A generalized few-state model for the first hyperpolarizability

The properties of molecules depend on their chemical structure and thus structure–property relations help design molecules with desired properties. Few-state models are often used to interpret experimental observations of non- linear optical properties. Not only the magnitude, but also the relative orientation of the transition dipole moment vectors is needed for few-state models of the non-linear optical properties. The effect of the relative orientation of the transition dipole moment vectors is called dipole alignment and this effect has previously been studied for multiphoton absorption properties. However, so far no such studies are reported for the first hyperpolarizability. Here we present a generalized few-state model for the static and dynamic first hyperpolarizability β, accounting for the effect of dipole alignment. The formulae derived in this work are general in the sense that they can be used for any few-state model, i.e. two-state model, three-state model or in general an n-state model. Based on the formulae, we formulate minimization and maximization criteria for the alignment of transition dipole moment vectors. We demonstrate the importance of dipole alignment by applying the formulae to the static first hyperpolarizability of ortho-, meta- and para-nitroaniline. The formulae and the analysis provide new ways to understand structure–property relationship for β and can hence be used to fine-tune the magnitude of β in a molecule.