the first tens of pi- coseconds is mainly due to a efficient reaction of the sub-population in close vicinity, diffu- sional separation becomes dominant after ~ 100 ps and finally leads to free ions and radicals.