To determine the total electronic configuration, the term symbols and to characterize the transition dipole moment, the product rules for symmetry species are beneficial. They are based on the fact that (with the only constraint that the symmetry species is not degenerated) a direct multiplication of characters is possible. This operation yields a row of characters identical to the characters for some other irreducible representation of the group. This is shown for a combination of symmetry species within point group C_{2v}.
C_{2v}, 2mm 


































To get the product of the symmetry species of A_{2} and B_{1} we just multiply the characters of the two rows:
C_{2v}  E  C_{2}  σ_{v}(xz)  σ'_{v}(yz) 
A_{2}  1  1  1  1 
B_{1}  1  1  1  1 
A_{2} x B_{1}  1  1  1  1 
The characters recieved for the product A_{2} x B_{1 } are identical to the entries for symmetry species B_{2} listed in the original character table of group C_{2v}. Such products have been tabulated in literature, e.g. by Herzberg, Molecular Spectroscopy, Volume III. Below, we present the multiplication table for point group C_{2v}.
C_{2v}  A_{1}  A_{2}  B_{1}  B_{2} 
A_{1}  A_{1}  A_{2}  B_{1}  B_{2} 
A_{2}  A_{2}  A_{1}  B_{2}  B_{1} 
B_{1}  B_{1}  B_{2}  A_{1}  A_{2} 
B_{2}  B_{2}  B_{1}  A_{2}  A_{1} 
Note that
A_{1} x A_{1} = A_{1}
B_{1} x B_{1} = A_{1}...., 
i.e. the square of any not degenerated symmetry species is totally symmetric. 
B_{1} x
A_{1} = B_{1}
B_{2} x A_{1} = B_{2}...., 
i.e. if a totally symmetric species is one factor, the other factor and the resulting product are identical. 
The last table summarizes the rules for not degenerated symmetry species. Using Mulliken's notation with A,' , g, +, B, ", u and −, there is an obvious similarity with the sign rules known from algebraic multiplication.
i:  g x g = g  u x u = g  g x u = u x g = u 
σ_{h}:  (') x (') = (')  ('') x ('') = (')  (') x ('') = ('') 
C_{p}:  A x A = A  B x B = A  A x B = B x A = B 
C_{2}:  1 x 1 = 1  2 x 2 = 1  1 x 2 = 2 x 1 = 2 
Outlook: In another symmetry group (e.g. D_{2h}), the product rules may remind of a cyclic substitution.
D_{2h}:  1 x 2 = 3  2 x 3 = 1  3 x 1 = 2 
Thus B_{2g} x B_{3u }= B_{1u}
Auf diesem Webangebot gilt die Datenschutzerklärung der TU Braunschweig mit Ausnahme der Abschnitte VI, VII und VIII.