Hyperpolarizability

The hyperpolarizability, a nonlinear-optical property of a molecule, is the second order electric susceptibility per unit volume. The hyperpolarizability can be calculated using quantum chemical calculations developed in several software packages. See nonlinear optics.

Definition and higher orders
The linear electric polarizability $$\alpha$$ in isotropic media is defined as the ratio of the induced dipole moment $$\mathbf{p}$$ of an atom to the electric field $$\mathbf{E}$$ that produces this dipole moment.

Therefore, the dipole moment is:
 * $$\mathbf{p}=\alpha \mathbf{E}$$

In an isotropic medium $$\mathbf{p}$$ is in the same direction as $$\mathbf{E}$$, i.e. $$\alpha$$ is a scalar. In an anisotropic medium $$\mathbf{p}$$ and $$\mathbf{E}$$ can be in different directions and the polarisability is now a tensor.

The total density of induced polarization is the product of the number density of molecules multiplied by the dipole moment of each molecule, i.e.:


 * $$\mathbf{P} = \rho \mathbf{p} = \rho \alpha \mathbf{E} = \varepsilon_0 \chi \mathbf{E},$$

where $$\rho$$ is the concentration, $$\varepsilon_0$$ is the vacuum permittivity, and $$\chi$$ is the electric susceptibility. In a nonlinear optical medium, the polarization density is written as a series expansion in powers of the applied electric field, and the coefficients are termed the non-linear susceptibility:


 * $$\mathbf{P}(t) = \varepsilon_0 \left( \chi^{(1)} \mathbf{E}(t) + \chi^{(2)} \mathbf{E}^2(t) + \chi^{(3)} \mathbf{E}^3(t) + \ldots \right),$$

where the coefficients χ(n) are the n-th-order susceptibilities of the medium, and the presence of such a term is generally referred to as an n-th-order nonlinearity. In isotropic media $$\chi^{(n)}$$ is zero for even n, and is a scalar for odd n. In general, χ(n) is an (n + 1)-th-rank tensor. It is natural to perform the same expansion for the non-linear molecular dipole moment:


 * $$\mathbf{p}(t) = \alpha^{(1)} \mathbf{E}(t) + \alpha^{(2)} \mathbf{E}^2(t) + \alpha^{(3)} \mathbf{E}^3(t) + \ldots ,$$

i.e. the n-th-order susceptibility for an ensemble of molecules is simply related to the n-th-order hyperpolarizability for a single molecule by:


 * $$\alpha^{(n)}=\frac{\varepsilon_0}{\rho} \chi^{(n)} .$$

With this definition $$\alpha^{(1)}$$ is equal to $$\alpha$$ defined above for the linear polarizability. Often $$\alpha^{(2)}$$ is given the symbol $$\beta$$ and $$\alpha^{(3)}$$ is given the symbol $$\gamma$$. However, care is needed because some authors take out the factor $$\varepsilon_0$$ from $$\alpha^{(n)}$$, so that $$\mathbf{p}=\varepsilon_0\sum_n\alpha^{(n)} \mathbf{E}^n$$ and hence $$\alpha^{(n)}=\chi^{(n)}/\rho$$, which is convenient because then the (hyper-)polarizability may be accurately called the (nonlinear-)susceptibility per molecule, but at the same time inconvenient because of the inconsistency with the usual linear polarisability definition above.