6.5: Polarizability

Atomic polarizability in a static field

We observe that, when an electric field is applied to an atom or molecule, it displaces charges in response to the field to lower the total energy. The distribution of bound electrons within a molecule will distort away from their equilibrium configuration, or polarize, thereby inducing a dipole moment, even in those atoms of molecules that don’t have a dipole in the absence of the field. This effect is illustrated in Figure 6.5.1 for the case of an atom, showing how application of a field leads to the displacement of electron density \(\rho _e\) in the opposite direction of the field, resulting a shift of electronic charge density from a mean position \(R_0\) to \(R\). The relatively massive nucleus is hardly affected relative to the light electrons such that its position and thereby the center of positive nuclear charge remains at \(R_0\). This results in an induced dipole moment that depends on the degree that electronic density is displaced, the charges on the nuclei, and the magnitude of the field. Specifically, the microscopic factor that describes the proportionality between the strength of the applied electric field \(\mathcal{E}_{\text{ext}}\) and the induced dipole moment \(\mu _{\text{ind}}\) is the atomic or molecular polarizability:

\[\mathbf{\mu }_{\text{ind}} = \alpha \mathcal{E}_{\text{ext}} %eq57%LTeq01\]

The SI units for \(\alpha \) are \(\text{Cm}^2 \text{V}^{-1}\), but it is typically quoted as the polarizability volume \(\alpha /4\pi \varepsilon _0\) in units of \(\text{m}^3\).

Let’s calculate the polarizability classically by describing what happens to the charges on the application of the field. The classical dipole moment is calculated by integrating over charge densities for electrons and nuclei, but we want to just illustrate the concept with a simple classical approach. Let’s imagine that an atom’s electronic charge \(q_e\) is distributed uniformly through a sphere of radius \(a\) and that the nucleus is a point charge \(q_N\). Before applying the external field \(\mathcal{E}_{\text{ext}}\), the nucleus and sphere are centered at \(R_0\). The internal electric field experienced by the nucleus as a result of electronic charge density is given by

\[\mathcal{E}_e (R) =  - \frac{1}{4\pi \varepsilon _0}\frac{q_e ( R - R_0)}{a^3} %eq58%LTeq02\]

We will consider small perturbative shifts in the position of the electron density, so that applying a field does not change the shape of the spherical electron density, but merely displaces it from \(R_0\). Then, the restoring force that the electrons exert on the nucleus is \(F_{e \to N}(R) = q_N\mathcal{E}_e(R)\),  which we can invert to obtain the force that the nucleus exerts on the electrons

\[F_{N \to e} =  - F_{e \to N} =  - \frac{1}{4\pi \varepsilon _0}\frac{q_e q_N}{a^3} (R - R_0) %eq59%LTeq03\]

Note the linear form of this equation \(F =  - \kappa( R - R_0)\), where \(\kappa \) is the force constant for displacing the electron density:

\[\kappa  = \frac{ - q_e q_N}{4\pi \varepsilon _0a^3} %eq60%LTeq04\]

If we now apply a uniform external electric field to the atom, its force on the electrons is

\[F_{\mathcal{E} \to e} = q_e\mathcal{E}_{\text{ext}} %eq61%LTeq05\]

We can now solve for the displacement in the electronic charge density relative to the nucleus by finding the condition where the forces on the electronic charge from the external field and from the nucleus match each other:  \(F_{N \to e}\left( R \right) = F_{\mathcal{E} \to e}\) or

\[ - q_N\mathcal{E}_e (R) = q_e\mathcal{E}_{\text{ext}} %eq62%LTeq06\]

Substituting eq. (6.5.2) into this expression gives us an expression for the displacement of the electron density as a linear function of the applied external field:

\[R - R_0 = \frac{4\pi \varepsilon _0a^3}{q_N}\mathcal{E}_{\text{ext}} %eq63%LTeq07\]

We also see the intuitive observation that the displacement is inversely proportional to the nuclear charge. That is, a higher positive nuclear charge makes it that much harder to displace the electrons from their equilibrium position.

From the displacement in eq. (6.5.7), we can calculate the dipole moment induced by the external field. In practice, it is proper to find the difference in electronic charge density before and after the application of the field, but we are after simple insights. We can recognize that \(q_e\) and \(q_N\) are, generally, not the charges of the bare particles, but effective charges that describe a valence electron interacting with a screened nucleus and the external field. As a result, it’s effective for calculating the dipole to assume that \(q_e =  - q_N = q\).  We also assume that the position and magnitude of \(q_N\) don’t change during application of the field.  In that case we can assume that the induced dipole is proportional to \(q_e\): \(\mu _{\text{ind}} = q_e(R - R_0)\).  Then, from eq. (6.5.7) we find

\[\mu _{\text{ind}} = 4\pi \varepsilon _0a^3\mathcal{E}_{\text{ext}} %eq64%LTeq08\]

As anticipated from eq. (6.5.1), we find a linear proportionality between the applied electric field and the induced dipole moment. The corresponding atomic polarizability is given by

\[\alpha  = 4\pi \varepsilon _0a^3 %eq65%LTeq09\]

This result confirms that the ability to distort the electron density of an atom is proportional to the volume of space over which it is distributed, in this case \(\alpha  = 3\varepsilon _0V_{\text{atom}}\).

How well does this model do? We first have to ask, which type of radius is appropriate?  Comparing the van der Waals radius (\(a=1.83\, \text{Å}\)) and covalent radius (\(a= 0.96 \, \text{Å}\)) for atomic argon results in polarizability volumes of \(\alpha /4\pi \varepsilon _0 = 6.13 \, \text{Å} \kern-.1em^3\) and \(0.88 \, \text{Å} \kern-.1em^3\), which can be contrasted with the experimental value of \(1.66\, \text{Å} \kern-.1em^3\).\(\bibref{Cite1} %in-text-citation\) The same comparison done for hydrogen, with van der Waals radius (\(1.20 \, \text{Å}\)) and Bohr radius (\(0.53 \, \text{Å}\)), leads to polarizability volumes of \(1.73 \, \text{Å} \kern-.1em^3\) and \(0.15 \, \text{Å} \kern-.1em^3\), compared to the experimental values of \(0.67 \, \text{Å} \kern-.1em^3\).\(\bibref{Cite1} %in-text-citation\) The model is satisfying in the sense that it puts us in within the correct order of magnitude and illustrates certain physical trends, but it has enough uncertainty and variation from atom to atom that that it’s not very useful quantitatively. A perhaps more useful application is to invert this calculation to interpret the effective radii of the electron density that corresponds to the experimental polarizability. In that case, we find that \(a = 1.18 \, \text{Å}\) and \(0.87 \, \text{Å}\) for argon and hydrogen respectively.

The extension of such classical models to molecular polarizabilities adds additional concerns that make them impractical. In addition to the variation of polarizability with different atoms and the types of bonds that they participate in, the polarizability of a molecule is anisotropic and can depend significantly on nuclear configuration. For instance, if one changes a bond length the molecular polarizability can change linearly with displacement in an analogous manner to our earlier discussion of induced dipole moments: \(\alpha  = \alpha _0 + \alpha' Q\), where \(\alpha _0\) is a static polarizability and \(\alpha' \) is a polarizability derivative.  Even so, within similar types of molecules, the molecular polarizability does correlate strongly with molecular volumes, and models that model polarizability as classical induced dipoles remain an active area of research.

Polarizability of the driven oscillator

We have established the phenomenology for the polarizability, however in the context of a static field. For light-matter interactions, the dynamic electromagnetic field can induce dipole moments in non-polar and non-absorbing media which oscillates with the periodicity of the field. These oscillating dipoles can in turn become sources for further interactions, such as Raman or light scattering. The polarizability is similarly closely related to the dielectric susceptibility and index of refraction that govern the optical and electrical properties of condensed matter, as we will see later. Critical to all these topics is the frequency dependence of the polarizability, which we can address with the driven harmonic oscillator model. We will equate the oscillator coordinate with the degree of electronic charge displacement \(Q = R - R_0\) and define the particle as a single electron with mass \(m_e\) and charge \(q\).

The force constant \(\kappa \) in eq. (6.5.4) can be rewritten with eq. (6.5.9) as \(\kappa  = q^2 / \alpha \), which allows us to calculate the resonance frequency for the bound electron cloud directly from a polarizability using \(\omega _0 = \sqrt {\kappa /m_e} \), or

\[\omega _0 = \frac{q}{\sqrt {\alpha m_e} } %eq66%LTeq10\]

Let’s calculate \(\omega _e\). Taking a representative polarizability volume for atoms and small molecules of \(\alpha= 2.0 \, \text{Å} \kern-.1em^3\), we find a resonance frequency of \(\omega _0/2\pi  = 1.8 \times 10^{15} \, \text{Hz}\). This is equivalent to a period of \(0.5 \, \text{fs}\), or absorption at a deep-UV wavelength of \(167 \, \text{nm}\). Despite the limitations of classical thinking, this calculation reveals an important observation. The intrinsic resonances associated with polarizing electrons in molecules are typically of very high frequency, reflecting the extremely rapid dynamics of electrons that aren’t highly delocalized. These frequencies exceed the most common range for molecular spectroscopy in the UV, visible, infrared and terahertz/microwave regimes that we use for studying the structure and dynamics of nuclei, bonds, and molecular valence electrons. Thus, within this atomic polarizability model, the influence of optical and infrared radiation on polarizable electrons in matter is typically in a regime where \(\omega  \ll \omega _e\).

For the interaction of the light and electron, we consider the electric field to have the same form as before; however, in this case we will take the force that the field exerts on the electron to be calculated from eq. (6.5.5), or

\[F_{\text{ext}}(t) = q\mathcal{E}_0\cos \omega t %eq67%LTeq11\]

Thus, with the substitution of \(q\) for \(\mu'\!\), all of the expressions that redrived for the driven oscillator response remain the same. The frequency dependent displacement of the oscillator \(\tilde Q(\omega )\) in eq. (6.3.5) allows us to write the induced dipole moment as

\[\tilde \mu _{\text{ind}}(t) = \frac{q^2}{m_e} \frac{1}{\left( \omega ^2 - \omega _0^2 \right) + 2 \mathrm{i} \omega \gamma }\mathcal{E}_{\text{ext}}(t) %eq68%LTeq12\]

If we next compare this with the definition of the polarizability in eq. (6.5.1), we find that the polarizability is given by the complex quantity

\[\tilde \alpha (\omega ) = \frac{q^2}{m_e}\frac{1}{\left( \omega ^2 - \omega _0^2 \right) + 2\mathrm{i}\omega \gamma } %eq69%LTeq13\]

Where the real and imaginary parts of the polarizability \(\tilde \alpha (\omega ) = \alpha' \! (\omega ) + \mathrm{i} \alpha '' \! (\omega )\) are the same as those given for the susceptibility in eq. (6.4.8) with the substitution of \(q\) for \(\mu' \!\). Note, when the complex polarizability is evaluated at zero frequency (the static limit), this expression recovers eq. (6.5.10).

So, what is the difference between the polarizability and susceptibility? Looking forward our discussion of condensed matter, \(\alpha \) is an intrinsic molecular property whereas \(\chi \) is a macroscopic property on scales much larger than the wavelength of light. In a dilute gas of \(N\) non-interacting molecules, \(\chi \) is just proportional to \(N\alpha \), but in condensed matter \(\chi \) will also need to account for the interactions of the polarizable molecules with each other and the medium they are embedded in. Furthermore, the microscopic induced dipole in our model, \(\mu \), will be replaced by the macroscopic polarization \(P = \varepsilon _0\chi \mathcal{E}\), a dipole moment density.

More generally, the polarizability of electrons in polyatomic molecules and condensed matter is also influenced by changing nuclear positions, which we have not addressed here. Therefore the expression in eq. (6.5.12) can be interpreted more broadly. The resonance frequency \(\omega _0\) can be interpreted as any coordinate for bound charges and their natural resonance frequency that gives rise to a displacement of electron density from their average configuration, such as the vibration of molecules.

In discussing the classical driven oscillator, we have invoked two notations to represent how much charges within molecules are displaced when driven by an oscillating electric field. We used the dipole derivatives \(\mu ' \! = \partial \mu /\partial Q\) in discussing absorption spectroscopy and the magnitude of the displaced electronic charge \(q\) in discussing the polarizability \(\alpha \). Both have units of coulombs (\(\text{C}\)); what is the difference? Within the context of this model there isn’t a significant difference, but the differences will become more apparent as we look at the molecular factors governing the interactions, whether we are near a molecular resonance, and in nonlinear spectroscopy. In practice one language is more useful for discussing resonant interactions, whereas the other is more common for off-resonant interactions.

This can be illustrated by combining both forms of the interaction in considering the light-matter interaction potential \(V =  - \mu \mathcal{E}\). For absorption, the force on this oscillator depends on \( - \partial V/\partial Q\) and thereby \(\mu ' \! \). This is equivalent to saying that the molecule’s dipole moment varies about an average value \(\mu _0\) as the oscillator vibrates even in the absence of a field: \(\mu  = \mu _0 + \mu ' Q\).   In the case of the polarizability there is an additional contribution that arises with the field present—the induced dipole moment \(\mu _{\text{ind}} = \alpha \mathcal{E}\). Combining all of these factors, we find that the magnitude of the interaction potential is

\[\begin{align}\begin{aligned}
  V =&  - \mu \mathcal{E} \\
   =&  - \left( \mu _0 + \mu ' \! Q + \mu _{\text{ind}} \right)\mathcal{E} \\
   =&  - \mu _0\mathcal{E} - \mu ' \! Q\mathcal{E} - \alpha \mathcal{E}^2 \\
\end{aligned}\end{align} %eq70%LTeq14\]

The second term in the interaction energy is proportional to \(\mu ' \) and scales linearly in \(\mathcal{E}\) and \(Q\). It results in resonant absorption at a natural frequency governed by the molecule’s potential energy surface. The final term that depends on \(\alpha \) scales as \(\mathcal{E}^2\) because the electric field serves first to induce a dipole moment then exchanges energy with the oscillator via this induced dipole. This term, which describes Raman spectroscopy and light scattering, is most likely to dominate when we have strong electric fields and are far from resonance.

References