The theory of molecular nonlinear optics based on the sum-over-states (SOS) model is reviewed. The interaction of radiation with a single wtpisolated molecule is treated by first-order perturbation theory, and expressions are derived for the linear () polarizability and nonlinear (, ) molecular hyperpolarizabilities in terms of the properties of the molecular states and the electric dipole transition moments for light-induced transitions between them. Scale invariance is used to estimate fundamental limits for these polarizabilities. The crucial role of the spatial symmetry of both the single molecules and their ordering in dense media, and the transition from the single molecule to the dense medium case (susceptibilities , , ), is discussed. For example, for , symmetry determines whether a molecule can support second-order nonlinear processes or not. For asymmetric molecules, examples of the frequency dispersion based on a two-level model (ground state and one excited state) are the simplest possible for and examples of the resulting frequency dispersion are given. The third-order susceptibility is too complicated to yield simple results in terms of symmetry properties. It will be shown that whereas a two-level model suffices for asymmetric molecules, symmetric molecules require a minimum of three levels in order to describe effects such as two-photon absorption. The frequency dispersion of the third-order susceptibility will be shown and the importance of one and two-photon transitions will be discussed.
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