Theory Of Spectroscopy

THEORY :-
         Vibrational spectroscopy of molecules can be relatively complicated. Quantum mechanics requires that only certain well-defined frequencies and atomic displacements are allowed. These are known as the normal modes of vibration of the molecule. A linear molecule with N atoms has 3N - 5 normal modes, and a non-linear molecule has 3N - 6 normal modes of vibration. There are several types of motion that contribute to the normal modes. Some examples are:

•    stretching motion between two bonded atoms;
•    bending motion between three atoms connected by two bonds;
•    Out-of-plan deformation modes that change an otherwise planar structure into a non-planar one.
        
          Infrared spectroscopy allows one to characterize vibrations in molecules by measuring the absorption of light of certain energies that correspond to the vibrational excitation of the molecule from v = 0  v = 1 (or higher) states. As indicated above, not all of the normal modes of vibration can be excited by infrared radiation. There are selection rules that govern the ability of a molecule to be detected by infrared spectroscopy.
         
       The Raman Effect was originally observed in 1928. It is due to the interaction of the electromagnetic field of the incident radiation, Ei, with a molecule. The electric field may induce an electric dipole in the molecule, given by

                                  p =  Ei          ...................(1)


where  is referred to as the polarizability of the molecule and p is the induced dipole. The electric field due to the incident radiation is a time-varying quantity of the form

                                  Ei = Eo cos (2 i t)                    .........................(2)


For a vibrating molecule, the polarizability is also a time-varying term that depends on the vibrational frequency of the molecule, 
vib
                    =  o +  vib cos(2 vib t)                                                                         (3)


Multiplication of these two time-varying terms, Ei and , gives rise to a cross product term of the form:
                                          
         This cross term in the induced dipole represents light that can be scattered at both higher and lower energy than the Rayleigh (elastic) scattering of the incident radiation. The incremental difference from the frequency of the incident radiation,  i, are by the vibrational frequencies of the molecule,  vib. These lines are referred to as the "anti-Stokes" and "Stokes" lines, respectively. The ratio of the intensity of the Raman anti-Stokes and Stokes lines is predicted to be

                                        

      The Boltzmann exponential factor is the dominant term in equation (5), which makes the anti-Stokes features of the spectra much weaker than the corresponding Stokes lines.


         Infrared spectroscopy and Raman spectroscopy are complementary techniques, because the selection rules are different. For example, homonuclear diatomic molecules do not have an infrared absorption spectrum, because they have no dipole moment, but do have a Raman spectrum, because stretching and contraction of the bond changes the interactions between electrons and nuclei, thereby changing the molecular polarizability. For highly symmetric polyatomic molecules possessing a center of inversion (such as benzene) it is observed that bands that are active in the IR spectrum are not active in the Raman spectrum (and vice-versa). In molecules with little or no symmetry, modes are likely to be active in both infrared and Raman spectroscopy.