The absorption lineshape

Lee S-Y 1998 Forward and inverse transforms between the absorption lineshape and Raman excitation profiles XVith int. Conf on Raman Spectroscopy ed A M Heyns (New York Wiley) pp 48-51... 

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 ... 

In die Fourier transform of a real time series, die peakshapes in the real and imaginary halves of die spectrum differ. Ideally, the real spectrum corresponds to an absorption lineshape, and die imaginary spectrum to a dispersion lineshape, as illustrated in Figure 3.20. The absorption lineshape is equivalent to a pure peakshape such as a Lorentzian or Gaussian, whereas die dispersion lineshape is a little like a derivative. 

Both Langhoff and Robinson 58> and Delory and Trie 59> have considered the effects on the emission decay curves and on the absorption lineshapes of variations with energy in the density of states and in the coupling strengths of the dense manifold with the intensity-carrying discrete state. The former workers also extended their treatment to include coupling with a second manifold, the radiation field continuum. 

For both NMR spectroscopy in the solid state and EPR spectroscopy in the liquid state it is possible to determine a fairly large number of parameters s . After solving this first problem, one has then to determine the absorption lineshapes in terms of this information. The main aim of this chapter is to show the algorithm of Chapter III at work. However, before launching into this illustration, we would like to mention the alternative approaches currently being used in the field of magnetic resonance. 

We have found that up to constant factors, the absorption lineshape is determined by the power spectriiin that characterizes the coordinate that couples to the external... 

Having found a relationship between the absorption lineshape associated with a periodically modulated system variable and the power spectrum of this variable, we now consider a specific example. Consider a harmonic oscillator which is randomly perturbed so that its frequency changes in time as ... 

A standard experimental probe of this motion is infrared spectroscopy. We may use the results of Sections 7,5 and 8.2.3 to examine the effect of interaction with the thermal environment on the absorption lineshape. The simplest model for the coupling of a molecular system to the radiation field is expressed by a term —fi S in the Hamiltonian, where is the molecular dipole, and (t) is the oscillating electric field (see Section 3.1). For a one-dimensional oscillator, assuming that /r is proportional to the oscillator displacement from its equilibrium position and taking cos((uZ), we find that the coupling of the oscillator to the thermal environment and the radiation field can be modeled by Eq. (8.31) supplemented by a term (F/ni where F denotes the radiation induced driving force. We can... 

In Section 6.2,3 we have seen that a simple quantum mechanical theory based on the golden rule yields an expression for the absorption lineshape that is given essentially by the Fourier transform of the relevant dipole correlation function (/ii(0)/ii(Z)). Assuming again that ix is proportional to the displacement x of the oscillator from its equilibrium position we have... 

A very important result of the theory of quantum dynamics is the connection between the time evolution in a given spectral region and the absorption lineshape into the same region. That such a cormection exists is to be expected, because the time evolution is detennined by the distribution of initial amplitudes among exact eigenstates according to Eq. (2.6), while the absorption process, in principle, prepares these initial amplitudes in the spectral region of interest. 

The absorption lineshape corresponds to the photon-energy dependence of the rate at which the photon is absorbed by the molecule. We consider absorption under conditions where it is a linear process, that is, where the rate at which the molecular system absorbs energy from the radiation field at frequency co is proportional to the radiation intensity (number of photons) at this frequency. Under such conditions it is enough to consider the rate of absorption from a single photon state and to use the... 

We are interested in the rate at which the dressed state 0 = g, k, or rather the probability that the system remains in this state, decays because of its coupling to the state Is, vac and through it to the continuum /. The absorption lineshape is this rate, displayed as a function of > = fo. This rate is evaluated in Appendix 9B and leads to the following expression for the absorption lineshape... 

Show that under the same model assumptions used above the absorption lineshape is Lorentzian and the decay rate of state l after it is initially prepared is exponential. Also show that the decay rate is Vg/h and the width of the Lorentzian is F, with... 

For the initial value problem with i/f(z = 0) = s) we got an exponential decay with the characteristic relaxation rate kg = Yg/h. For the absorption lineshape into state (s) we got a Lorentzian with linewidth given by the same T,. There appears to be a fundamental relationship between the lifetime... 

The model (9.73)—(9.75) was presented as an initial value problem We were interested in the rate at which a system in state 0) decays into the continua L and R and have used the steady-state analysis as a trick. The same approach can be more directly applied to genuine steady state processes such as energy resolved (also referred to as continuous wave ) absorption and scattering. Consider, for example, the absorption lineshape problem defined by Fig. 9.4. We may identify state 0) as the photon-dressed ground state, state 1) as a zero-photon excited state and the continua R and L with the radiative and nonradiative decay channels, respectively. The interactions Fyo and correspond to radiative (e.g. dipole) coupling elements between the zero photon excited state 11 and the ground state (or other lower molecular states) dressed by one photon. The radiative quantum yield is given by the flux ratio Yr = Jq r/(Jq r Jq l) = Tis/(Fijj -F F1/,). 

Note that in such spectroscopic or scattering processes the pumping state 0) represents a particular state of energy Eq out of a continuous manifold. In most cases this state belongs to one of the manifolds R and L. For example, in the absorption lineshape problem this photon-dressed ground state is one particular state of the radiative (R) continuum of such states. 

Appendix 9B Evaluation of the absorption lineshape for the model of Figs 9.2 and 9.3... 

Recalling that Eq = Eg + hu> we find that the absorption lineshape is given by... 

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 T 0 limit of Eq. (12.55b) written for the absorption lineshape of a diatomic molecule with a single vibrational mode a. 

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... 

Solution The absorption lineshape is given by Eq. (18.11). Without invoking yet the Condon approximation it reads... 

We next consider the effect of thermal relaxation on the absorption lineshape. We start with the Bloch equations in the form (10.184),... 

In Section 9.3 we have seen that there is in principle a close relationship between an absorption lineshape and the underlying dynamics of a molecule excited to the corresponding spectral region. The discussion in the previous section however has taught us that life is less simple In many systems the absorption lineshape is an average over many individual molecules that experience different local environments,... 

Whereas the absorption lineshape is always positive, the dispersion lineshape has positive and negative parts it also extends further. 

Two peaks in Fv at Q] jzJ]2, are expected these are just the two lines of the spin 1 doublet. Note that the two peaks have opposite signs - that is they are anti-phase in Fv In addition, since these are cosine modulated we expect the absorption lineshape (see section 7.2). The form of the cross-peak multiplet can be predicted by "multiplying together" the Fl and F2 multiplets, just as was done for the diagonal-peak multiplet. The result is shown opposite. This characteristic pattern of positive and negative peaks that constitutes the crosspeak is know as an anti-phase square array. 

Thus the real part shows the dispersion mode lineshape, and the imaginary part shows the absorption lineshape. The 90° phase shift simply swaps over the real and imaginary parts. 

In general this is a mixture of the absorption and dispersion lineshape. If we want just the absorption lineshape we need to somehow set to zero, which is easily done by multiplying S((d) by a phase factor exp(i0j). 

The absorption lineshape cr(co) is given by the linear response to the stationary external field. To that end, we need to evaluate the density matrix to first order in Linl. We then have... 

In conclusion, in this section we presented the formal expressions for the absorption lineshape [Eq. (70)] and for spontaneous Raman and fluorescence spectroscopy. For the latter, we derived Liouville space expressions in the time and the frequency domain [Eqs. (74) and (75)], an ordinary correlation function expression [Eq. (76)], and, finally, the factorization approximation resulted in Eqs. (77) and (78). The factorization approximation is expected to hold in many cases for steady-state experiments and for time-resolved experiments with low temporal resolution. It is possible to observe a time-dependent shift of spontaneous emission lineshapes using picosecond excitation and detection [66-68]. This shift arises from the reorganization process of the solvent and also from vibrational relaxation that occurs during the t2 time interval. A proper treatment of these effects requires going beyond the... 

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