In centrosymmetric media, the second order susceptibility becomes zero due to the inherent symmetry of the material, which prevents any change when transitioning from position vector r to -r. This phenomenon is a fundamental aspect of nonlinear optics, as detailed in the lecture on nonlinear spectroscopy and the textbook "Introduction to Nonlinear Optics." The symmetry of the media ensures that nonlinear optical effects, which rely on second order susceptibility, do not manifest.
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