Lorentzian envelope

Dashed lines correspond to the two components of the Lorentzian envelope fitted to the spectrum (smoothed solid line). Linear solid lines represent fluorescence scattering and are taken as ground levels (taken from Ref. 252). 

Fig. 1 Calculated Raman spectra from A alkylpyridinium and B alkyltrimethylammonium. The hydrocarbon tail-length was varied between Cj and C, for each surfactant. For a better visualization, lorentzian envelopes with a fixed line-width of 8 1/cm for all Raman bands were used. The symmetric CH stretching mode for both species is marked using a vertical line and labelled with the corresponding frequency, i.e. 2875 1/cm...

Fig. 3 Comparision between lorentzian-envelop generated Raman spectra, using a fixed line-width of 8 1/cm from heptylpyridinium and the Raman spectrum from a polycrystalline sample of dodecylpyridinium-bromide. A shows the C-C stretching spectral range and B corresponds to the C-H stretching range. The excitation source was the 488 nm line from an Ar ion laser, the scanning rate was 2 l/cm/2s...

Among them, Li-i-HP can be considered a benchmark model system [29, 30] because its low number of electrons makes possible to calculate accurate PES s. Its electronic spectrum has been meassured by Polanyi and coworkers [22], and has been recently very nicely reproduced using purely adiabatic PES s [31]. In the simulation of the spectrum[31], the transition lines were artificially dressed by lorentzians which widths were fitted to better reproduce the experimental envelop. The physical origin of such widths is the decay of the quasibound states of the excited electronic states through electronic predissociation (EP) towards the ground electronic state. This EP process is the result of the non-adiabatic cou-... 

Fig. 2. Comparison of the theoretical (a,b) and experimental (c) results for the first PE band of benzene, corresponding to the X2Eig ground state of Bz+. In the calculation, the line spectra, obtained from the Lanczos algorithm, were convoluted with Lorentzians of width FWHM = 20meV to yield the smooth envelope. For the parameter values, see Table 1. (a) Pure three-mode JT spectrum without mode v2. (b) Convolution of JT spectrum of (a) with a Poisson distribution corresponding to v2. (c) Experimental recording of Ref. [6].

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. 

When y - A2 the equivalent of the microscopic time y is = y/ -Decoupling effects are present when Uj = F. To obtain an approximate value of F we can use the experimental data as follows. First, we evaluate the value of decay of the oscillation envelopes of the angular velocity autocorrelation function as a function of Equation (14) shows that this is, approximately, a Lorentzian, the linewidth of which provides the approximate expression for F. The agreement with the numerical decoupling effect is quantitatively good when the ratio ai/uf is assumed to be equal to 8.5. Simple Markovian models cannot account for decoupling effects. 

Figure 2. Different possibilities that can occur when plotting amplitudes in dependence of the difference between normal mode frequencies 0) and internal mode frequencies 0), . The dashed line indicates the enveloping Lorentzian (bell-shaped) curve that can be expected in the case of a physically well-defined amplitude.

If amplitudes /Inn plotted as a function of Am n. then the distribution of amplitude points should be enveloped by the Lorentzian (bell-shaped) curve of Figure 2 similar to the one describing the liiie shape of spectroscopic bands [9] since this curve complies with expectations (60) - (62). 

In the absence of the continuum these molecular states would be the exact molecular eigenstates [see Eq. (4.27)]. (b) Each m) state can be considered a resonance since it is broadened into a Lorentzian-like envelope of width y as a result of the weak coupling between. y) and /) to the quasi-continuum. The spacing between the resonance states. 

As shown in figure 4.9, the time evolution P f) shows three distinct time dependencies, each characterized by either 7, F, or e (1) The Lorentzian-like envelope, with width F for the complete absorption spectrum transforms into an exponential decay with rate constant F/ft, which is for the short-time decay of PXt)- (2) The set of resonances, separated on average by an energy e, transform into a set of oscillations (i.e., recurrences) whose periods are approximately elh. (3) The envelope of each individual resonance also transforms into an exponential-like decay, characterized by the rate 7/ft, which corresponds to leakage from the sparse i) — /) subspace into the quasi-continuum ). The recurrences described above in (2) are damped out by this slow decay. 

The rate of the initial decay of P (t) is related to the width of the overall absorption envelope for the 8727 to 8927 cm > frequency range of the experimental spectrum. If a Fourier transform were taken of a Lorentzian /(w) for only the principal peak in the spectrum, the decay rate would be significantly smaller. 

If Cj(0 is found from Eq. (4.32) and inserted into Eq. (4.25), the absorption spectrum becomes the structureless Lorentzian band envelope ... 

The statistical limit does not require that jj ) be coupled to all the other zero-order states. All that is required for application of the statistical limit is that a sufficiently large number of the ) states are coupled to j ), so that IVR appears to be irreversible. Thus, the experimental observation of a Lorentzian band envelope in an absorption spectrum does not necessarily imply that the initially prepared zero-order state is coupled to all the remaining zero-order states so that a microcanonical ensemble is formed. What is required is that the effective density of states multiplied by the resolution be much larger than one. 

As will be discussed in chapter 6, of fundamental importance in the theory of unimolecular reactions is the concept of a microcanonical ensemble, for which every zero-order state within an energy interval AE is populated with an equal probability. Thus, it is relevant to know the time required for an initially prepared zero-order state j) to relax to a microcanonical ensemble. Because of low resolution and/or a large number of states coupled to i), an experimental absorption spectrum may have a Lorentzian-like band envelope. However, as discussed in the preceding sections, this does not necessarily mean that all zero-order states are coupled to r) within the time scale given by the line width. Thus, it is somewhat unfortunate that the observation of a Lorentzian band envelope is called the statistical limit. In general, one expects a hierarchy of couplings between the zero-order states and it may be exceedingly difficult to identify from an absorption spectrum the time required for IVR to form a micro-canonical ensemble. 

P(n,t) calculated in this manner for the n = 3 overtone of benzene (Lu and Hase, 1988b, 1989) is plotted in Figure 4.13, where it is compared with the quantum result. The classical P(n,t) decays exponentially to zero without the recurrences seen in the quantum calculation. It is these recurrences which give the structure in the absorption spectrum. The spectrum calculated from the classical exponential P(n,t) is a smooth Lorentzian band envelope with fwhm of 85 cm L... 

For a practical calculation on a finite-size system, we substitute a Lorentzian of width r for the delta function, which should be large enough to envelop at least a few vibrational modes. 

A comparison between the measured ° and calculated second band of the photodetachment spectrum of N02 is shown in Fig. 1. The theoretical line spectrum has been generated by employing the linear vibronic-coupling model with parameters determined by ab initio calculations which are more accurate than those previously available. Subsequently, the line spectrum has been convoluted with Lorentzians of suitable width to account for the finite experimental resolution and thermal line broadening effects. The experimental spectral envelope is seen to be very well reproduced by the theory. The theory predicts, moreover, a sub-structure of vibronic lines under most of the peaks of the spectral envelope, which could not be resolved experimentally. 

Figure 2. Survey and detailed XP spectra of Cls, Ols, P2p, S2p, and Zn2p of a commercial purified ZnDTP, frozen on sputtered gold and used as reference compound. In each detailed spectrum, the points are the original data the line between the points is the envelope of the model Gaussian-Lorentzian product functions (dotted lines) used in the curve-fitting routine. The P2p and S2p signals are fitted with two components to...

For a Lorentzian line where the envelope decays as we use it as the match filter (the apodization function). 

Figure 2.3. Simple spectra and interferograms (a) two infinitesimally narrow lines of equal intensity (b) two infinitesimally narrow lines of unequal intensity note that the amplitude of the beat signal in the interferogram never goes to zero (c) Lorentzian band centered at the mean of the lines in (a) and (c) the frequency of the interferogram is identical to (a) and (b) and the envelope decays exponentially (d) Lorentzian band at the same wavenumber as (c) but of twice the width the exponent of the decay for the interferogram has a value double that of the exponent for (c). denotes the Fourier transform.

The spectra in Figure 2.3c and d both have Lorentzian profiles and yield sinusoidal interferograms with an exponentially decaying envelope. The narrower the width of the spectral band, the greater is the width of the envelope of the interferogram. For a monochromatic source, the envelope of the interferogram will have an infinitely large width (i.e., it will be a pure cosine wave). Conversely, for broadband spectral sources, the decay is very rapid. 

The signal-to-noise ratio of a spectrum can be optimized by multiplying the experimental signal by a weighting function which matches the experimental decay envelope matched filtration . The Fourier transform of an exponential decay with time constant T is a Lorentzian line shape with a full width at half height of H nT) Hz. Thus to obtain the best signal-to-noise ratio for a Lorentzian line of width W Hz the experimental data should be multiplied by a decaying exponential of time constant ll 7tW) s. This... 

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