Intermolecular forces — van der Waals, hydrogen bonding, dispersion

The weak forces that hold molecules together — permanent dipoles, induced dipoles, and dispersion from quantum fluctuations — gathered into one 1/r⁶ picture and made tangible with the Lennard-Jones potential.

Opening

If chapter 10 dealt with the polarity and polarizability of a molecule, this chapter looks at how those two ideas show up as forces between one molecule and another. A covalent bond is a strong force worth hundreds of kJ/mol, but water boiling at 100 °C, argon condensing into a liquid, and a gecko sticking to the ceiling are all the work of forces about a hundred times weaker — intermolecular forces. By the end of this chapter you will be able to state in one line that the “van der Waals force” is really a bundle of three distinct contributions, why all three fall off as 1/r61/r^6, and why the hydrogen bond occupies a special place in that hierarchy.

Main 1 — The three faces of the van der Waals force

“Van der Waals force” is really a name lumping together three different attractive contributions.

  • Dipole–dipole (Keesom): two permanent dipoles μ⃗1,μ⃗2\vec\mu_1, \vec\mu_2 attract while aligning their orientations. Boltzmann-averaging over all orientations gives Udd∝−∣μ⃗1∣2∣μ⃗2∣2/(kBT⋅r6)U_{\text{dd}} \propto -|\vec\mu_1|^2 |\vec\mu_2|^2 / (k_B T \cdot r^6).
  • Dipole–induced dipole (Debye): a permanent dipole polarizes the electron cloud of a neighboring molecule. Udi∝−μ2α/r6U_{\text{di}} \propto -\mu^2 \alpha / r^6.
  • Induced–induced dipole (London dispersion): even nonpolar molecules attract through instantaneous quantum fluctuations of their electron clouds. UL∝−α1α2/r6U_{\text{L}} \propto -\alpha_1 \alpha_2 / r^6. It is the only attraction acting between noble-gas atoms, and the cohesion of liquid argon, the boiling points of alkanes, and a gecko’s footpad are all its doing.

The key point is that all three contributions fall off as 1/r61/r^6. At short range, by contrast, the electron clouds overlap and Pauli repulsion kicks in; that part is much steeper, modeled empirically as 1/r121/r^{12}.

Main 2 — The Lennard-Jones 12-6 potential

What summarizes all of this in one line is the Lennard-Jones potential.

V(r)=4ε ⁣[(σr)12−(σr)6]V(r) = 4\varepsilon\!\left[\left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6}\right]

Here ε\varepsilon (epsilon) is the depth of the well and σ\sigma (sigma) is the distance at which V=0V = 0. Differentiating the two terms and finding the minimum, the equilibrium separation is

re=21/6σ≈1.122 σr_e = 2^{1/6}\sigma \approx 1.122\,\sigma

and the depth there is exactly V(re)=−εV(r_e) = -\varepsilon. Representative values: argon ε/kB≈120\varepsilon/k_B \approx 120 K, σ≈0.34\sigma \approx 0.34 nm; methane ε/kB≈148\varepsilon/k_B \approx 148 K, σ≈0.38\sigma \approx 0.38 nm; water sits around ε/kB≈78\varepsilon/k_B \approx 78 K, σ≈0.32\sigma \approx 0.32 nm with extra Coulomb terms added for hydrogen bonding. Just two terms — an r12r^{12} repulsion and an r6r^6 attraction — become the standard starting point of molecular dynamics, from gas condensation all the way to protein-folding simulations.

Main 3 — Hydrogen bonds: special and not so special

When hydrogen bonds to a strongly electronegative atom like F, O, or N, that hydrogen becomes electron-deficient and carries a large permanent dipole — and being small enough, it can approach a neighboring molecule’s electronegative atom closely. The hydrogen bond formed this way is much stronger than an ordinary van der Waals force — 10–40 kJ/mol versus about 1 kJ/mol for a typical London interaction. It is also directional: the hydrogen must lie roughly along the N–H⋯O line.

What hydrogen bonds are responsible for: water’s high boiling point (100 °C, against −161-161 °C for CH4_4 of comparable molecular mass), the stability of the DNA double helix, the secondary structure of proteins, and the fact that ice is less dense than water. Energetically the hydrogen bond sits between a true covalent bond (about 400 kJ/mol) and an ordinary van der Waals force (about 1 kJ/mol) — weak compared to bonds, but strong compared to most non-bonded interactions. That “medium strength” is exactly where life works: too strong and the bonds never break, so information cannot be read; too weak and structure cannot be held.

In Python

# Plot the Lennard-Jones potential for argon, methane, and krypton.
# (eps/k_B, sigma) in units of (K, nm).
import numpy as np
import matplotlib.pyplot as plt

kB = 1.380649e-23  # J/K

species = {
    "Ar": (120.0, 0.34),
    "CH4": (148.0, 0.38),
    "Kr": (164.0, 0.36),
}

r = np.linspace(0.30, 1.00, 400)  # nm

for name, (eps_over_kB, sigma) in species.items():
    eps = eps_over_kB * kB                       # J
    V = 4 * eps * ((sigma / r)**12 - (sigma / r)**6)
    plt.plot(r, V / kB, label=f"{name} (eps/k_B={eps_over_kB:.0f} K)")
    r_e = 2**(1/6) * sigma
    plt.plot(r_e, -eps_over_kB, "o", color="0.3")
    print(f"{name}: r_e = {r_e:.3f} nm, well depth = {eps_over_kB:.0f} K")

plt.axhline(0, color="0.6", lw=0.8)
plt.xlabel("r / nm"); plt.ylabel("V(r) / k_B  [K]")
plt.ylim(-200, 200)
plt.title("Lennard-Jones 12-6 potential")
plt.legend(); plt.tight_layout(); plt.show()

If all three curves pass through a minimum of depth −ε-\varepsilon at re=21/6σr_e = 2^{1/6}\sigma, then with a two-term recipe — ”1/r121/r^{12} repulsion plus 1/r61/r^6 attraction” — you have the cohesive tendency of three molecules in hand.

To the next chapter

Chapter 12: Electronic states of solids widens the view from molecules to a solid where 102310^{23} atoms are arranged in a regular lattice. When the bonding/antibonding pairs of chapter 6 and the Hückel ladder of chapter 7 grow infinitely long, they become a band, and whether that band is filled or not separates metals, semiconductors, and insulators.