8.2 London dispersion: the origin of the attraction

The Lennard-Jones potential of the previous lesson took its $-1/r^6$ attraction as given. That term has a genuine microscopic origin, and it is quantum mechanical: even molecules with no permanent charge, no dipole, and no chemical bond attract one another. This lesson explains why, and where the exponent $6$ comes from.

Fluctuating dipoles

A neutral atom has, on time-average, no dipole moment. But its electron cloud is in constant zero-point motion, so at any instant the centre of negative charge is displaced from the nucleus: the atom carries a fleeting, fluctuating dipole $\mathbf{p}_1(t)$ . This instantaneous dipole produces an electric field at a neighbour a distance $r$ away, and that field polarises the neighbour, inducing a second dipole $\mathbf{p}_2$ aligned to be attracted back. The two fluctuations are correlated — the induced dipole always orients to lower the energy — so the interaction does not average to zero even though each dipole alone does.

▶ Why the attraction scales as 1/r⁶ Derivation

The field of a dipole $\mathbf{p}_1$ falls off as $E \sim p_1/r^3$ . It induces in the neighbour, of polarisability $\alpha$ , a dipole

p_2 \;=\; \alpha E \;\sim\; \frac{\alpha\,p_1}{r^3}.

The interaction energy of the original dipole with this induced one is itself of order $-p_1 p_2/r^3$ (dipole–dipole energies scale as $1/r^3$ ). Substituting,

U \;\sim\; -\frac{p_1\,p_2}{r^3} \;\sim\; -\frac{\alpha\,p_1^2}{r^6}.

The mean-square fluctuation $\langle p_1^2\rangle$ is set by the atom’s polarisability and a characteristic excitation energy, leaving the temperature-independent result

U(r) \;=\; -\frac{C_6}{r^6}.

where

$\alpha$: molecular polarisability C·m²/V
$C_6$: dispersion coefficient J·m⁶
$r$: separation m

Two powers of $1/r^3$ — one for the field that reaches the neighbour, one for the energy of the induced dipole back at the source — combine to the $1/r^6$ law. The coefficient $C_6$ grows with polarisability, which is why larger, softer atoms (more easily distorted electron clouds) attract more strongly and condense at higher temperatures.

What it explains

This dispersion attraction is universal: it acts between all molecules, polar or not, because every electron cloud fluctuates. It is the only cohesive force available to the noble gases and to non-polar molecules, and it is what allows argon or methane to condense at all. It is also the weak, ubiquitous background beneath the stronger, directional interactions — hydrogen bonds, permanent-dipole forces — that appear in more complex molecules.

⏳ The history — London dispersion and the cohesion of the inert gases

Into the 1930s the cohesion of the inert gases — helium, neon, argon — was a genuine puzzle. These atoms have no permanent dipole, no chemical bond, no evident mechanism for mutual attraction, yet they condense to liquids and even solids when cold enough.

Fritz London supplied the answer in 1930 from quantum mechanics. The zero-point motion of an atom’s electrons gives it a fluctuating instantaneous dipole; this induces a correlated dipole in a neighbour through its polarisability, and the correlation yields a net attractive $-C_6/r^6$ tail. The force now bears his name.

The Lennard-Jones potential pairs London’s physically-derived $1/r^6$ attraction with a $1/r^{12}$ repulsion that John Lennard-Jones adopted in 1924 largely for algebraic convenience — $12 = 2\times 6$ . The exponent $12$ is not derivable from first principles (the true repulsion is closer to exponential), but the form is so tractable that the Lennard-Jones potential remains the workhorse of molecular simulation a century later.