Optical absorption lineshapes

In many applications we use models that are more explicit about the nature of the initial and final states involved in this transition. A common model (see Chapter 12) is a two-level system that interacts with its thermal environment. The lineshape of interest then corresponds to the photon-induced transition from state 1 to state 2, dressed by states of the thermal environment. The initial and final states are now z) = 11, a and f = 2, a where a and a are states of the bath. Equation (6.21) can then be rewritten as  

5 The form (6.23) relies on a weak system-bath couphng, whereupon the energies are written as additive contributions, for example, Ei -I- Ba, of these subsystems. 

The simplest model for the time-dependent dipole correlation function is an exponentially decaying function, (/i(z)/t(O)) exp(—F Z ). This form leads to a Lorentzian lineshape 

This is a useful model but we should keep in mind its limitations (1) It cannot describe the correct dynamics near Z = 0 because the function exp(—T /1) is not analytic at that point. Also, it does not obey the fundamental identity (6.72), and therefore corresponds to a high temperature approximation. 

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Problem 18.2. A well-known result from the theory of optical absorption lineshapes is that the integrated lineshape associated with the transition between two quantum levels is equal, up to known numerical factors, to the squared radiative coupling element between these levels. For example, using Eq. (18.9) or (18.10) yields / dcoLlai ) o< /zi,2l. Show that, under the Condon approximation, the integrated absorption lineshape of an overall transition between two vibronic manifolds of two electronic states 1 and 2 is also proportional to the squared radiative electronic coupling l/xp2p. 

Cornil et al. have studied the optical absorption spectra of PPV oligomers containing from two to five phenyl/phenylene rings and analysed the extent to which the vibronic couplings affect the lineshape of the spectra11. It is useful to set first the theoretical... 

Kramers-Kronig (KK) transformation of the reflection spectra. This provides the optical absorption "(cu) semiexperimentally and allows a thorough analysis of the various relaxation mechanisms creating the absorption lineshape (2.102), (2.111) of an ideal finite crystal in its phonon bath. This method is currently used. However, two major difficulties often obscure the credibility of the results ... 

A direct consequence of the observation that Eqs. (12.55) provide also golden-rule expressions for transition rates between molecular electronic states in the shifted parallel harmonic potential surfaces model, is that the same theory can be applied to the calculation of optical absorption spectra. The electronic absorption lineshape expresses the photon-frequency dependent transition rate from the molecular ground state dressed by a photon, g) = g, hco ), to an electronically excited state without a photon, x). This absorption is broadened by electronic-vibrational coupling, and the resulting spectrum is sometimes referred to as the Franck-Condon envelope of the absorption lineshape. To see how this spectrum is obtained from the present formalism we start from the Hamiltonian (12.7) in which states L and R are replaced by g) and x) and Vlr becomes Pgx—the coupling between molecule and radiation field. The modes a represent intramolecular as well as intermolecular vibrational motions that couple to the electronic transition... 

Consider now the absorption lineshape, which, as discussed above, corresponds to an optical transition between states 1 and 2. What is measured is the extinction... 

We further wish to emphasize the formal relationship that exists between the optical lineshape and the elements of the molecular density matrix. The general expression for the absorption lineshape is [44] ... 

With a similar setup as used by Ippen et al. for pump-probe experiments, except for an intensity stabilizer in both beams, we performed experiments on the electronic origin at 6027 A and vibronic transitions at 5933 and 5767 A. The results of these experiments are shown in Fig. 22. Except for minor details, the transient on the purely electronic transition is in agreement with our expectation that the singlet excited state is long lived (19.5 ns) on a picosecond time scale. The transient on the 261 cm vibration confirms what was already known from the optical absorption spectrum, namely, that it is very short lived. From the near Lorentzian lineshape at low temperature we calculate a 3.3 ps relaxation time in... 

Various optical data (absorption, polarized light absorption, circular dichroism spectra at several different temperatures and photochemical holebuming experiments) are simulated with a version of the model described in Section 2. The detailed results are reported in refs. 3,4 and 9-12. A typical low temperature absorption lineshape is compared to the experimental spectrum in Fig. 1. Other optical spectra are reproduced with a similar quality agreement. 

The optical absorption or excitation lineshape of dopant molecules in crystalline matrices at low temperatures has been investigated both experimentally and theo-... 

Figure 2.15. Spectra of the damping y(u>) and of the absorption c"(w) (with arbitrary absorption units) derived from our model (2.127)—(2.130), at temperatures ranging between 3 and 70 K. At low temperatures we may distinguish in the spectrum pure acoustical effects below the threshold, and combined effects (acoustical + optical) above the threshold. At high temperatures, both branches contribute to yield the broad and asymmetrical lineshape. The energy origin has been chosen at the unperturbed exciton band botton. So the absorption spectra show a red shift even at low temperatures, which should be considered when comparing the model with the experimental spectra of Figs. 2.12-13.

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