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We have designed a hyper-Rayleigh scattering scheme to measure six scalar invariants of the squared hyperpolarizability tensor ␤ 2 . Our theoretical approach expresses the rotational invariants of the irreducible ␤ components as scalars, which eliminates the need for difficult frame transformations. We applied our scheme to several conjugated chiral molecules and found that there are significant Kleinman-disallowed pseudotensor contributions to their hyperpolarizability. These components, alongdoi:10.1364/josab.17.001531 fatcat:ogr6jelrwvdlxdn5lwsfz5bixa