= Bulk electric-dipole selection rule
{title2=$\chi^{(2)}=0$}
Suppose a homogeneous bulk material is unchanged when all spatial coordinates are reversed. Both <electric polarization> and <electric field> reverse under this inversion, so a local <electric dipole> response must obey $P(-E)=-P(E)$. Substituting the <nonlinear optical susceptibility> expansion gives
$$
P(-E)+P(E)=2\epsilon_0\left(\chi^{(2)}E^2+\chi^{(4)}E^4+\cdots\right)=0.
$$
For every sufficiently small $E$, all even coefficients vanish. In the full <tensor> relation, two reversed input <electric fields> leave their product unchanged while the output <electric polarization> changes sign, forcing every bulk electric-dipole component of $\chi^{(2)}$ to vanish.
The qualification matters: the argument assumes inversion <symmetry>, a local bulk <electric dipole> response, and no applied bias that breaks that <symmetry>. A surface breaks bulk inversion <symmetry>; spatially varying or magnetic responses also lie outside this local electric-dipole model. Thus detecting <second-harmonic generation> at a surface or through a weaker magnetic contribution does not contradict the bulk selection rule. Conversely, absence of inversion <symmetry> permits some quadratic components but does not guarantee a nonzero component in every beam geometry.
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