Transformation Properties of Tensors w.r.t. Isometric Transformations

Typical examples for vectors of this type are the electrical dipole moment M e)f( ) or any vector of the type Xr( ) = ZkXk( ), where the sum is to be extended over a set of nuclei equivalent w.r.t. ( ). Equation (3.27) follows immediately from the properties of T NCf) 3 and has been proved earlier for the expectation value of the electric dipole operator 

The coupling operator of an irreducible operator (A, ( )) of rank s for a semirigid model in a multipole field (Esa) may be taken in the form 

Further important mathematical tools are the two fundamental formulae from rotation group theory (s GIN) 

33) and (3.33 ) express the transformation behavior of the rotation group matrices w.r.t. rotations of the laboratory system ei1 and the frame system ef, respectively. The second Eq. (3.33 ) assumes for the case H . ( %) the special form 

For electric and magnetic dipole moment and the electric and magnetic quadrupole operator the last equation yields 

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