The optical absorption lineshape

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 

We have already seen that this form of electron-phonon coupling expresses shifts in the vibrational modes equilibrium positions upon electronic transition, a standard model in molecular spectroscopy. Applying the polaron transformation to get a Hamiltonian equivalent to (12.27) and (12.29), then using Eq. (12.34) with 2 = Eg = Eg + flM and 1 = E leads to the electronic absorption lineshape in the form 2-abs( w) l(gv //e - vv 2 5( g + dui+ vib(v)-Fviblv )) 

A word of caution is needed here. The golden-rule expression, Eq. (12.33) or (12,43), was obtained for the rate of decay of a level interacting with a continuous manifold (Section 9,1), not as a perturbation theory result but under certain conditions (in particular a dense manifold of final states) that are not usually satisfied for optical absorption, A similar expression is obtained in the weak coupling limit using time-dependent perturbation theory, in which case other conditions are not 

This expression can be interpreted as a decay rate of level g, 0) into the manifold x, v ) only if this manifold is (1) continuous or at least dense enough, and (2) satisfies other requirements specified in Section 9.1. Nevertheless, Eq. (12.59) can be used as a lineshape expression even when that manifold is sparse, leading to the zero temperature limit of (12.57) 

It displays a superposition of lines that correspond to the excitation of different numbers of vibrational quanta during the electronic transition (hence the name multiphonon transition rate). The relative line intensities are determined by the corresponding Franck-Condon factors. The fact that the lines appear as 5 functions results from using perturbation theory in the derivation of this expression. In reality each line will be broadened and simplest theory (see Section 9.3) yields a Lorentzian lineshape. 

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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 ... 

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. 

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... 

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

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] ... 

The probe absorption coefficient a v) can now be calculated by incorporating the (Gaussian) inhomogeneous distribution and convoluting with the homogeneous lineshape. In the hmit of low optical density a( >)/< 1, where / is the crystal length, (A ) is found to be proportional to... 

Fig. 29. Shown here are experimental data points and theoretical curves for the optical nutation in pentacene at 1.8 K. In each plot the vertical axis corresponds to relative absorption (arbitrary units) and r = 0 is taken to be the switching time of the EO pulse. The top plot considers only inhomogeneous lineshape averaging. The bottom plot considers averaging over both the inhomogeneous line shape and the laser beam spatial profile. The fit is quite good in the latter case and provides values of 27.3 1.3 ns for the nutation time and 45 2 ns for T2.

In this model, the symbols have the following meaning. ( )R(a3) ao(l)HD(a3) is the optical rotation produced by the vapour and depends on the number of absorption lengths ao and on the lineshape function D(o)) which takes the form of a Doppler-broadened dispersion curve for magnetic and electric field induced rotation ( )r will depend on the strength of the field and D(oa) on the direction and type of field (see table 2). The transmitted intensity 1 - Ij exp[-aoG(o))] where the lineshape function G(o)) for a single spectral component can usually be accurately described by a Doppler-broadened Lorentzian curve. Finally the terms B and C in equation (53) represent respectively the finite extinction ratio of the polarisers and a laser independent... 

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. 

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