Linear absorption

Fig. 5.3 shows the result for the calculation of the linear absorption line shape function for the Q_y band absorption of a single Pheo in ethanol solution. The position of the maximum was taken as a parameter, while the linewidth itself was calculated without any free parameters.

The perfect agreement with the experimental curve indicates the very good quality of the en-ergy gap function calculation. The difference between calculated and measured curve at the high energy wing is due to a vibrational progression. The vibrational progression is included within the mixed quantum-classical calculation, but with a classical amplitude. This classical amplitude is a factor 1/f_MN(ω_vib,T) smaller than the quantum mechanically calculated am-plitude of the vibrational progression (cf. Sec. 3.6.2). It is important to mention that the factor f_MN(ω_vib,T) depends explicitly on the vibrational frequency ω_vib and on the temperature T (cf. Eq. 3.94 in Sec. 3.6.2). In Fig. 5.4, the absorption linewidths for the Qy and the Qx band are shown. This results were calculated utilizing 80 ns of MD simulation (8 times 10 ns started with the same initial conformation).

The solution of the TDSE from the MD data and the respective averaging of the expansion co-efficients have been explained in Sec. 4.5. The averaged expansion coefficient can be interpreted as the respective S-operator matrix elements (cf. Sec. 3.6) and be inserted into the equation for the linear absorption cross section (Eq. 3.73).

When the mixed quantum-classical vibrational progression is to be adjusted by the correc-tion factors f_MN(ω_vib,T), the single peaks have to be fitted to Gaussian functions. Inhomo-geneous broadening dominates against homoInhomo-geneous broadening, thus the line shapes can be approximated as Gaussian shaped. After the fitting, the contributions of the main peak and the satellites can be seperated.

The upper panels in Fig. 5.4 show the results of the respective calculation with Eqs. 3.73 and 3.79 for a single Pheo in ethanol. The red lines show different Gaussian fits for the calculated absorption curves.

The Qy and Qx peaks are mainly Gaussian line shaped. Only the wings of the absorption curves have to be fitted with at least one additional Gaussian function to receive the mixed quantum-classically calculated result. When looking at the wings of the Q_x peak in the right upper panel of Fig. 5.4, on both sides of the main peak a vibrational progression is observable.

This vibrational progression is also there for the Qypeak in the left upper panel of Fig. 5.4, but

Figure 5.4:Linear absorption of Pheo in ethanol, computed from 80 ns of MD simulation. Left panels: Qy

band, right panels: Qxband. The lower panels show the lower parts of the respective absorption band in detail. Red dotted line: single Gaussian fit for the lineshape. Red full line: fit containing two Gaussian functions for the main peak and one Gaussian function for each vibrational satellite. Red dashed line:

fit containing four Gaussian functions for the main peak and one Gaussian function for each vibrational satellite (only for the Qyband). The maximum of the main peaks was taken as parameter from [22].

5.1 The single Pheo molecule in ethanol solution it is much weaker.

In what follows, a very simplified picture is considered: if the vibrational progression is caused by a single vibronic coordinate that couples to the transition, this will result in a main peak with equidistant vibrational satellites on both sides of the main peak (cf. Sec. 3.6.2), with a distance in energy of ¯hω_vib. In this approximation a Gaussian shaped main peak will be accompanied by Gaussian shaped vibrational progression peaks that have the same linewidth.

However, in the experimental and in the calculated spectra the distance between the peaks is not equidistant. Furthermore, the linewidths are not the same.

Vibrational progression

The fitting of the main and of the two vibrational peaks on the high energy wing (quantum me-chanically calculated vibrational progression occurs on the high energy wing, cf. Eq. 3.84) with Gaussian functions enables to separate the single peaks and to quantify their respective ampli-tudes. The classically derived amplitude of the vibrational progression can then be corrected utilizing the factor fMN derived in Sec. 3.6.2 (cf. [154]).

The vibrational progressions of the Q_xand the Q_yband in Fig. 5.4 and the respective fits can be observed in much more details in the two lower panels. It can be seen that the calculation result can be fitted properly with one Gaussian function for each satellite and two Gaussian functions for the Q_x main peak as well as with four Gaussian functions for the Q_ymain peak.

For the fitting of the Q_ymain peak, more Gaussian functions are necessary, since the vibrational satellites have a much lower amplitude. If the amplitude of the vibrational peak is small, the contibution of the main peak at the frequency of the vibrational peak becomes of importance.

Moreover, if the frequency is far away from the resonance energy (15015 cm⁻¹for the Q_yband), the single Gaussian fit approximates the respective part of the lineshape rather inadequately.

It has to be stated here that on the one hand, the fitting of the main peak with more Gaussian functions yields a better agreement with the calculated result. But on the other hand, for weak vibrational progression (as for the Qy band), it is not clear which part of the total absorption cross section belongs to which peak (main peak or vibrational satellite).

In Fig. 5.5, the Gaussian fits shown in the Fig. 5.4 are utilized to calculate the Pheo (in ethanol) absorption spectrum in the optical range (except the Soret band). The positions of the main peaks were used as fit-parameters from the experiment [22]. The maximum of Q_ymain band intensity was normalized to 1. The ratio between the Qy band and the Qxband intensity was computed from the oscillator strengths that were calculated with DFT (cf. Sec. 4.2). In order to correct the mixed quantum-classical vibrational progression, the vibrational satellites were increased by the frequency-dependent factor f₀₁(ω_vib,T) (cf. Sec. 3.6.2). This was done for all the four fitted vibrational peaks. The underlying assumption is that the two vibrational progression peaks of each band (Q_x and Q_y band) consist of a variety of vibrational modes coupling to the electronic transition, not only the two vibrational modes that correspond to the two visible vibrational satellites. Thus, the whole vibrational progression is given as a first order vibrational progression of all the vibrational modes that couple to the electronic transition. The respective result matches the experimentally measured absorption curve [22]

quite well.

In Fig. 5.5, the dashed green line shows a spectrum which was computed by overestimating the vibrational satellites by a factor of 2. This was done to demonstrate that the linewidths of the first vibrational satellites have the same linewidths as the measured vibrational satellites.

Those are broader than the Q_x and Q_y main peaks. This indicates that not only two, but a variety of vibrational modes couple to the electronic transitions. The two second vibrational satellites (the second satellite of the Qymain peak and the one of the Qx main peak) are both multiplied by the factor f₀₁, not by the factor f₀₂. Nevertheless, the second visible vibrational satellite for both of the electronic transitions is modelled quite well. This indicates that this second visible satellite in the experiment consists of two contributions. The first contribution is

Figure 5.5:Linear absorption of Pheo in ethanol. The only parameters utilized for the calculated curves are the maxima of the two peak positions taken from [22]. Black line: experimental result. Red lines:

single Gaussian fit functions from the calculated results including the Qy band, the Qxband, and all vibrational satellites (cf. Fig. 5.4). Dashed green line: sum over the red lines. Dotted green line: same result as dashed green line, but the vibrational satellites are multiplied with a factor of 2. This makes it possible to compare the linewidths of the first vibrational satellites which match the experimental result quite well.

due to a vibrational progression of hydrogen vibrational modes (the wave number is about 3000 cm^-1) and the second contribution is due to a second order vibrational progression of carbon and nitrogen vibrational modes. This, however, can not be determined conclusively within this work, since the contributions of the first and the second order vibrational progressions can not be seperated.

Finally, the high energy wing of the Q_y band is to be discussed. Fig. 5.3 shows that the computed lineshape matches the measured lineshape, aside from the high energy wing of the peak. Considering the results of this section, it seems to be clear that low frequency vibrational modes as well couple to the electronic transition. The vibrational progression due to this low frequency modes thus affects the main peak itself. This low frequency vibrational progression can not be seperated from the main peak. Thus, it can not be amplified by a factor f01(ω_vib,T) in order to correct the mixed quantum-classical vibrational progression.