In quantum mechanics the basis for a spectroscopic selection rule is the value of the transition moment integral 
where  and  are the wave functions of the two states, "state 1" and "state 2", involved in the transition, and  is the transition moment operator. This integral represents the propagator (and thus the probability) of the transition between states 1 and 2; if the value of this integral is zero then the transition is "forbidden".
In practice, to determine a selection rule the integral itself does not need to be calculated: It is sufficient to determine the symmetry of the transition moment function  
If the transition moment function is symmetric over all of the totally symmetric representation of the point group to which the atom or molecule belongs, then the integrals value is (in general) not zero and the transition is' allowed. Otherwise, the transition is "forbidden".
The transition moment integral is zero if the transition moment function,  is anti-symmetric or odd, i.e.  holds. The symmetry of the transition moment function is the direct product of the parities of its three components. The symmetry characteristics of each component can be obtained from standard character tables. Rules for obtaining the symmetries of a direct product can be found in texts on character tables.