Hyperpolarizabilities

Hyperpolarizabilities are a class of nonlinear optical properties of molecules and materials. They describe the nonlinear response of the electric dipole moment of a system to an applied electric field. While linear polarizability describes how the dipole moment changes linearly with the field, hyperpolarizabilities capture higher-order terms in this relationship.

The first hyperpolarizability, denoted as beta ($\beta$), is responsible for second-order nonlinear optical phenomena such as second-harmonic generation (SHG) and the Pockels effect. For a molecule to exhibit a non-zero $\beta$, it must lack inversion symmetry. The second hyperpolarizability, gamma ($\gamma$), is responsible for third-order nonlinear optical effects like third-harmonic generation (THG) and the Kerr effect, and it can be non-zero even in centrosymmetric systems.

These properties are crucial in the design of materials for optical communications, data storage, and optical computing. They are typically calculated using quantum chemical methods and can be experimentally measured through techniques like electric-field induced second-harmonic generation (EFISHG) for $\beta$ and degenerate four-wave mixing (DFWM) for $\gamma$. The magnitude of hyperpolarizabilities is highly dependent on the molecular structure and electronic properties of the material.