# 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?
• Good question. There are several ways depending on the state of the molecule. For gas phase molecules Rayleigh scattering is used. Raman scattering also works. In these cases polarization of light is also important. A more detailed exposition on this will take time for me to compile. Dec 9, 2020 at 4:58
• Here is a starting point: pubs.acs.org/doi/full/10.1021/jp9845322 Jan 2, 2021 at 4:16
• @KarstenTheis - please consider taking a bit of time to make that into an answer. Jan 7, 2021 at 20:03
• @GeoffHutchison It's a bit far from my comfort zone but I gave it a shot. Jan 8, 2021 at 11:10
• Jan 8, 2021 at 11:18

The polarizability of molecules determines how a molecule interacts with other molecules (intermolecular interactions and chemical reactions), how it interacts with electromagnetic radiation (optical properties), and how it interacts with electrical fields (electrical properties). Thus, there are multiple ways to compare the polarizability of two molecules or samples, and to quantify it.

[OP] Is the refractive index the most common method of experimentally measuring the polarizability?

That is hard to say without a systematic search of all the polarizability data. A starting point for such a search might be the polarizability data provided by NIST, which includes references that describe methods used. In general, there are electrical techniques (static electrical field or slowly oscillating) and optical techniques (electromagnetic field). Light scattering depends on the electric polarizability of the medium, so any technique that is based on light scattering (Raman spectroscopy, X-ray diffraction, refraction) in principle could be used to determine polarizability.

[OP] Can it be measured directly, or must be through a relationship such as those above?

The refractive index can be measured directly. The polarizability, while it has direct impact on molecular properties like the strength of dispersion forces, is determined from other measurements.

[OP] Does this relationship hold for oriented (anisotropic) polarizabilities, or only averaged (isotropic) ones?

As written in the question, the Lorentz-Lorenz equation is for the isotropic case. The literature linked to below also shows the more general equation.

[OP] 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?

The refractive index is frequency-dependent while the polarizability is not. You measure the refractive index at multiple wavelengths and extrapolate to infinitive wavelength. Alternatively, you can do a non-linear fit to the respective theoretical dispersion function.

Measurements of molecular polarizability are done at low concentrations in the gas phase, and yield the isotropic polarizability. Here is one paper that compares the experimental data to computational results for the iodine molecule. More complicated experiments involving polarized light are required to figure out the anisotropy, if any. For example, this paper from the same two labs compares the anisotropy of the polarizability of butadiene to computational results. An example of a quantumchemical computation of polarizability is here, for the hydrogen atom. In the solid phase with oriented molecules, the anisotropy of polarization is easier to measure (for example with cross-polarized filters as in the example below showing a plastic lens under mechanical stress): 