General dispersion formulas for molecular third-order nonlinear optical properties

For the electronic optical hyperpolarizabilities for molecules γω∥=γZZZZ(−ωσ;ω1, ω2,ω3) and γω⊥=γZXXZ(−ωσ;ω1, ω2,ω3), where Z and X are laboratory axes and ωσ=ω1+ω2+ω3, it is demonstrated that the following relationships exist: (1) γω∥/γ0∥ =1+Aω2L+⋅⋅⋅ , where ω2L=ω2σ+ ω21+ω22+ω23 and A is frequency independent; (2) γω⊥/γ0⊥=1+Bω2L+ ⋅⋅⋅ , where B=p+qa and p and q are frequency independent and a=(ωσω3−ω1ω2)/ω2L ; (3) (1)/(3) (γω∥/γω⊥)=1+Cω2L +⋅⋅⋅ , where C=r(1−6a) and r is frequency independent. In particular, for the four nonlinear optical processes: Kerr (K), degenerate four-wave mixing (D), electric-field-induced second-harmonic generation (E), and third-harmonic generation (T), the ratios (in the same order) are for B, 1:(1+k/2):(1+k/3):(1+k/6) (where k is frequency independent) and for C, 1:(−2):(−1):0.