# How are polarizabilities measured experimentally?

Most common spectroscopies that produce either a full spectrum, a tensor, or a scalar value have a specific instrument associated with them that is relatively self-contained and not custom. For example, (linear) IR spectra come from benchtop FT-IR spectrometers, NMR chemical shifts use a FT-NMR spectrometer (which is capable of many experiments), and so on. One spectroscopic property that serves as a benchmark for electronic structure theory and is important for designing optical materials is the (electric dipole) polarizability, which "is defined as the ratio of the induced dipole moment $\mathbf{p}$ of (a molecule) to the electric field $\mathbf{E}$ that produces this dipole moment":

$$\mathbf{p} = \alpha \cdot \mathbf{E}.$$

Physically, this corresponds to how easy it is to shift a system's charge density or electron cloud around. More explicitly, it is a symmetric rank-2 tensor:

$$\begin{bmatrix} p_{1} \\ p_{2} \\ p_{3} \end{bmatrix} = \begin{bmatrix} \alpha_{11} & \alpha_{12} & \alpha_{13} \\ \alpha_{21} & \alpha_{22} & \alpha_{23} \\ \alpha_{31} & \alpha_{32} & \alpha_{33} \end{bmatrix} \begin{bmatrix} E_{1} \\ E_{2} \\ E_{3} \end{bmatrix},$$

as the applied electric field may have 3 independent Cartesian coordinates, and the system being measured may have some anisotropic response to this field. In some cases, calculating the tensor may be time-consuming, but formulation is well-understood and there are standardized methods for doing so. However, it is not obvious to me how it is measured experimentally. There is the Lorentz-Lorenz equation (rearranged),

$$\alpha = \frac{n^2 - 1}{n^2 + 2} \frac{3}{4\pi N},$$

which relates the refractive index $n$ of a system or medium to its polarizability. There is also the Clausius-Mossoti relation, which relates the dielectric constant $\epsilon$ to the polarizability, as the refractive index and (complex) dielectric constant are related:

$$n \propto \sqrt{\epsilon}.$$

This implies to me that if you can measure the refractive index with a refractometer, then that is the preferred way of measuring the polarizability.

• Is the refractive index the most common method of experimentally measuring the polarizability? Can it be measured directly, or must be through a relationship such as those above?
• Does this relationship hold for oriented (anisotropic) polarizabilities, or only averaged (isotropic) ones?
• The refractive index and the polarizability are frequency-dependent properties. How is the static (frequency-independent) polarizability measured? Or, is it even possible without extrapolation from multiple frequency-dependent data points?