Florian Vogelbacher

thanks a lot for your answer. and How about Mirror symmetry?

thank you

   This is one fundamental result in nonlinear optics. The idea is that the second order susceptibility is a pure antisymmetric tensor that when two spatial indices commute it takes value zero. In any case it is convenient to know the physical details for such definition reading the section 10 of a classical text as

  L.D. Landau and E.M. Lifshitz, Electrodynamics of Continous Media. Pergamon. New York. (1960)

  Dear Florian,

  The second order susceptibility can only occur in crystals that don't present spatial inversion symmetry. This is what separate them from other approaches as the third order susceptibility. The physical resons associated to the electric polarization are not simple to show and it is better to go to text. Microscopically it is due, very basically, to the destruction of two photons of given only one of double frequency and the potential must be invariant under the spatial transformation x-> -x.

    I have reading my post and I see that it is not very well written.Sorry.

   The second order susceptibility can only occur in crystals that don't present spatial inversion symmetry. This is what separate them from other approaches as the third order susceptibility. The physical reasons associated to the electric polarization are not simple to show and it is better to go to a text on the subject. Microscopically it is due, very basically, to the destruction of two photons of a given frequency to only one photon of double frequency and the potential must be invariant under the spatial transformation x-> -x for avoiding this susceptibility. In fact, the elements of the Coxeter subgroup of the point group gives this information directly.

Dr. Florian Vogelbacher

Thank you. 

Yes, im working on  ( P 6 m 2) space group and D3h point Group.

Best regards