Optical susceptibility, nonlinear second order origin

Lalama, S.J., Garito, A.F. Origin of the nonlinear second-order optical susceptibility of organic systems. Phys. Rev. A 20, 1179-1194 (1979)... 

X and x are the second and third-order nonlinear optical susceptibilities. Two important manifestations of optical nonlinearities are harmonic generation and refractive fndex modulation by electric and optical fields. Their origin can conveniently be explained by considering a plane wave propagation through the nonlinear medium. The polarization is then given by... 

In isotropic media the reflection of all vectors at the origin should not change the nonlinear susceptibility. This yields Xijk = —Xijk which could be only fulfilled by Xijk = 0. In all media with an inversion center the second-order susceptibility tensor vanishes This means, for instance, that optical frequency doubling in gases is not possible. 

---