As is pointed in this question, second harmonic generation is forbidden in the bulk of the materials possessing centrosymmetry. In some papers it is said that in the dipolar approximation the SHG vanishes, and if we consider magnetic dipoles or electric quadrupole terms or other higher order multipole terms, they can generate second harmonic.
I was not sure what do they mean by this. How do we consider various multipole terms of the field here? I know that the optical nonlinearities are expressed with the expansion of the polarization vector of the material (which means dipole): $$P=P_0+\epsilon_0 \chi^{(1)} E+\epsilon_0 \chi^{(2)} E^2 +\epsilon_0 \chi^{(3)} E^3 +...$$
but don't know how the similar expansions can exist for the higher order multipoles.
