Theoretical study on the geometry dependence of the second hyperpolarizability of the allyl cation based on a numerical Liouville three-type analysis

The second hyperpolarizability (γ) of the allyl cation is analyzed in terms of the virtual excitation processes by the numerical Liouville approach (NLA). The influence of molecular geometry on γ is investigated. The results indicate that molecular geometry affects the magnitude and sign of γ and that three types of virtual excitation process exhibit remarkably different dependence on the molecular geometry. A negative γ for C2v geometry without bond alternation is shown to be converted to a positive γ for bond-alternated geometry. Implications of the results are discussed in relation to a polymeric system with a charged defect.