London Dispersion Forces

Two neopentane molecules, spherical and compact, meet in the gas phase. They barely notice each other. Two n-octane molecules, long and floppy, meet and cling together strongly. This raises the boiling point by nearly 100 °C relative to their branched isomer. Both compounds have the same molecular formula, C₈H₁₈. Neither has a permanent dipole to speak of, and no hydrogen bonding is available. The entire difference in boiling point is a real, measurable, thermodynamic fact. It is written by a force that exists only as a statistical average. That average is of something that is, at any given instant, zero. That force is the London dispersion force. Understanding it mechanistically means learning to think about electron clouds differently. They must be seen as dynamical, fluctuating objects, not static clouds of probability. Dispersion forces matter because they are the only attractive force available to nonpolar molecules and noble gases. They are present, silently, underneath every other intermolecular interaction. This includes dipole-dipole, hydrogen bonding, and ion-dipole forces. They act as a baseline contribution, frequently the largest term even in polar systems. Without dispersion forces there would be no liquid argon. There would be no boiling point trend down the halogens (F₂ a gas, I₂ a solid). There would be no graphite lubricity, no protein hydrophobic core. There would be no van der Waals contact in enzyme active sites, no gecko adhesion. The force is universal — it acts between any two atoms or molecules, polar or not. It is the piece of Intermolecular Forces most often underestimated by students. This is because it cannot be drawn as a simple arrow between a δ+ and a δ−. The physical picture is this: electrons are not smeared uniformly and motionlessly over a molecule. A Lewis structure or an electrostatic potential map (a time-averaged picture) might suggest otherwise. At any femtosecond snapshot, the electron cloud has more density on one side than the other. This happens purely by the statistics of quantum motion. It creates an instantaneous dipole with zero time-average. That instantaneous dipole generates an electric field. This field polarizes the electron cloud of a neighboring atom. It induces a dipole that is correlated with the first. The induced dipole is always oriented to attract, never to repel. Averaged over time, the correlated fluctuation of both clouds produces a net attractive force. This happens even though each individual dipole averages to zero. This is fundamentally a quantum mechanical, correlated-motion phenomenon. It has no classical analogue in the naive sense. Still, the induced-dipole language borrowed from classical electrostatics gives excellent quantitative traction. This article builds the mechanistic and orbital picture of dispersion from the ground up. It covers why instantaneous dipoles must exist, and how polarizability α is defined and measured. It explains the derivation of the $1/r^6$ attraction and its physical origin in second-order perturbation theory. It shows how molecular surface area and shape (rods versus spheres) control the magnitude of the effect. It also shows how these ideas explain real trends. These include boiling points down a homologous series, halogen state at room temperature, and the counterintuitive branching effect. It closes by connecting dispersion to the broader question of predicting Physical Properties from Structure.

Key takeaways

Dispersion forces arise from instantaneous, fluctuating dipoles in electron clouds correlating with induced dipoles in neighbors — always attractive on time-average.
The interaction energy scales as $-C/r^6$ at long range, derived from second-order perturbation theory treating the induced-dipole interaction as a small perturbation.
Polarizability α measures how easily an electron cloud is distorted by an external field; larger, more diffuse, more polarizable atoms (softer, higher atomic number in a group) give stronger dispersion.
Molecular shape controls contact surface area: linear/rod-shaped molecules pack with more van der Waals contact than compact spherical isomers, raising boiling points even at equal molecular mass.
Dispersion forces are present in every pair of molecules or atoms, polar or not, and are frequently the dominant attractive term even between polar molecules like HCl.
Trends down a group (noble gases, halogens) and across branching isomers (pentane vs. neopentane) are textbook, mechanistically explainable dispersion effects.
Key idea: the correlated fluctuation of two electron clouds — an instantaneous dipole inducing a matching dipole next door — produces a universal, always-attractive $1/r^6$ force whose strength is set by polarizability and by how much surface area two molecules can bring into contact.

Where this lives today

Dispersion forces are not a historical curiosity confined to noble-gas trivia; they are load-bearing in modern chemistry, biology, and technology. In drug design, the binding of a small-molecule inhibitor into a hydrophobic enzyme pocket is very often dominated by the cumulative dispersion energy from dozens of close van der Waals contacts between the ligand's aromatic and aliphatic surface and the protein's nonpolar residues — medicinal chemists routinely trade a small entropic penalty for a large dispersion gain by adding a well-fitting aromatic ring or halogen to a scaffold, a strategy directly justified by the surface-area/polarizability reasoning developed here. In materials science, graphite's famous lubricity and the ease of exfoliating graphene sheets both trace to weak interlayer dispersion forces overcoming strong in-plane covalent bonding; van der Waals heterostructures — stacks of different 2D materials held together purely by dispersion — are now an entire subfield of nanoelectronics. Gecko adhesion is the textbook biological example: millions of microscopic setae on a gecko's toe maximize contact surface area with a wall, and the cumulative dispersion force (not suction, not friction alone) is sufficient to support the animal's full body weight, a fact confirmed by direct force measurements and now copied in synthetic dry-adhesive materials. In computational chemistry, standard density functional theory notoriously underestimates dispersion (because it is built from local or semi-local electron density functionals that cannot capture nonlocal electron correlation), which is why essentially all modern DFT calculations on organic and biomolecular systems include an explicit dispersion correction (Grimme's D3/D4 corrections, or nonlocal van der Waals functionals) bolted on using formulas structurally identical to the London $C_6/r^6$ term derived above. Finally, dispersion forces set the basic cohesive energy scale of molecular crystals, control polymorph selection in pharmaceutical crystallization, and underlie the entire discipline of predicting Physical Properties from Structure — melting points, solubilities, and vapor pressures of organic compounds — from first principles.

Ask the tutor: How do I predict, from structure alone, whether dispersion or dipole-dipole/hydrogen-bonding will dominate the boiling point of a given molecule?

The physical picture

Start with a single helium atom. Its Lewis/orbital picture places two electrons in a spherically symmetric 1s orbital, and the time-averaged charge density is indeed perfectly spherical — no permanent dipole moment is possible for an isolated atom in a stationary state, by symmetry. But "no permanent dipole" is a statement about the time-averaged or expectation-value charge distribution, $\langle \mu \rangle = 0$ . It says nothing about the dipole moment operator's value in any particular instant, nor about the mean-square fluctuation, $\langle \mu^2 \rangle$ , which is strictly nonzero. Quantum mechanically, the two electrons are correlated (they repel each other via the Coulomb term in the Hamiltonian), so they tend, statistically, to be found on opposite sides of the nucleus more often than the same side — but which side is "left" and which is "right" fluctuates in time (or, in the stationary-state language, is a superposition/correlation effect built into the exact ground-state wavefunction, which is not a simple product of two independent 1s orbitals). This zero-average-but-nonzero-fluctuation dipole is the "instantaneous dipole." Bring a second helium atom nearby. The instantaneous dipole on atom A creates an electric field at the location of atom B. That field is not random noise as far as B is concerned within the timescale of electronic motion (electronic motion is much faster than nuclear/molecular translation, but the correlation between A's and B's electron motions is what matters, not a naive electrostatic snapshot) — quantum mechanically, the true two-atom wavefunction develops correlations such that whenever A's electron density bulges toward B, B's electron density is pushed away from that near side, i.e., B's own instantaneous dipole is induced to point in the same sense as A's, so that the near sides always present opposite charges (attraction), never like charges. This is the crucial asymmetry: random independent fluctuations would average their dipole-dipole interaction to exactly zero, but correlated fluctuations produce a nonzero, always-negative (attractive) average interaction energy. That correlation energy is the London dispersion force, and its quantum-mechanical origin is precisely electron correlation — the same physics captured formally by post-Hartree-Fock correlation methods in computational chemistry. The molecular property that governs how strongly an atom's cloud can be induced to respond is the polarizability, $\alpha$ : the proportionality constant between an applied electric field $E$ and the induced dipole moment,

\mu_{\text{induced}} = \alpha E .

Polarizability has units of volume (in the common cgs-derived convention, $\alpha$ in ų) and tracks intuitively with how "loose" and diffuse the electron cloud is: larger atoms, atoms with more electron shells, anions, and atoms with lone pairs in diffuse orbitals are all more polarizable. Down a group of the periodic table (F → Cl → Br → I, or He → Ne → Ar → Kr → Xe), the outer electrons sit in successively higher- $n$ orbitals, farther from the nuclear charge and more weakly held, hence more easily distorted — polarizability rises sharply, and with it, dispersion strength and boiling point.

Figure

Instantaneous dipole on atom A inducing a correlated dipole on atom B

Two molecules only feel this attraction where their van der Waals surfaces can approach closely, so the total dispersion energy between two molecules is, to a good first approximation, a sum over all close atom-atom contacts — which is why shape matters enormously. A long floppy chain (n-octane) can lie alongside another chain over its whole length, maximizing the number of close atom-atom contacts and hence total dispersion energy, while a compact, nearly spherical isomer (2,2,3,3-tetramethylbutane, or neopentane's C8 relatives) presents a small, low-surface-area silhouette to its neighbors — fewer contacts, weaker net attraction, lower boiling point, despite identical molecular formula and identical total electron count (identical total polarizability, roughly).

Figure

Contact surface area: linear chain versus compact sphere of the same formula

The core ideas

The instantaneous-dipole/induced-dipole picture is quantified by treating the interaction as a perturbation on two isolated, non-interacting atoms/molecules using second-order perturbation theory. The central quantitative result — the London formula — for two identical species at large separation $r$ is

U_{\text{disp}}(r) = -\frac{3}{4}\,\frac{\alpha^2 I}{(4\pi\varepsilon_0)^2\, r^6},

where:

$\alpha$ is the (static) polarizability of each species (identical for like species),
$I$ is the first ionization energy of the species (used as a proxy for the characteristic excitation energy scale that appears in the sum-over-states),
$r$ is the center-to-center separation,
$(4\pi\varepsilon_0)^2$ carries the SI units when $\alpha$ is expressed in the SI polarizability convention (C·m²·V⁻¹); many textbooks instead use $\alpha$ in volume units (ų) where the $4\pi\varepsilon_0$ factors are absorbed.

For two different species, the generalized (approximate, Slater–Kirkwood-style) form is

U_{\text{disp}}(r) = -\frac{3}{2}\,\frac{\alpha_A \alpha_B}{(4\pi\varepsilon_0)^2\, r^6}\cdot\frac{I_A I_B}{I_A + I_B},

where:

$\alpha_A, \alpha_B$ are the polarizabilities of the two partners,
$I_A, I_B$ are their ionization energies,
the harmonic-mean-like combination $I_AI_B/(I_A+I_B)$ reduces correctly to $I/2$ when $I_A = I_B = I$ , recovering the identical-species formula above.

The overwhelmingly important qualitative content of both equations, and the one every student must internalize, is the scaling:

U_{\text{disp}} \propto -\frac{\alpha_A\alpha_B}{r^6}.

Reading: the attraction grows with the product of polarizabilities (bigger, softer electron clouds interact more strongly) and collapses extremely fast with distance — doubling $r$ cuts the interaction by a factor of 64 — which is why dispersion is fundamentally a contact force: it matters enormously at van der Waals contact distance and is negligible a few bond lengths further out. This steep $1/r^6$ falloff, contrasted with the $1/r$ falloff of a bare Coulomb interaction between ions, is why van der Waals radii and close molecular packing (crystal engineering, protein folding, drug-receptor fit) are so sensitive to small changes in shape and distance.

Derivation AWhy the instantaneous dipole exists: electron correlation in two coupled oscillators

Goal: show, using the simplest possible quantum model — two coupled one-dimensional harmonic oscillators representing the "electron on a spring" for each of two atoms — that the exact ground-state energy of the coupled system is lower than the sum of the two isolated oscillator ground-state energies, and that this energy lowering scales as $-1/r^6$ . This is the textbook Drude-oscillator derivation of dispersion, and it is fully solvable in closed form.

Step 1 — model each atom as a charge on a spring. Represent atom A by an electron of charge $-e$ and effective mass $m$ tethered to its nucleus by a harmonic restoring force with natural frequency $\omega_0$ (this mimics the real binding of an electron in an atomic orbital, whose "spring constant" is related to $\omega_0$ via the polarizability, $\alpha = e^2/(m\omega_0^2)$ , a standard classical-oscillator model of polarizability). Let $x_A$ be its displacement from equilibrium along the line joining the two atoms. Atom B, at distance $r$ away, is modeled identically with coordinate $x_B$ and the same $\omega_0, m$ . Isolated, the Hamiltonian is simply

H_0 = \frac{p_A^2}{2m} + \frac{1}{2}m\omega_0^2 x_A^2 + \frac{p_B^2}{2m} + \frac{1}{2}m\omega_0^2 x_B^2,

with ground-state energy $E_0 = \hbar\omega_0$ (two independent oscillators, each contributing $\tfrac12\hbar\omega_0$ — wait, each contributes $\tfrac12\hbar\omega_0$ , so $E_0 = \hbar\omega_0$ total, for the two-oscillator system with no coupling).

Step 2 — add the dipole-dipole coupling. When electron A is displaced by $x_A$ , atom A carries an instantaneous dipole $\mu_A = -ex_A$ ; likewise $\mu_B = -ex_B$ for B. Two dipoles aligned along the line connecting them (the head-to-tail geometry) interact via the classical dipole-dipole potential,

V_{\text{dd}} = \frac{1}{4\pi\varepsilon_0}\frac{\mu_A\mu_B \,(1 - 3\cos^2\theta)}{r^3},

which for the collinear arrangement ( $\theta = 0$ ) becomes

V_{\text{dd}} = -\frac{2}{4\pi\varepsilon_0}\frac{\mu_A\mu_B}{r^3} = -\frac{2e^2}{4\pi\varepsilon_0 r^3}\,x_Ax_B.

The full coupled Hamiltonian is $H = H_0 + V_{\text{dd}}$ , with a coupling constant $k' \equiv 2e^2/(4\pi\varepsilon_0 r^3)$ so that $V_{\text{dd}} = -k' x_A x_B$ .

Step 3 — diagonalize by normal modes. Introduce symmetric and antisymmetric combinations,

x_+ = \frac{x_A+x_B}{\sqrt2}, \qquad x_- = \frac{x_A-x_B}{\sqrt2}.

Substituting, the cross term $-k'x_Ax_B$ becomes $-\tfrac{k'}{2}(x_+^2 - x_-^2)$ , so the Hamiltonian separates exactly into two independent oscillators,

H = \left[\frac{p_+^2}{2m} + \frac{1}{2}\!\left(m\omega_0^2 - k'\right)x_+^2\right] + \left[\frac{p_-^2}{2m} + \frac{1}{2}\!\left(m\omega_0^2 + k'\right)x_-^2\right].

Each bracket is a harmonic oscillator with shifted force constant, giving new frequencies

\omega_\pm = \omega_0\sqrt{1 \mp \frac{k'}{m\omega_0^2}} \approx \omega_0\left(1 \mp \frac{k'}{2m\omega_0^2} - \frac{k'^2}{8m^2\omega_0^4} + \cdots\right),

using the binomial expansion $\sqrt{1+x}\approx 1+\tfrac{x}{2}-\tfrac{x^2}{8}$ for small $k'/(m\omega_0^2)$ (valid because dispersion coupling is weak at chemically relevant distances).

Step 4 — sum the zero-point energies and extract the $r^{-6}$ term. The exact ground-state energy of the coupled system is $E = \tfrac12\hbar(\omega_+ + \omega_-)$ . Adding the two expansions, the terms linear in $k'$ cancel (one has $-k'/2m\omega_0^2$ , the other $+k'/2m\omega_0^2$ ), and the leading surviving correction is quadratic in $k'$ :

E \approx \hbar\omega_0\left(1 - \frac{k'^2}{8m^2\omega_0^4}\right) = \hbar\omega_0 - \frac{\hbar k'^2}{8m^2\omega_0^3}.

The interaction energy is the shift relative to the uncoupled system, $\Delta E = E - \hbar\omega_0$ :

\Delta E = -\frac{\hbar k'^2}{8m^2\omega_0^3}.

Substituting $k' = 2e^2/(4\pi\varepsilon_0 r^3)$ ,

\Delta E = -\frac{\hbar}{8m^2\omega_0^3}\cdot\frac{4e^4}{(4\pi\varepsilon_0)^2 r^6} = -\frac{\hbar e^4}{2 m^2\omega_0^3 (4\pi\varepsilon_0)^2 r^6}.

Using $\alpha = e^2/(m\omega_0^2)$ (the classical Drude polarizability) to eliminate $e^2/m$ , and identifying $\hbar\omega_0$ with a characteristic excitation energy scale $\sim I$ (ionization energy), this becomes, up to the numerical prefactor that a fully quantum treatment refines to $3/4$ ,

\Delta E \sim -\frac{\alpha^2 \hbar\omega_0}{(4\pi\varepsilon_0)^2 r^6} \sim -\frac{3}{4}\frac{\alpha^2 I}{(4\pi\varepsilon_0)^2 r^6},

recovering exactly the London dispersion formula. The key physical content revealed by this derivation: the energy lowering is a zero-point (quantum) effect — it survives at $T=0$ and does not require any thermal population of excited states; it exists purely because coupling two oscillators splits their frequencies asymmetrically, and the symmetric combination softens (lower $\omega_+$ , lower zero-point energy) more than the antisymmetric combination stiffens, so the net zero-point energy always decreases. This asymmetric splitting is the rigorous version of "correlated fluctuation always attracts."

Go deeper · Derivation B · 2 minSecond-order perturbation theory: why the sign must be negative

Derivation BSecond-order perturbation theory: why the sign must be negative

Goal: confirm, using standard second-order Rayleigh–Schrödinger perturbation theory applied to two atoms in their unperturbed ground states, that the dispersion energy correction is necessarily negative (attractive) to all orders in this treatment, not just in the coupled-oscillator toy model.

Step 1 — set up the perturbation. Let the unperturbed system be atom A in its ground state $|0_A\rangle$ and atom B in its ground state $|0_B\rangle$ , with unperturbed energy $E_0^{(0)} = E_{0_A} + E_{0_B}$ . The perturbation is the full electrostatic dipole-dipole operator between the instantaneous charge distributions,

\hat V = \frac{1}{4\pi\varepsilon_0 r^3}\Big[\hat{\boldsymbol\mu}_A\cdot\hat{\boldsymbol\mu}_B - 3(\hat{\boldsymbol\mu}_A\cdot\hat n)(\hat{\boldsymbol\mu}_B\cdot\hat n)\Big],

where $\hat{\boldsymbol\mu}_A, \hat{\boldsymbol\mu}_B$ are the (operator-valued, fluctuating) dipole moment operators of each atom and $\hat n$ is the unit vector along the line joining them. Crucially, $\langle 0_A|\hat{\boldsymbol\mu}_A|0_A\rangle = 0$ (no permanent dipole in an atomic ground state), so first-order perturbation theory gives

E^{(1)} = \langle 0_A 0_B|\hat V|0_A0_B\rangle = 0,

because $\hat V$ is linear in each atom's dipole operator individually, and each such matrix element vanishes between identical ground states. This proves rigorously that there is no first-order (classical, permanent-dipole) electrostatic energy between two ground-state atoms — any interaction must arise at second order or beyond.

Step 2 — write the second-order correction. Standard non-degenerate perturbation theory gives

E^{(2)} = \sum_{n\neq 0}\frac{\left|\langle n|\hat V|0\right|^2}{E_0^{(0)} - E_n^{(0)}},

where the sum runs over all excited states $|n\rangle = |m_A\rangle\otimes|k_B\rangle$ of the combined two-atom system (excluding the ground state), and $E_n^{(0)} = E_{m_A} + E_{k_B}$ . Every excited-state energy $E_n^{(0)}$ is, by definition of "ground state," strictly greater than $E_0^{(0)}$ , so the denominator $E_0^{(0)} - E_n^{(0)}$ is strictly negative for every single term in the sum.

Step 3 — read off the sign. The numerator $|\langle n|\hat V|0\rangle|^2$ is manifestly non-negative (it is a modulus squared), and at least one such matrix element must be nonzero (dipole-allowed transitions exist on both atoms, since real atoms are polarizable). Therefore every term in the sum is

\frac{(\text{non-negative})}{(\text{strictly negative})} \le 0,

and the sum, $E^{(2)}$ , is strictly negative. This is a fully general, model-independent proof: the leading nonvanishing correction to the energy of two ground-state atoms from the induced dipole-dipole interaction is always attractive, to second order, regardless of the detailed atomic structure. There is no analogous repulsive second-order dispersion term — the sign is fixed by the structure of perturbation theory itself (ground state has the lowest energy denominator possible, so all virtual excitations lower the energy).

Step 4 — recover the $r^{-6}$ dependence and connect to $\alpha$ . Since $\hat V \propto 1/r^3$ , and $E^{(2)}$ is quadratic in $\hat V$ (it appears as $|\langle n|\hat V|0\rangle|^2$ ), the entire sum scales as

E^{(2)} \propto \frac{1}{r^6}.

Performing the angular sum over the dipole-dipole tensor and invoking the standard dipole polarizability sum rule,

\alpha_A = \frac{2}{3}\sum_{m\neq 0}\frac{|\langle m|\hat\mu_A|0\rangle|^2}{E_m - E_0},

one can show (by a Casimir–Polder-style manipulation converting the double sum over excited states of both atoms into a single frequency integral over each atom's dynamic polarizability $\alpha(i\omega)$ evaluated at imaginary frequency) that the closed-form result is precisely the London/Casimir–Polder integral,

U_{\text{disp}}(r) = -\frac{3\hbar}{\pi r^6}\int_0^\infty \alpha_A(i\omega)\,\alpha_B(i\omega)\,d\omega,

which reduces to the simple London formula above when each $\alpha(i\omega)$ is approximated by a single-term (single-oscillator) model with characteristic frequency $\omega_0 \sim I/\hbar$ . This confirms both the sign (always attractive) and the power law ( $r^{-6}$ ) from first principles, independent of the crude coupled-oscillator picture, and shows that the coupled-oscillator derivation is not a coincidence but a special case of the exact quantum-mechanical result.

Worked examples

Example 1 — Foundational: ranking noble gas boiling points from polarizability. The noble gases He, Ne, Ar, Kr, Xe have no permanent dipoles, no bonds, and interact purely by dispersion. Their polarizabilities (in ų) are approximately He: 0.21, Ne: 0.40, Ar: 1.64, Kr: 2.48, Xe: 4.04, and their first ionization energies (eV) are He: 24.6, Ne: 21.6, Ar: 15.8, Kr: 14.0, Xe: 12.1. Using the qualitative scaling $U_{\text{disp}} \propto -\alpha^2 I$ atom-pairwise (the exact prefactor is not needed for ranking, only the trend of the product $\alpha^2 I$ ), compute relative dispersion strengths:

\alpha^2 I(\text{He}) = (0.21)^2(24.6) = 1.08,

\alpha^2 I(\text{Ne}) = (0.40)^2(21.6) = 3.46,

\alpha^2 I(\text{Ar}) = (1.64)^2(15.8) = 42.5,

\alpha^2 I(\text{Kr}) = (2.48)^2(14.0) = 86.1,

\alpha^2 I(\text{Xe}) = (4.04)^2(12.1) = 197.5.

The trend He < Ne < Ar < Kr < Xe in $\alpha^2 I$ matches exactly the experimental boiling point order: He (4.2 K) < Ne (27.1 K) < Ar (87.3 K) < Kr (119.9 K) < Xe (165.0 K). Note the interesting internal detail: polarizability rises faster down the group than ionization energy falls, so the product is dominated by the $\alpha^2$ term — this is the general rule, which is why polarizability (a size/diffuseness measure) is usually cited as the single most important predictor, with ionization energy as a secondary refinement.

Limiting check: as atomic number increases without bound (hypothetically), $\alpha$ increases roughly as the cube of atomic radius while $I$ decreases only slowly (bounded below by relativistic and shell-structure effects); the product $\alpha^2I$ therefore grows essentially monotonically, consistent with the observed unbroken monotonic trend in boiling points down the noble gas group — there is no group-18 exception.

Example 2 — Applied: pentane isomers and the branching effect on boiling point. Consider the three isomers of pentane, C₅H₁₂: n-pentane (linear), isopentane (2-methylbutane, one branch), and neopentane (2,2-dimethylpropane, maximally branched/most spherical). All three have identical molecular formula, hence essentially identical total electron count and very similar total (isotropic) polarizability. Yet their boiling points differ substantially: n-pentane 36.1 °C, isopentane 27.8 °C, neopentane 9.5 °C. Mechanistically: dispersion energy is not simply "total polarizability" — it is a sum over close atom-atom contacts achievable when two molecules pack together, weighted by $1/r^6$ so only genuinely close contacts matter. n-Pentane is a flexible rod that can align side-by-side with a neighbor, bringing many CH₂/CH₃ groups within van der Waals contact distance simultaneously along its whole length — maximal contact area. Neopentane is a compact, nearly spherical molecule (four methyl groups radiating from a central quaternary carbon); two neopentane molecules approaching each other can only achieve a small, roughly point-like contact patch no matter how they orient, because the molecule presents a convex, low-surface-area silhouette in every direction. Isopentane, with one branch, is intermediate in shape and intermediate in boiling point, exactly as observed. Quantitatively, the surface areas (solvent-accessible surface area, roughly) scale as n-pentane > isopentane > neopentane, tracking the boiling point order precisely — this is the standard explanation given by the empirical correlation between molecular surface area and dispersion-driven boiling point within a set of isomers of fixed formula.

Limiting check: in the hypothetical limit of an infinitely long, fully extended n-alkane chain compared with a maximally compact isomer of the same formula (not physically realizable beyond a certain chain length, but useful as a thought experiment), the boiling point gap would grow without bound as chain length increases, since surface contact area for the linear chain grows linearly with chain length while the compact isomer's surface area grows only as (chain length)^(2/3) — consistent with the empirically observed widening boiling-point gap between linear and highly branched alkanes as the carbon count increases (e.g., compare octane/2,2,3,3-tetramethylbutane, gap ≈ 20 °C, versus the ≈27 °C pentane/neopentane gap being roughly comparable per-carbon at this chain length but continuing to widen for longer chains).

Example 3 — Exam/competition-level: estimating a Hamaker-style dispersion energy between two nonidentical, non-round-number atoms and checking units. Estimate the leading-order dispersion interaction energy between one Ar atom and one Xe atom held at a center-to-center separation of $r = 4.00$ Å, using the heteronuclear London formula, given: $\alpha_{\text{Ar}} = 1.64\times10^{-30}\ \text{m}^3$ (in the volume-polarizability convention, related to the SI polarizability by $\alpha_{SI} = 4\pi\varepsilon_0\alpha_{\text{vol}}$ ), $\alpha_{\text{Xe}} = 4.04\times10^{-30}\ \text{m}^3$ , $I_{\text{Ar}} = 15.8\ \text{eV}$ , $I_{\text{Xe}} = 12.1\ \text{eV}$ .

Step 1. In the volume-polarizability convention, the heteronuclear London formula simplifies (the $4\pi\varepsilon_0$ factors cancel against the definition of volume polarizability) to

U_{\text{disp}}(r) = -\frac{3}{2}\cdot\frac{1}{4\pi\varepsilon_0}\cdot\frac{\alpha_{\text{Ar}}\alpha_{\text{Xe}}}{r^6}\cdot\frac{I_{\text{Ar}}I_{\text{Xe}}}{I_{\text{Ar}}+I_{\text{Xe}}}.

(Different textbooks place the $4\pi\varepsilon_0$ differently depending on whether $\alpha$ is quoted in ų or in SI $\text{C}\,\cdot\,\text{m}^2\text{V}^{-1}$ ; we keep it explicit here to track units carefully.)

Step 2 — compute the harmonic-mean-like ionization factor.

\frac{I_{\text{Ar}}I_{\text{Xe}}}{I_{\text{Ar}}+I_{\text{Xe}}} = \frac{(15.8)(12.1)}{15.8+12.1}\ \text{eV} = \frac{191.2}{27.9}\ \text{eV} = 6.85\ \text{eV}.

Converting to joules: $6.85\ \text{eV} \times 1.602\times10^{-19}\ \text{J/eV} = 1.097\times10^{-18}\ \text{J}$ .

Step 3 — compute $r^6$ .

r = 4.00\ \text{Å} = 4.00\times10^{-10}\ \text{m}, \qquad r^6 = (4.00\times10^{-10})^6 = 4.10\times10^{-59}\ \text{m}^6.

Step 4 — compute the polarizability product.

\alpha_{\text{Ar}}\alpha_{\text{Xe}} = (1.64\times10^{-30})(4.04\times10^{-30}) = 6.63\times10^{-60}\ \text{m}^6.

Step 5 — assemble, tracking the prefactor. Using the volume-polarizability convention where the working formula (absorbing $4\pi\varepsilon_0$ appropriately for atomic-unit-style ų inputs converted to SI as above) is

U_{\text{disp}} = -\frac{3}{2}\cdot\frac{\alpha_{\text{Ar}}\alpha_{\text{Xe}}}{r^6}\cdot\left(\frac{I_{\text{Ar}}I_{\text{Xe}}}{I_{\text{Ar}}+I_{\text{Xe}}}\right)\Big/(4\pi\varepsilon_0)^{0},

taking the standard simplified atomic-physics form (units chosen so the ų/eV/Å inputs combine directly, as is conventional in this estimate, giving energy in eV once $\alpha$ is in ų and $r$ in Å):

U_{\text{disp}} = -\frac{3}{2}\cdot\frac{(1.64)(4.04)}{(4.00)^6}\cdot 6.85\ \text{eV} = -\frac{3}{2}\cdot\frac{6.63}{4096}\cdot 6.85\ \text{eV}.

U_{\text{disp}} = -\frac{3}{2}\times1.619\times10^{-3}\times6.85\ \text{eV} = -1.66\times10^{-2}\ \text{eV}.

Converting, $U_{\text{disp}} \approx -1.66\times10^{-2}\times1.602\times10^{-19}\ \text{J} = -2.66\times10^{-21}\ \text{J}$ per pair, or about $-1.60\ \text{kJ/mol}$ when scaled by Avogadro's number — a modest but entirely real attraction, comparable in order of magnitude to $k_BT$ at room temperature ( $k_BT \approx 0.0257$ eV at 298 K), which is exactly why noble gases condense to liquids only at cryogenic temperatures where thermal agitation ( $k_BT$ ) can no longer overwhelm this weak attraction.

Limiting check: setting $I_{\text{Ar}} = I_{\text{Xe}} = I$ recovers the homonuclear factor $I^2/2I = I/2$ , matching the identical-species formula's $\tfrac34\alpha^2 I$ prefactor structure exactly (since $\tfrac32\cdot\tfrac{I}{2} = \tfrac34 I$ ) — confirming internal consistency between the heteronuclear and homonuclear forms.

Limiting cases, scope & common traps

Situation	What actually happens	Why
Two nonpolar molecules, no dipole	Only dispersion acts; still attractive, still governs bp/mp	Instantaneous-dipole correlation is universal, present between any two atoms
Polar molecules (e.g., HCl, acetone)	Dispersion still present underneath dipole-dipole forces, often the larger term	Dispersion scales with the whole polarizable electron cloud, not just the permanent dipole; for many small polar molecules dispersion contributes >50% of total attraction
Very short range ( $r$ near or below van der Waals contact)	Attraction reverses to steep repulsion	Pauli/exchange repulsion from overlapping filled orbitals dominates; real intermolecular potentials (Lennard-Jones) combine $-1/r^6$ attraction with $+1/r^{12}$ repulsion
Very long range ( $r$ beyond a few nm, especially for large/conducting objects)	True $1/r^6$ law breaks down; retardation sets in	Finite speed of light means the induced field lags the inducing dipole; the Casimir–Polder retarded regime instead falls off as $1/r^7$
Comparing isomers of identical formula	Boiling point differences arise almost entirely from shape/surface-area, not from different total polarizability	Total electron count (hence total α) is nearly identical; contact geometry differs sharply
Very large, ovate molecules (e.g. long-chain fatty acids, graphite sheets)	Dispersion can dominate over stronger-per-contact but less numerous hydrogen bonds	Summing many weak contacts over a huge contact area can exceed the energy of a few strong, localized H-bonds

The chemist behind it

Origins · 2 minFritz Wolfgang London and how this was discovered

Fritz Wolfgang London

1900–1954 · Theoretical physics and quantum chemistry, Germany → France → United States (Duke University)

Fritz London trained in philosophy before turning to physics, earning his doctorate in Munich and working through the tumultuous years of early quantum mechanics in Zurich, Berlin, and Rome. Together with Walter Heitler, he produced in 1927 the first genuinely quantum-mechanical treatment of the covalent bond in H₂ — the Heitler–London calculation, foundational to all of Intermolecular Forces theory and to valence bond theory generally — showing for the first time that electron exchange and correlation, not classical electrostatics, hold a covalent bond together. London's 1930 paper "Zur Theorie und Systematik der Molekularkräfte" applied the newly matured tools of quantum perturbation theory to the old puzzle of why all molecules, polar or not, attract one another at short range; his second-order perturbative treatment of the induced-dipole/induced-dipole interaction produced exactly the $-C/r^6$ law derived above and gave the effect its universally recognized name. Forced to leave Germany in 1933 under the Nazi racial laws, London worked in Oxford and Paris before settling at Duke University in the United States, where he later turned to a second major body of work — with his brother Heinz London, the phenomenological theory of superconductivity (the London equations, 1935) — making him one of the rare physicists to have two entirely distinct, eponymous, foundational contributions across quantum chemistry and condensed matter physics.

History

- 1927 — Heitler and London give the first quantum-mechanical (valence-bond) treatment of the H₂ covalent bond, establishing electron correlation as a bonding mechanism.

- 1930 — Fritz London publishes the perturbation-theoretic derivation of the universal

-C/r^6

attraction between any two atoms or molecules, naming what we now call the London dispersion force.

- 1937 — J. E. Lennard-Jones and collaborators combine the

-1/r^6

attractive term with an empirical

+1/r^{12}

repulsive term into the now-standard Lennard-Jones potential used throughout molecular simulation.

- 1948 — Hendrik Casimir and Dirk Polder extend the theory to long-range retarded dispersion forces, predicting the

1/r^7

falloff that becomes dominant when the electromagnetic field's finite propagation speed matters (the basis of the macroscopic Casimir effect).

- 1956 — Evgeny Lifshitz develops a fully macroscopic, continuum electrodynamic theory of dispersion forces between bulk materials (Lifshitz theory), underlying modern treatments of colloid stability and nanoscale adhesion.

Do polar molecules also experience London dispersion forces, or only dipole-dipole forces?

Polar molecules experience both. Every molecule, polar or not, has a fluctuating electron cloud and nonzero polarizability, so dispersion is always present as a baseline contribution. For many small polar molecules — HCl is the classic example — dispersion actually contributes more to the total attractive energy than the permanent dipole-dipole term does, because the dipole-dipole interaction, while individually stronger per favorable orientation, is partly averaged out by molecular rotation and tumbling, whereas dispersion (arising from correlated fluctuations, not orientation-dependent alignment of permanent dipoles) survives full rotational averaging. Only for strongly polar, hydrogen-bonding molecules like water does the permanent-dipole/hydrogen-bonding contribution clearly overtake dispersion as the dominant term.

Why does the dispersion energy fall off as 1/r⁶ specifically, rather than 1/r³ like a simple dipole-dipole interaction?

A single, static dipole-dipole interaction indeed falls off as $1/r^3$ . But dispersion is a second-order effect: the instantaneous dipole on atom A (itself already a fluctuating, zero-average quantity) must first generate a field that then induces a correlated dipole on atom B, and the resulting interaction energy involves the product of both dipole moments interacting through the $1/r^3$ dipole-dipole coupling. Since the whole process is quadratic in the $1/r^3$ coupling (appearing as $|\langle n|\hat V|0\rangle|^2$ in second-order perturbation theory, and $\hat V \propto 1/r^3$ ), the net energy scales as $(1/r^3)^2 = 1/r^6$ .

Is polarizability the same thing as molecular size?

Closely related but not identical. Polarizability does correlate strongly with molecular volume and electron count — bigger molecules with more electrons are usually more polarizable — but the precise relationship also depends on how tightly the electrons are held. A small, easily distorted anion can be more polarizable than a larger but very electronegative, tightly-bound neutral molecule. Within a single group of the periodic table, increasing principal quantum number of the outer electrons (going down the group) increases polarizability faster than simple volume scaling alone would suggest, because outer electrons in higher shells are both more numerous and more weakly bound to the nucleus.

How do dispersion forces explain why oil and water don't mix (the hydrophobic effect)?

The hydrophobic effect is subtler than "oil molecules attract each other via dispersion, so they clump" — although that attraction is real and does contribute. The larger effect is entropic: water molecules around a nonpolar solute must maintain their hydrogen-bond network, forming an ordered "cage" (clathrate-like shell) that costs entropy; aggregating nonpolar solutes together reduces the total ordered-water surface area and releases that entropic penalty. Dispersion attraction between the nonpolar solute molecules themselves is a real, additional favorable contribution to aggregation, but the dominant driving force for hydrophobic aggregation in water specifically is usually the entropic release from disrupting less ordered-water structure, not dispersion energy alone.

Why do standard DFT calculations need an explicit "dispersion correction" added on?

Standard exchange-correlation functionals in density functional theory (like common GGA or hybrid functionals) are built from the local electron density and its gradient at each point in space. But dispersion is fundamentally a nonlocal correlation effect — it requires simultaneous knowledge of fluctuations at two well-separated regions of space (one atom's instantaneous dipole and another, distant atom's induced response) that a local or semi-local functional simply cannot see. This is why methods like Grimme's DFT-D3/D4 add an explicit empirical $-C_6/r^6$ term (parametrized per atom pair, structurally identical to the classical London formula) directly onto the DFT energy, and why fully nonlocal van der Waals density functionals (vdW-DF) were developed as a more first-principles alternative.

Do dispersion forces ever become repulsive at any distance?

The dispersion term itself, derived from second-order perturbation theory on well-separated ground-state atoms, is always attractive (negative) at every separation where the perturbative expansion is valid. What changes sign is the total intermolecular potential once you also include Pauli exchange repulsion, which arises when the electron clouds begin to overlap directly (not just interact through long-range induced fields) and grows extremely steeply, roughly as $1/r^{12}$ in the empirical Lennard-Jones model. The combination of the two — attractive dispersion at longer range, repulsive exchange at short range — is what produces the characteristic potential well with a minimum at the van der Waals contact distance, not a sign change in the dispersion term itself.

Why does chain branching lower boiling point but often raise melting point, or does it always lower both?

It is not a universal rule for melting points, unlike the fairly reliable trend for boiling points. Boiling point tracks liquid-phase intermolecular attraction, which is well predicted by the surface-area/dispersion argument given here: branching reduces achievable contact area, lowering boiling point. Melting point additionally depends on how efficiently the solid molecules can pack into a crystal lattice; highly symmetric, compact, spherical molecules like neopentane can sometimes pack into unusually favorable, high-symmetry crystal lattices (neopentane's solid-state behavior is in fact anomalous, showing plastic-crystal/rotator phases), occasionally giving melting points that do not follow the same simple monotonic trend as boiling points. Always check both shape-driven liquid-state dispersion and lattice-packing efficiency separately when predicting melting point trends.