Abstract: The dynamical Lie algebraic (DLA) formulation is used to describe the nonlinear optical process for the model system of substituted benzenes in a Pariser-Parr-Pople(PPP) Hamiltonian. It is shown how to generate the dynamical Lie algebra for the given Hamiltonian, and then the evolution operator for the system is expressed as a function of group parameters in terms of the elements of the dynamical algebra. The group parameters can be determined with a set of coupled nonlinear differential equations. The statistical average of the molecular electronic polarization can be determined according to the density operator formalism in statistical mechanics, and the hyperpolarizability of the molecules is thus obtained readily. An expression for the hyperpolarizabilities of the di-substituted benzenes as a function of the on-site energy is derived. Compared with other quantum calculations, DLA method appears to provide an effective method for the calculation of the hyperpolarizability of conjugated organic molecules.

Key words: Dynamical Lie algebraic method, Second-order optical nonlinearity, Statistical mechanics, Substituted benzene system