Chromophore line shapes

It is easiest to formulate this problem in the case of a single high-frequency vibrational mode, or chromophore, so let us consider this situation first. For the absorption line shape, which involves only the ground and excited state of the chromophore, a cmcial element is the 0 —> 1 transition frequency and its dependence on the classical bath coordinates. Second, one needs (in the case of IR spectroscopy) the projection of the transition dipole in the direction p of the electric field axis. This projection can depend on bath coordinates in two ways. 

First, as the molecule on which the chromophore sits rotates, this projection will change. Second, the magnitude of the transition dipole may depend on bath coordinates, which in analogy with gas-phase spectroscopy is called a non-Condon effect For water, as we will see, this latter dependence is very important [13, 14]. In principle there are off-diagonal terms in the Hamiltonian in this truncated two-state Hilbert space, which depend on the bath coordinates and which lead to vibrational energy relaxation [4]. In practice it is usually too difficult to treat both the spectral diffusion and vibrational relaxation problems at the same time, and so one usually adds the effects of this relaxation phenomenologically, and the lifetime 7j can either be calculated separately or determined from experiment. Within this approach the line shape can be written as [92 94]... 

From a theoretical perspective, since the designation of the lab-fixed axes is arbitrary, what is relevant is the relative orientation of the polarizations of the excitation and scattered light. Thus the line shape for excitation light polarized along axis p, and scattered light polarized along axis q (p or q denote X, Y, or Z axes in the lab frame) is called Ipq(co). When p = q this is lyy, and when p q this is IVH. Mixed quantum/classical formulae for Ipq(co) are identical to those for the IR spectmm, except mPi is replaced by apqP which is the pq tensor element of the transition polarizability for chromophore i. Thus we have, for example [6],... 

We wanted to extend this approach to include dynamical effects on line shapes. As discussed earlier, for this approach one needs a trajectory co t) for the transition frequency for a single chromophore. One could extract a water cluster around the HOD molecule at every time step in an MD simulation and then perform an ab initio calculation, but this would entail millions of such calculations, which is not feasible. Within the Born Oppenheimer approximation the OH stretch potential is a functional of the nuclear coordinates of all the bath atoms, as is the OH transition frequency. Of course we do not know the functional. Suppose that the transition frequency is (approximately) a function of a one or more collective coordinates of these nuclear positions. A priori we do not know which collective coordinates to choose, or what the function is. We explored several such possibilities, and one collective coordinate that worked reasonably well was simply the electric field from all the bath atoms (assuming the point charges as assigned in the simulation potential) on the H atom of the HOD molecule, in the direction of the OH bond. 

The Franck-Condon factors of polarizable chromophores in Eq. [153] can be used to generate the complete vibrational/solvent optical envelopes according to Eqs. [132] and [134]. The solvent-induced line shapes as given by Eq. [153] are close to Gaussian functions in the vicinity of the band maximum and switch to a Lorentzian form on their wings. A finite parameter ai leads to asymmetric bands with differing absorption and emission widths. The functions in Eq. [153] can thus be used either for a band shape analysis of polarizable optical chromophores or as probe functions for a general band shape analysis of asymmetric optical lines. 

A typical absorption experiment involves a very large number of individual chro-mophore molecules, and the absorption line shape in a low-temperature solid is usually inhomogeneously broadened. This means that the line shape simply reflects the distribution of possible transition frequencies for the many chromophores, which is due to a distribution of local environments. Thus in this case it is clear that one can learn something about the structure of the solid from an analysis of the inhomogeneous lineshape [1-3]. 

For inhomogeneously broadened line shapes it necessarily follows that no information about time-dependent fluctuations of the chromophore s transition frequency (which I will call spectral dynamics) can be obtained from the line shape itself. This does not mean that such dynamic fluctuations do not occur it simply means that either their amplitude is much smaller than the inhomogeneous line width or that their time scale is much longer than the inverse of the inhomogeneous line width. In either case these dynamic fluctuations are of great interest because they result from time-dependent changes in the local environments of chromophores, and hence can provide information about solid-state dynamics. 

The experimental techniques of fluorescence line narrowing and hole burning were invented, in part, to access this dynamic information. They each involve selective excitation by a narrow-band laser of a nearly resonant subset of chromophores. The resulting fluorescence line shape or hole shape reflects the spectral dynamics of the members of this subset, unobscured by the other chromophores. In a similar vein, in the time-domain photon echo experiment, after the application of a short pulse the inhomogeneous dephasing of all of the chromophores is then rephased by a second pulse, and so the echo decay again reflects only transition frequency fluctuations. 

In fact, single molecule spectroscopy (SMS) experiments have recently become a reality. The first experiments were performed on pentacene (the chromophore) in a p-terphenyl crystal [8-10]. I will focus here on the experiments of Ambrose, Basche, and Moemer [9, 10], which involved repeated fluorescence excitation spectrum scans of the same chromophore. For each chromophore molecule they found an identical (except for its center frequency) Lorentzian line shape whose line width is determined by fast phonon-induced fluctuations (and by the excited state lifetime), as discussed above. However, for each of a number of different chromophore molecules Moemer and coworkers found that the chromophore s center frequency changed from scan to scan, reflecting spectral dynamics on the time scale of many seconds The transition frequencies of each of the chromophores seemed to sample a nearly infinite number of possible values. Plotting the transition frequency as a function of time produces what has been called a spectral diffusion trajectory (although the frequency fluctuations are not necessarily diffusive ). These fascinating and totally... 

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