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The requirement is that $\chi^{(2)}$ be non-centrosymmetric. That's a bit different than having a particular parity. The states involved must be neither odd nor even; the parity must be mixed. That way the dipole matrix element exists between all three intermediate states involved in calculation of the susceptibility.

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If you define $C:q\rightarrow -q$, $P:(x,y,z)\rightarrow (-x,-y,-z)$ and $T:t\rightarrow -t$, then all the Maxwell eaquations are invariant under $C$, $P$, $T$ or any combination of them. To see this you just have to notice how the transformations act on sources and coordinates. The charge conjugation acts non trivially as \begin{align} C\rho&=-\rho,\\ ...

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A basic postulate in elementary particle theories is CPT invariance. Also the weak interaction is the only fundamental interaction that breaks parity-symmetry, and similarly, the only one to break CP-symmetry. ...... The laws of nature were long thought to remain the same under mirror reflection, the reversal of one spatial axis. The results of an ...

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