The intermolecular interaction that is dependent on the inverse cube of distance between the molecules is:

We begin by recalling how the potential energy $$U$$ of different kinds of intermolecular forces varies with the centre-to-centre separation $$r$$ of the two interacting species. The power of $$r$$ in the denominator is the feature that differentiates one type of interaction from another.

For two ions carrying integral charges $$q_1$$ and $$q_2$$, Coulomb’s law gives the potential energy formula first:

$$U_{$$ ion-ion $$} \;=\;\dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1q_2}{r}.$$

Because only a single factor of $$r$$ appears in the denominator, an ion-ion interaction is said to be proportional to $$r^{-1}$$.

Next, for an ion of charge $$q$$ placed near a polar molecule of permanent dipole moment $$\mu$$, the ion-dipole potential energy is obtained from electrostatics as:

$$U_{$$ ion-dipole $$} \;=\;-\dfrac{1}{4\pi\varepsilon_0}\dfrac{q\mu\cos\theta}{r^{2}}.$$

Here $$\theta$$ is the angle between the dipole axis and the ion-molecule line. The power of $$r$$ is now 2, so an ion-dipole interaction varies as $$r^{-2}$$.

For two molecules each possessing a permanent dipole moment $$\mu_1$$ and $$\mu_2$$, the classical electrostatic expression (for fixed orientations) is:

$$U_{$$ dipole-dipole $$} \;=\;-\dfrac{1}{4\pi\varepsilon_0}\dfrac{\mu_1\mu_2\,(2\cos\theta_1\cos\theta_2-\sin\theta_1\sin\theta_2\cos\phi)}{r^{3}}.$$

Because the leading denominator is $$r^{3}$$, a permanent dipole-dipole interaction is proportional to $$r^{-3}$$. A hydrogen bond is essentially a strong, highly directed dipole-dipole interaction that inherits this same distance dependence; the presence of the hydrogen atom bonded to a highly electronegative element (such as O, N or F) merely strengthens the interaction without changing the $$r^{-3}$$ dependence.

Finally, London dispersion forces (instantaneous dipole-induced dipole) have a quantum-mechanical origin. The London formula gives:

$$U_{\text{London}} \;=\;-\dfrac{C}{r^{6}},$$

so such forces vary as $$r^{-6}$$.

Collecting these results:

  • Ion-ion  $$\propto r^{-1}$$
  • Ion-dipole  $$\propto r^{-2}$$
  • Dipole-dipole (hydrogen bond)  $$\propto r^{-3}$$
  • London force  $$\propto r^{-6}$$

Because the question asks for the interaction that depends on the inverse cube of the distance, we must choose the hydrogen bond, which is a specialised dipole-dipole interaction and therefore follows the $$r^{-3}$$ law.

Hence, the correct answer is Option A.