Symmetry elements second-harmonic generation

On the assumption of total symmetry of the tensor of third-order nonlinear polarizability c(— co coi, cog, cog), its non-zero and independent elements are the same as those of Table 12. Direct theoretical calculations of c = c(0 0,0,0) have been performed for the atoms of inert gases and some simple molecules. Values of the tensor elements = c(— cu cu, 0,0) have been determined for numerous molecules from static Kerr effect studies and values of c = c(— cd ot>,coi — col) from measurements of optical birefringence induced by laser li t. Measurements of second-harmonic generation by gases in the presence of a static electric field yield the tensor elements c " = c( — 2co co, to, 0), which can also be obtained from second-harmonic scattering in centro-symmetric liquids. The elements of the tensor c = c(— 3co co, co, co)... 

In a recent account, Radhakrishnan has discussed molecular structure, symmetry, and shape as the design elements in the fabrication of molecular crystals for second harmonic generation and has highlighted the role of computational modeling. 

The assumption, known as Kleinman symmetry, that the quadratic (and higher-order) response function is symmetric imder interchange of any pair of operators is only true at zero frequency, as is easily verified from O Eq. 5.31. Note that the quadratic response function is symmetric when permuting the operators B and C at co = co, which is the case for, for example. Second Harmonic Generation j jjk -2(o-,co,(o) = Pikj -2(o-,(o,co). Kleinman symmetry is often assumed in calculations of the electric dipole hyperpolarizabUity at low frequencies where it is approximately vahd. This reduces the number of independent tensor elements and thus the computational effort... 

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