Selection Rules

Although there are a very large number of energy levels in molecules, transitions are allowed only between specific levels. Several conditions have to met in addition to the primary transition energy requirement, ie,

ΔE = nhν                                                                                                                      (12.4)

Normally n = 1 and only one photon is absorbed. However, with advances in laser techniques, multiphoton absorption has become possible. In addition to energy conservation in an absorption or emission process, angular momentum has to be conserved during a transition. Photons (particles of light) possess an angular momentum of ћ. During an electronic transition, for example, a molecule may get exited from an S state to a P state. In the S state, angular momentum is zero and in the P state, angular momentum is 1 (in units of ћ ). The transition probability P_{i→ f}   from a state  ψ_i to a state  is given by  ψ_f    is given

P_i→f α <ψ_i│µ │ψ_f>│²                                                                                                                                          ( 12.5 )

Where µ is the transition dipole operator. Fluctuations in molecular charge densities, dipole moment, polarizability and so on are responsible for this “transition moment” operator.

Another feature that contributes to the intensity of a spectral line is the population of the energy levels. The lower levels are usually more populated than the higher levels and the intensities of absorption lines from lower levels are usually higher. The relative population of energy levels is governed by the Boltzmann distribution at equilibrium.

N_upper/N_lower α exp (-ΔE/k_BT)                                                                                              (12.6)

Here,  ΔE = E _upper – E _lower, k_B = Boltzmann constant = 1.38 x 10 ^-16 erg/K and T = absolute temperature.  When ΔE is large, the number of molecules in the upper level, N _upper is very small.

Most of spectroscopy deals with absorption of light.  The amount of light absorbed depends on the path length l of the sample (ie the linear extent through which the light travels through the sample), the concentration of the absorbing molecules, c, contained in the sample and the molar absorption coefficient Є.   The Beer-Lambert law which relates the absorbed intensity of light, I, to the incident intensity I_o is given by

I/I_o =  10 ^{– Єcl}                                                                                                                   (12.7)

The quantity I/I_o is called transmittance.  This law is very useful in determining unknown concentrations of molecules in samples when  Є is already determined by previous experiments.  A block diagram of an absorption spectrometer is shown in Fig. 12.2.