Van der Waals Forces

The ionic, covalent, and metallic bonds all involve electron transfer or sharing. A weaker attraction acts between every pair of atoms and molecules, including the closed-shell noble gases that neither share nor transfer electrons. This van der Waals attraction holds argon and methane crystals together, condenses the noble gases at low temperature, and sets the short-range shape of every intermolecular potential. It has three physically distinct sources, all decaying as $1/ r^{6}$ .

The three contributions

Write $p$ for a permanent electric dipole moment and $α$ for the electric polarizability, the constant relating an induced dipole to the field that induces it, $p_{ind} = α E$ . The three long-range attractions between two molecules a distance $r$ apart are the following.

Keesom (orientation) energy — between two permanent dipoles. Each dipole torques the other into a favorable alignment; thermal motion partly randomizes the orientation, and the Boltzmann average of the dipole-dipole energy leaves a net attraction $U_{K} = - \frac{1}{3 k _{B} T} \frac{p _{1}^{2} p _{2}^{2}}{( 4 π ϵ _{0} ) ^{2} r ^{6}} .$ It is temperature-dependent because it relies on thermal averaging.
Debye (induction) energy — between a permanent dipole and the dipole it induces in a polarizable neighbor, $U_{D} = - \frac{p _{1}^{2} α _{2} + p _{2}^{2} α _{1}}{( 4 π ϵ _{0} ) ^{2} r ^{6}} .$ It survives at $T = 0$ and needs only one of the molecules to be polar.
London (dispersion) energy — between the instantaneous dipoles of two neutral, nonpolar atoms, present even when both permanent moments vanish. It is the only van der Waals term available to the noble gases and is usually the largest of the three for small molecules.

Only the London term requires quantum mechanics; the other two follow from classical electrostatics plus thermal averaging. The remainder of the lesson derives the London energy, because it is the term that bonds the noble-gas solids.

The London dispersion energy

A noble-gas atom has zero average dipole moment, but its electron cloud fluctuates, so its instantaneous moment is nonzero. A momentary dipole on one atom produces a field that polarizes the neighbor, and the induced dipole is correlated with the original in just the way that lowers the total energy. The correlation of zero-average fluctuations produces a nonzero average attraction.

A fluctuating instantaneous dipole on the left atom polarizes the right atom; the induced dipole points so that the pair attracts, and the correlation persists on average even though each moment averages to zero.

The cleanest derivation models each atom as a one-dimensional charge $- e$ on a spring of natural frequency $ω_{0}$ , the Drude oscillator. With the atoms on the $x$ axis a distance $r$ apart and displacements $x_{1}, x_{2}$ along the line of centers, the two oscillators couple through the dipole-dipole interaction, giving the Hamiltonian

\hat{H} = i = 1, 2 \sum (\frac{p _{i}^{2}}{2 m} + \frac{1}{2} m ω_{0}^{2} x_{i}^{2}) - \frac{2 k e ^{2}}{r ^{3}} x_{1} x_{2} .

The normal coordinates $x_{\pm} = (x_{1} \pm x_{2}) / 2$ diagonalize the coupled system into two independent oscillators with frequencies

ω_{\pm} = ω_{0} 1 \mp \frac{2 k e ^{2}}{m ω _{0}^{2} r ^{3}} .

The zero-point energy of the coupled pair is $\frac{1}{2} ℏ (ω_{+} + ω_{-})$ . Expanding the square roots to second order in the small coupling $β = 2 k e^{2} / (m ω_{0}^{2} r^{3})$ , the linear terms cancel in the sum and the shift relative to the two uncoupled atoms is

Δ E = \frac{1}{2} ℏ (ω_{+} + ω_{-} - 2 ω_{0}) \approx - \frac{ℏ ω _{0}}{8} β^{2} = - \frac{ℏ ω _{0} α ^{2}}{2 r ^{6}} \equiv - \frac{C _{6}}{r ^{6}},

using the Drude polarizability $α = e^{2} / (m ω_{0}^{2})$ (in Gaussian units, where $k = 1$ ). The energy is negative — an attraction — and scales as $1/ r^{6}$ . The coefficient $C_{6} = \frac{1}{2} ℏ ω_{0} α^{2}$ is set by the polarizability and a characteristic transition energy $ℏ ω_{0}$ of order the atom's ionization energy.

The same result follows from second-order perturbation theory with the dipole-dipole interaction $V \propto 1/ r^{3}$ as the perturbation. The first-order shift $⟨ 0∣ V ∣0 ⟩$ vanishes because each atom has no permanent moment. The second-order shift is a sum over virtual excitations of both atoms,

Δ E^{(2)} = n \neq = 0 \sum \frac{∣ ⟨ n ∣ V ∣0 ⟩ ∣ ^{2}}{E _{0} - E _{n}},

in which every denominator $E_{0} - E_{n}$ is negative because the ground state is lowest. The sum is therefore negative, $∣ V ∣^{2} \propto 1/ r^{6}$ , and the dispersion attraction is recovered with the same $1/ r^{6}$ dependence.

Second-order perturbation theory: the dipole-dipole coupling admits no first-order shift, and every virtual excitation to a state above the ground state contributes a negative term, so the net energy shift is an attraction.

The Lennard-Jones potential

At short range the electron clouds overlap and the exclusion principle produces a steep repulsion. Combining it with the London attraction gives a total potential. The repulsion has no simple closed form, but the twelfth power is a computationally convenient choice that makes the whole potential a function of $r^{- 6}$ and its square.

Setting $d U / d r = 0$ locates the minimum at $r_{m} = 2^{1/6} σ \approx 1.122 σ$ , where $U (r_{m}) = - ϵ$ . The two parameters $ϵ$ and $σ$ are fit to gas-phase data (second virial coefficient, viscosity) and then used to predict the solid.

The Lennard-Jones potential in reduced units: the depth epsilon sits at the minimum near 1.12 sigma, and sigma is the inner separation where the curve crosses zero between the repulsive wall and the attractive tail.

Cohesive energy of the noble-gas crystals

The noble gases (except helium) crystallize in the face-centered cubic structure. The cohesive energy is the sum of Lennard-Jones pair energies over the lattice. With nearest-neighbor spacing $R$ and each pair separation written as $p_{ij} R$ for dimensionless $p_{ij}$ , the total energy of $N$ atoms is

U_{tot} = 2 N ϵ [A_{12} (\frac{σ}{R})^{12} - A_{6} (\frac{σ}{R})^{6}], A_{n} = j \sum^{'} p_{ij}^{- n},

where the primed sums run over all neighbors of a fixed atom. For the FCC lattice the lattice sums converge to $A_{12} = 12.13$ and $A_{6} = 14.45$ . Minimizing over $R$ gives the equilibrium spacing and cohesive energy per atom,

\frac{R _{0}}{σ} = (\frac{2 A _{12}}{A _{6}})^{1/6} = 1.09, \frac{U _{tot} ( R _{0} )}{N} = - \frac{A _{6}^{2}}{2 A _{12}} ϵ = - 8.6 ϵ .

The predicted nearest-neighbor spacing $R_{0} = 1.09 σ$ is independent of the element, a nontrivial prediction that the measured spacings confirm.

Predicted (8.6 epsilon) versus measured cohesive energy per atom for the heavy noble-gas solids; agreement improves down the column as the atoms grow heavier and zero-point corrections shrink.

The Lennard-Jones fit reproduces the noble-gas solids to a few percent with two parameters. The residual overbinding is systematic and understood: the atoms are not fixed at $R_{0}$ but oscillate with zero-point energy, which the classical lattice sum ignores. That zero-point contribution is largest for the lightest atom, and for helium it is so large — larger than the shallow Lennard-Jones well — that helium does not solidify at atmospheric pressure at any temperature, and its liquid is the setting for superfluidity.

The van der Waals bond completes the catalog of molecular bonding. The overlap repulsion and the pair potentials assembled here also fix the crystal-structure problem: the same lattice sums, applied to the Coulomb energy of ions rather than the London energy of neutral atoms, give the Madelung constant and the cohesive energy of ionic solids, the subject of the structure of solids.

Summary

Van der Waals attraction has three $1/ r^{6}$ contributions: the temperature- dependent Keesom orientation energy between permanent dipoles, the Debye induction energy between a permanent and an induced dipole, and the London dispersion energy between fluctuating instantaneous dipoles.
The London energy follows from the coupled Drude-oscillator model as $U_{L} = - \frac{1}{2} ℏ ω_{0} α^{2} / r^{6}$ , and equivalently from second-order perturbation theory, where every virtual-excitation term lowers the energy.
The Lennard-Jones potential $U = 4 ϵ [(σ / r)^{12} - (σ / r)^{6}]$ combines the London tail with an empirical $r^{- 12}$ repulsion; its minimum is at $r_{m} = 2^{1/6} σ$ with depth $ϵ$ .
Summing over the FCC lattice gives $R_{0} = 1.09 σ$ and a cohesive energy $8.6 ϵ$ per atom, matching argon ( $R_{0} = 0.371 nm$ , $0.089 eV/atom$ ) to within the neglected zero-point energy.