Which intermolecular interaction is dependent on the inverse cube of distance?
2

The ion-ion or charge-charge interaction is given by

$$
V_{\text{cc}} = \frac{Z_{i}Z_{j}e^{2}}{4\pi\epsilon_{0}} \frac{1}{|\mathbf{R}_{i} - \mathbf{R}_{j}|}
$$

and is therefore proportional to $R^{-1}$.

3

The ion-dipole or charge-(permanent) dipole interaction is given by

$$
V_{\text{cd}} = \frac{Z_{i}e}{4\pi\epsilon_{0}} \frac{\mathbf{R}_{ij} \cdot \vec{\mu}_{j}}{R_{ij}^{3}}
$$

and is therefore proportional to $R^{-2}$.

4

"London force" is often meant to encapsulate any of the van der Waals forces (here, here), but in particular the dispersion force. All of the above have a $R^{-6}$ dependence.



Here is a general table that covers each of the interactions so far:

\begin{array}{lll}
\hline
\text{type of molecular unit} &amp; \text{type of force} &amp; n~\text{in}~R^{-n} \\
\hline
\text{ions}                        &amp; \text{coulombic}               &amp; 1 \\
\text{ion - polar molecule}        &amp; \text{ion - dipole}            &amp; 2 \\
\text{two polar molecules}         &amp; \text{dipole - dipole}         &amp; 3 \\
\text{ion - nonpolar molecule}     &amp; \text{ion - induced dipole}    &amp; 4 \\
\text{polar and nonpolar molecule} &amp; \text{dipole - induced dipole} &amp; 6 \\
\text{nonpolar molecules}          &amp; \text{dispersion}              &amp; 6 \\
\hline
\end{array}

1

From Wikipedia:


  The hydrogen bond is often described as an electrostatic dipole-dipole interaction. However, it also has some features of covalent bonding: it is directional and strong, produces interatomic distances shorter than the sum of the van der Waals radii, and usually involves a limited number of interaction partners, which can be interpreted as a type of valence. These covalent features are more substantial when acceptors bind hydrogens from more electronegative donors.


This implies that hydrogen bonding interactions may not behave exactly as $R^{-3}$. Consider the water dimer:



which is a prototypical example of a hydrogen-bonded complex. From Stone, using density functional-based symmetry-adapted perturbation theory (SAPT-DFT) in the aug-cc-pVQZ basis,
\begin{array}{lr}
\hline
\text{Type of interaction} &amp; \text{Energy (kJ/mol)} \\
\hline
\text{Electrostatic}                 &amp; -33.4 \\
\text{Exchange-repulsion}            &amp;  31.5 \\
\text{Dispersion}                    &amp; -12.2 \\
\text{Induction}                     &amp;  -8.0 \\
\text{Charge transfer}               &amp;  -5.9 \\
\text{Exchange-dispersion}           &amp;   2.3 \\
\text{Exchange-induction}            &amp;   7.8 \\
\delta_{\text{HF}}~\text{correction} &amp;  -3.4 \\
\hline
\text{Total}                         &amp; -21.5 \\
\hline
\end{array}
the electrostatic component is clearly the largest contributor, but after taking the absolute value of all interaction terms, is about 32% of the total interaction. Another approach for decomposing the interaction energy is based on absolutely localized molecular orbitals (ALMO-EDA), which at the $\omega$B97M-V/def2-QZVPP level gives
\begin{array}{lr}
\hline
\text{Type of interaction} &amp; \text{Energy (kJ/mol)} \\
\hline
\text{Electrostatics}  &amp; -65.7 \\
\text{Pauli repulsion} &amp;  65.0 \\
\text{Dispersion}      &amp;  -7.7 \\
\text{Polarization}    &amp;  -4.6 \\
\text{Charge transfer} &amp;  -7.9 \\
\hline
\text{Total}           &amp; -20.9 \\
\hline
\end{array}
SAPT-DFT and ALMO-EDA are in good agreement, though charge transfer is more important in ALMO-EDA primarily due to the lackluster definition of charge transfer within SAPT. Electrostatics are now about 44% of the total interaction. For reference, here is how the additional terms decay (Stone, Table 1.2):
\begin{array}{lccl}
\hline
\text{Contribution} &amp; \text{Additive?} &amp; \text{Sign} &amp; \text{Comment} \\
\hline
\textbf{Long-range}~(U \sim R^{-n}) &amp; &amp; &amp; \\
\text{Electrostatic} &amp; \text{Yes}     &amp; \pm &amp; \text{Strong orientation dependence} \\
\text{Induction}     &amp; \text{No}      &amp; -   &amp;                                      \\
\text{Dispersion}    &amp; \text{approx.} &amp; -   &amp; \text{Always present}                \\
\text{Resonance}     &amp; \text{No}      &amp; \pm &amp; \text{Degenerate states only}        \\
\text{Magnetic}      &amp; \text{Yes}     &amp; \pm &amp; \text{Very small}                    \\
\hline
\textbf{Short-range}~(U \sim e^{-\alpha R}) &amp; &amp; &amp; \\
\text{Exchange-repulsion}  &amp; \text{approx.} &amp; + &amp; \text{Dominates at very short range} \\
\text{Exchange-induction}  &amp; \text{approx.} &amp; - &amp;                                      \\
\text{Exchange-dispersion} &amp; \text{approx.} &amp; - &amp;                                      \\
\text{Charge transfer}     &amp; \text{No}      &amp; - &amp; \text{Donor-acceptor interaction}    \\
\hline
\end{array}

Answer

Although components of hydrogen bonding interactions may have a $R^{-3}$ distance dependence, there are significant contributors to the hydrogen bond interaction other than electrostatics that do not decay as $R^{-3}$; in general, it is a complex interaction and its definition is still hotly debated. For that reason, the best answer is (3), the ion-dipole interaction.

References


Masahiro Yamamoto, Kyoto University: charge-charge, charge-dipole, dipole-charge, dipole-dipole interaction
Stephen Lower, Simon Fraser University: Chem1 Virtual Textbook, Interactions between Molecular Units
Anthony Stone: The Theory of Intermolecular Forces
ALMO-EDA calculations: geometry is B97-D/def2-TZVP, EDA input as comment