Lorentzian distribution resonant transition energies

Still assuming that a Lorentzian distribution of vibrational energies and the dipole approximation are employed. In this expression is the IR transition momenL Mu is the Raman transition probability, is the resonant mode frequency and is the natural line width of the transition. Since sum-frequency active modes must be both IR- and Raman-active, any vibrational mode that has an inversion centre cannot be sum-frequency-active. This result coupled with the coherent nature of sum-frequency generation precludes any sum-frequency response from bulk isotropic media. 

The energy distribution of the emitted radiation and that of the resonant absorption cross section have identical lineshapes. Both are Lorentzian lines centered at the nuclear transition energy E, with fullwidth at half height given by the uncertainty principle energy width of the nuclear excited state. This width, the natural linewidth, is defined by... 

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