The general failure of Kleinman symmetry in practical nonlinear optical applications

Evidence is presented to refute the general applicability of the overall permutation symmetry condition originally proposed by Kleinman in describing the non-resonant nonlinear optical properties of the overwhelming majority of practical materials and systems. Kramers–Kronig dispersion relations demonstrate that the range of validity of Kleinman symmetry is substantially more restrictive than is widely assumed. These findings are supported by numerous experimental observations of effects explicitly forbidden by Kleinman symmetry but allowed by the dispersion relations. Considerations of molecular symmetry coupled with the Kramers–Kronig dispersion relations yield alternative simple and general macroscopic symmetry relations that are consistent with experimental observations.