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Far from relevant resonances, when frequency dispersion of the nonlinear response can be neglected, a susceptibility can acquire full permutation symmetry among spatial indices. For second order this reduces a general symmetric-input tensor's 18 components to 10 before material point-group reductions. Mere transparency does not establish this approximation. The name refers to D. A. Kleinman.^[ref]

For a second-order response, $P_i^{(2)}({\omega }_3)={ϵ}_0{\sum }_{jk}{\chi }_{ijk}^{(2)}({\omega }_3;{\omega }_1,{\omega }_2)E_j({\omega }_1)E_k({\omega }_2)$, with the appropriate frequency conservation and negative-frequency convention for difference-frequency processes. Interchanging the two input field-frequency pairs gives intrinsic permutation symmetry of nonlinear susceptibility:For second-harmonic generation the input frequencies agree, so the last two Cartesian indices are symmetric and can be contracted into six input-pair labels. This reduces the unrestricted 27 Cartesian entries to at most 18 before crystal symmetries are imposed.

Crystal point-group transformations must leave the material tensor invariant and can force entries to vanish or become related. Under spatial inversion both electric field and polarization reverse, whereas the field product is even. A centrosymmetric bulk therefore has zero electric-dipole ${\chi }^{(2)}$. Surfaces and interfaces can break inversion symmetry, and higher-multipole processes are outside that simple prohibition.

Kleinman symmetry is a stronger approximate permutation symmetry used sufficiently far from resonances when dispersion can be neglected over the participating frequencies. It is not justified merely because two input frequencies are equal; resonant or appreciably dispersive media need the full frequency-dependent relations. These distinctions determine which polarization and propagation geometries have nonzero effective coupling, in addition to whether they satisfy phase matching.^[ref]

For a local electric-dipole response of order $n$, writeIts susceptibility is a rank-$(n+1)$ tensor. Initially it has $3^{n+1}$ components in three dimensions, before symmetries and frequency constraints are applied. nonlinear-susceptibility tensor symmetry involves several logically different restrictions.

Neumann's principle requires invariance under the material point group. For an operation $R$,For inversion, $R=-I$, so ${\chi }^{(n)}=(-1{)}^{n+1}{\chi }^{(n)}$. Thus all even orders vanish in a centrosymmetric material's local bulk electric-dipole response. In particular ${\chi }^{(2)}=0$ there, while ${\chi }^{(3)}$ is permitted. Surface symmetry breaking, an applied bias or nonlocal magnetic/quadrupole response lies outside that statement. A noncentrosymmetric crystal permits some second-order components, but other point-group operations can still eliminate a chosen component.

Intrinsic permutation symmetry follows from interchanging commuting input field factors, provided the associated frequency labels are interchanged too. At second order,For equal input frequencies this makes the last two spatial indices symmetric, reducing 27 components to 18 before material symmetries. At unequal frequencies it does not identify the two index arrangements with the arguments held fixed. Reality of the time-domain polarization separately givescausality imposes dispersion relations; neither arbitrary real coefficients nor full index symmetry should be inferred merely from the word lossless.

Kleinman symmetry is a stronger approximation far from relevant resonances when frequency dispersion can be neglected. A frequency-independent effective interaction energy makes its derivatives interchangeable, allowing full permutation of output and input spatial indices. At second order a fully symmetric rank-three tensor has $\left(\genfrac{}{}{0ex}{}{5}{3}\right)=10$ components. Close to resonance or across a dispersive band, that reduction can fail even when absorption at the measured frequencies is small. The original analysis distinguishes high-resonance electronic response from strongly dispersive lattice mechanisms and identifies additional symmetry constraints on the former.^[ref]

For an isotropic fourth-rank susceptibility,If the relevant input slots are fully interchangeable, the coefficients coincide and ${\chi }_{xxxx}^{(3)}=3{\chi }_{xxyy}^{(3)}$. With distinct or mixed-sign frequency slots, frequency arguments must be followed and isotropy alone leaves the three displayed structures. These distinctions explain why tensor contractions, pump polarizations and operating frequencies must be specified before comparing effective self- and cross-phase coefficients.