Lorentzian band shapes

Figure 3.17 presents ps-TR spectra of the olehnic C=C Raman band region (a) and the low wavenumber anti-Stokes and Stokes region (b) of Si-rra i-stilbene in chloroform solution obtained at selected time delays upto 100 ps. Inspection of Figure 3.17 (a) shows that the Raman bandwidths narrow and the band positions up-shift for the olehnic C=C stretch Raman band over the hrst 20-30 ps. Similarly, the ratios of the Raman intensity in the anti-Stokes and Stokes Raman bands in the low frequency region also vary noticeably in the hrst 20-30 ps. In order to better understand the time-dependent changes in the Raman band positions and anti-Stokes/Stokes intensity ratios, a least squares htting of Lorentzian band shapes to the spectral bands of interest was performed to determine the Raman band positions for the olehnic... 

The negative sign in equations 54-19 and 54-20 reflect the fact that the maximum second derivative is a negative value, which also agrees with Figure 54-1, and it also tells us that the magnitude of the second derivative decreases inversely as the cube of a (for the Normal band shape) and inversely as the fifth power of a (for the Lorentzian band shape), that is as the bandwidth of the absorbance band increases. This explains why the derivatives of the broad absorbance band decrease with respect to the narrow absorbance band as we see in Figure 54-1, and more so as the derivative order increases. 

In our previous chapter we derived the expressions for the first and second derivatives of both the Normal and Lorentzian band shapes [1]. For the following discussion, however, we will address only the Normal case, as we will see, the Lorentzian case will parallel it closely. 

It should be noted that due to er(Af)a a the Fourier transform of % (band-width determined by the dephasing constant of different Fourier transform bands are approximately proportional to the FCFs (0aij 0gg 2 (0ai/ 0go) 2- For the single displaced oscillator case, for example, I I2 = S e 2S, where S is the Huang-Rhys factor of this mode. 

Figure 2.22 Calculated and experimental VCD spectra of 18. Spectra of conformations a and b are calculated at the B3LYP/TZ2P level for S-18. Lorentzian band shapes are used (y = 4.0 crrr1). The spectrum of the equilibrium mixture of a and b is obtained using populations calculated from the B3LYP/TZ2P energy difference of a and b. The numbers indicate fundamental vibrational modes. Where fundamentals of a and b are not resolved only the number is shown.

Figure 1 The upper curve I(v) illustrates a Rayleigh line with a Lorentzian band shape. The broken curves shown are obtained as described in the text R(p) from Eq. (1), R (f>) from Eq. (2), and from Eq. (3) see text.

Lorentzian band shapes are generally assumed with a halfwidth at half-maximum ya for each vibrational mode a and... 

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