Deviations from Kleinman symmetry measured for several simple atoms and molecules

Deviations from Kleinman symmetry for the second hyperpolarizability tensors of some simple atoms and molecules (He, Ar, Kr, Xe, CH₄, CF₄, SF₆, H₂, D₂, N₂, O₂, CO₂, C₂{H}₆$, ${CHF}₃$, and ${C}₂H₄) have been measured down to the 0.1% level of accuracy at several frequencies far below the first strong electronic resonance, by means of dc-electric-field--induced second-harmonic generation (ESHG). The results of the experiment are discussed within the framework of several models. The observed Kleinman-symmetry deviations are small compared to the electronic dispersion, and appear to be dominated by electronic rather than vibrational contributions. However, vibrations are shown to play a role in the case of H₂, and the results observed for H₂ and D₂ differ significantly. The observed deviation for He is relatively large and should provide a sensitive test of ab initio calculations.