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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