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288                                                     P. LAZZERETTI ET AL.
                             5. Symmetry transformations of second-order density


                             The electron  density of a molecule in  the  presence of electric perturbation is a scalar
                             field with perturbation expansion  [6], [11]








                             Relaxing the  Einstein convention,  sums over repeated  Greek indices  are
                             made  explicit in this  Section, to  avoid  misunderstanding whenever  two  couples of
                             repeated indices  and  with         , appear in a formula, compare fo r (92) hereafter.
                             Introducing a basis set  of atomic functions, for the second-order term one defines
                             the expansion




                             For any symmetry operator        (rewritten  when operating on the domain of
                             basis functions  for instance, the rotation-reflexion about  the z-axis, with  matrix
                             representation




                             over a basis set  of Cartesian unit vectors, and belonging to a group G, one has





                             In the  transformed  coordinate system,

















                             Hence the  transformation law for  second-order density is
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