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

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

1. hydrogen bond
2. ion-ion interaction
3. ion-dipole interaction
4. London force

Inverse cube of distance I think means an ion-dipole interaction, since we know force due to a dipole on a charge is inversely proportional to the cube of distance. But I don't understand why the answer given is hydrogen bond. What is the distance dependence of the hydrogen bond? Shouldn't the answer be (3)?

## 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} & \text{type of force} & n~\text{in}~R^{-n} \\ \hline \text{ions} & \text{coulombic} & 1 \\ \text{ion - polar molecule} & \text{ion - dipole} & 2 \\ \text{two polar molecules} & \text{dipole - dipole} & 3 \\ \text{ion - nonpolar molecule} & \text{ion - induced dipole} & 4 \\ \text{polar and nonpolar molecule} & \text{dipole - induced dipole} & 6 \\ \text{nonpolar molecules} & \text{dispersion} & 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} & \text{Energy (kJ/mol)} \\ \hline \text{Electrostatic} & -33.4 \\ \text{Exchange-repulsion} & 31.5 \\ \text{Dispersion} & -12.2 \\ \text{Induction} & -8.0 \\ \text{Charge transfer} & -5.9 \\ \text{Exchange-dispersion} & 2.3 \\ \text{Exchange-induction} & 7.8 \\ \delta_{\text{HF}}~\text{correction} & -3.4 \\ \hline \text{Total} & -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} & \text{Energy (kJ/mol)} \\ \hline \text{Electrostatics} & -65.7 \\ \text{Pauli repulsion} & 65.0 \\ \text{Dispersion} & -7.7 \\ \text{Polarization} & -4.6 \\ \text{Charge transfer} & -7.9 \\ \hline \text{Total} & -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} & \text{Additive?} & \text{Sign} & \text{Comment} \\ \hline \textbf{Long-range}~(U \sim R^{-n}) & & & \\ \text{Electrostatic} & \text{Yes} & \pm & \text{Strong orientation dependence} \\ \text{Induction} & \text{No} & - & \\ \text{Dispersion} & \text{approx.} & - & \text{Always present} \\ \text{Resonance} & \text{No} & \pm & \text{Degenerate states only} \\ \text{Magnetic} & \text{Yes} & \pm & \text{Very small} \\ \hline \textbf{Short-range}~(U \sim e^{-\alpha R}) & & & \\ \text{Exchange-repulsion} & \text{approx.} & + & \text{Dominates at very short range} \\ \text{Exchange-induction} & \text{approx.} & - & \\ \text{Exchange-dispersion} & \text{approx.} & - & \\ \text{Charge transfer} & \text{No} & - & \text{Donor-acceptor interaction} \\ \hline \end{array}

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.