IR and Raman Transition Probabilities

In order to calculate the IR and Raman transition probabilities per unit time between the vibration state, i, to the vibration state, j, it is necessary to express in quantum mechanical terms the operator describing the interaction between the molecule and the electromagnetic radiation, which is given by 

the probability for the absorption or emission of electromagnetic radiation per unit time for the dipole transition is given by [59] 

In the case of the IR process, the dipole moment, p = p, is given by Equation 4.16 in the case of the Raman process, the dipole moment is given by p = aE. For molecules in the gas phase with a haphazard orientation and where the average of the square of the angular part is one, the probability for the absorption or emission of electromagnetic radiation per unit time for the dipole transition can be written as follows  

in the case of IR absorption neglecting the anharmonic terms, the transition probability can be expressed in the following form [56] 

With respect to the transition probability in the Raman case, it is determined by the following integrals [10]  

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