London Dispersion Force

The London dispersion force (also termed instantaneous dipole–induced dipole attraction or electrodynamic dispersion) is the universal, quantum-mechanical attractive component of intermolecular van der Waals forces that acts between all atoms and molecules—including completely nonpolar, spherically symmetric species such as the noble gases (He, Ne, Ar, Kr, Xe), homonuclear diatomics (H₂, N₂, O₂, Cl₂, Br₂, I₂), and nonpolar hydrocarbons (methane, lipid acyl chains, and aliphatic protein side chains).

Derived in 1930 by the German-American theoretical physicist Fritz London using second-order quantum-mechanical perturbation theory, the dispersion force resolves a fundamental paradox of classical electrostatics: why electrically neutral atoms with zero permanent electric dipole moment (μ = 0 D), zero quadrupole moment, and spherically symmetric time-averaged electron clouds still attract one another at medium range (V_disp(r) ∝ −1/r⁶) strongly enough to condense into liquids and solids. Across chemistry, geology, and structural biology, London dispersion forces govern the boiling points of nonpolar substances, the basal cleavage of layered phyllosilicate minerals (talc and graphite), base-stacking stability in the DNA double helix, van der Waals packing inside the hydrophobic core of proteins, and the dry fibrillar adhesion of gecko toe-pads.

🧬 Interactive Protein Folding Simulation: Pack nonpolar aliphatic and aromatic side chains via van der Waals dispersion and the hydrophobic effect in FoldLab.

Correlated Zero-Point Fluctuations and Instantaneous Dipoles

Even in its lowest-energy electronic ground state (Ψ₀) at absolute zero (0 K), the Heisenberg uncertainty principle (Δ*x* · Δ*p*_x ≥ ℏ/2) forbids the electrons of an atom or molecule from remaining frozen at fixed coordinates. Although the time-averaged expectation value of the dipole operator vanishes for a nonpolar atom (⟨*Ψ*₀|μ̂|*Ψ*₀⟩ = 0), at any given attosecond (~10⁻¹⁶ s) the instantaneous spatial positions of the electrons produce a non-zero instantaneous electric dipole moment μ₁(t).

This fluctuating dipole on atom 1 generates an oscillating electric field E₁(t) ∝ μ₁(t) / r³ at a neighboring atom 2 located a distance r away. That electric field polarizes the electron cloud of atom 2, inducing a synchronized induced dipole moment μ_2,ind(t) = α₂ E₁(t) ∝ α₂ μ₁(t) / r³ (where α₂ is the dipole polarizability of atom 2) oriented in phase to attract the instantaneous dipole of atom 1. Because the interaction energy between two dipoles scales as −*μ*₁ μ_2,ind / r³, substituting μ_2,ind ∝ α₂ μ₁ / r³ yields a net time-averaged attractive potential proportional to −1/r⁶.

Second-Order Perturbation Derivation and the London C₆ Coefficient

Modeling two isotropic three-dimensional quantum harmonic oscillators (atoms A and B with polarizabilities α′_A, α′_B and characteristic electronic excitation / first ionization energies I_A ≈ IE_1,A and I_B ≈ IE_1,B) coupled by the dipole-dipole Hamiltonian operator Ĥ′ = (e² / 4π*ε*₀r³)(x_Ax_B + y_Ay_B − 2*z*_Az_B), Fritz London derived the London dispersion potential:

V_London(r) = − C₆ / r⁶ = − (3 / 2) [ (I_A · I_B) / (I_A + I_B) ] · ( α′_A · α′_B / r⁶ )

For two identical atoms or molecules (A = B, with polarizability volume α′ = α / 4π*ε*₀ in ų and first ionization energy I in kJ mol⁻¹ or eV), the formula simplifies to:

V_London(r) = − (3 / 4) · ( I · (α′)² / r⁶ )

Why did Fritz London name this interaction the dispersion force? Because the exact same electronic oscillator strengths (f_0*k*) and transition frequencies (ω_0*k* ≈ I/ℏ) that govern the frequency-dependent dynamic polarizability α(ω)—and therefore the optical dispersion of the refractive index n(ω) of light passing through a transparent gas or crystal—govern the quantum-mechanical correlation of fluctuating dipoles! At very large separations (r > c/ω₀ ≈ 50–100 nm), the finite speed of light (c) introduces a phase delay between the fluctuating dipoles (the Casimir–Polder retardation effect, 1948), steepening the asymptotic decay from −1/r⁶ to −1/r⁷.

In physical chemistry, the total attractive van der Waals potential (V_vdW = −*C*_6,total / r⁶) between two neutral molecules at thermal equilibrium is the sum of three distinct −1/r⁶ contributions:

Look closely at the hydrogen chloride (HCl) column in the table above: even in a classic polar gas like HCl (μ = 1.08 D), London dispersion accounts for 81.4% of the total intermolecular van der Waals attraction, whereas permanent dipole–dipole (Keesom) orientation accounts for only 14.4%! Only in molecules with very small polarizabilities and large dipole moments capable of short-range hydrogen bonding (such as H₂O, NH₃, and HF) does electrostatic dipole orientation surpass London dispersion.

When two molecules approach closer than their equilibrium van der Waals contact distance (r_m = 2^1/6σ ≈ 3.3–4.0 Å), their filled valence electron clouds begin to overlap, triggering steep Pauli exchange repulsion (scaling exponentially as A exp(−*b r*) in the Buckingham potential, or approximated as +1/r¹²). Combining short-range Pauli repulsion (+1/r¹²) with medium-range London/van der Waals attraction (−1/r⁶) yields the Lennard-Jones (12–6) potential used across molecular dynamics simulations:

V_LJ(r) = 4 ε [ (σ / r)¹² − (σ / r)⁶ ]

where σ is the collision diameter at which V_LJ(σ) = 0, and ε is the potential well depth at the equilibrium contact minimum r_m = 2^1/6σ ≈ 1.122 σ.

1. Polarizability Volume (α′) and Molar Mass / Electron Count: Because London dispersion scales with the square of the polarizability ((α′)²), larger atoms and longer chains with more diffuse, easily distorted electron clouds experience dramatically stronger dispersion forces:

• Down the noble gases (He → Xe) and halogens (F₂ → Cl₂ → Br₂ → I₂), increasing α′ raises boiling points from 4.2 K (He) to 165 K (Xe) and turns gaseous F₂ (−188 °C) into liquid Br₂ (+58.8 °C) and solid I₂ (+184.3 °C).
• Across linear n-alkanes (C_nH_2*n*+2), each added –CH₂– methylene group adds ~4–5 kJ mol⁻¹ of additive dispersion enthalpy: C₁–C₄ (methane through butane) are gases at 298 K; C₅–C₁₇ (pentane through heptadecane, gasoline and kerosene) are liquids; and C₁₈+ (octadecane and paraffin waxes) are crystalline solids.

1. Molecular Shape and Surface Contact Area (n-Pentane vs. Neopentane): For constitutional isomers having the exact same molecular formula, molar mass, and total number of electrons (C₅H₁₂, M = 72.15 g mol⁻¹, 42 electrons), molecular geometry dictates how closely neighboring molecules can approach across their −1/r⁶ contact surface:

• n-Pentane (CH₃CH₂CH₂CH₂CH₃): Extended rod-like zigzag chain maximizes lateral intermolecular contact area ⟹ Boiling point = +36.1 °C (309.2 K); Δ*H*_vap = 25.8 kJ mol⁻¹.
• Isopentane (2-methylbutane, (CH₃)₂CHCH₂CH₃): Moderately branched ⟹ Boiling point = +27.8 °C (301.0 K); Δ*H*_vap = 24.6 kJ mol⁻¹.
• Neopentane (2,2-dimethylpropane, C(CH₃)₄): Compact, nearly spherical tetrahedral symmetry reduces surface-to-surface contact to point tangency ⟹ Boiling point = +9.5 °C (282.6 K, a gas at room temperature!); Δ*H*_vap = 22.7 kJ mol⁻¹.

• Protein Core Packing and Membrane Fluidity: While the hydrophobic effect (the entropic release of ordered clathrate water cages upon burying nonpolar surfaces) provides the primary thermodynamic push driving a polypeptide to collapse in water, once nonpolar side chains (leucine, isoleucine, valine, phenylalanine) are inside the dry protein interior, tight jigsaw-like London dispersion packing (packing density η ≈ 0.74, matching close-packed crystals) contributes hundreds of kilojoules per mole of favorable van der Waals enthalpy (Δ*H*_vdW < 0). Similarly, in cell membranes, saturated fatty-acyl chains pack straight in all-trans conformations with strong lateral London dispersion (favoring rigid gel phases), whereas cis C=C double bonds introduce a 30° rigid kink that increases intermolecular distance r (V ∝ −1/r⁶), lowering the lipid melting temperature (T_m) and preserving membrane fluidity in cold-water organisms.
• Base Stacking in the DNA Double Helix: Although Watson–Crick hydrogen bonds (two in A–T, three in G–C) provide the geometric specificity of base pairing, vertical π–π stacking of the flat, polarizable aromatic purine and pyrimidine rings at 3.4 Å separation—driven by London dispersion and hydrophobic desolvation—provides the majority of the thermodynamic free energy stabilizing the B-DNA double helix in aqueous solution.
• Gecko Dry Adhesion (Setal Spatulae): As demonstrated by Kellar Autumn and colleagues in 2000–2002, tokay geckos (Gekko gecko) climb vertical glass and ceilings without glue or suction cups by dividing each toe pad into ~500,000 keratinous hairs (setae), each branching into 100–1,000 nanoscale flat tips (spatulae, ~200 nm wide). Although a single spatula generates only ~10 nN of London dispersion force across a 0.3 nm contact gap, summing across ~10⁹ spatulae in simultaneous molecular contact supports over 100 N (~10 kg, more than 50 times the gecko's body weight).

If a single London dispersion contact is so weak (0.4 to 4 kJ mol⁻¹), how can dispersion forces hold solids and macromolecules together?

A single atom-to-atom London dispersion contact (such as two CH₄ molecules or two Ar atoms, ε ≈ 1.0–1.5 kJ mol⁻¹) is indeed weaker than thermal energy at room temperature (RT = 2.48 kJ mol⁻¹). However, unlike covalent bonds which saturate once a pair of valence orbitals is shared, London dispersion is pairwise additive across all contacting atom pairs along a macromolecular interface. In a polyethylene crystal (–CH₂–)_n, a protein hydrophobic core, or a graphene bilayer, hundreds of adjacent atom pairs sit simultaneously at their optimal van der Waals contact distance (~3.4–3.8 Å). When N = 100 methylene units align along two hydrocarbon chains, the cumulative dispersion enthalpy reaches 100 × 4 kJ mol⁻¹ ≈ 400 kJ mol⁻¹—exceeding the bond dissociation energy of a covalent C–C single bond (348 kJ mol⁻¹), so that heating polyethylene decomposes its covalent backbone before it can ever vaporize!

Why does carbon tetrachloride (CCl₄, μ = 0 D) boil at a higher temperature (+76.7 °C) than polar chloroform (CHCl₃, μ = 1.04 D, +61.2 °C) or fluoromethane (CH₃F, μ = 1.85 D, −78.4 °C)?

A common misconception is that polar molecules always have higher boiling points than nonpolar molecules. In reality, boiling point depends on the sum of Keesom (dipole–dipole), Debye (dipole–induced dipole), and London (dispersion) attractions. Fluoromethane (CH₃F) has a permanent dipole moment (μ = 1.85 D) nearly equal to water, yet it contains only 18 tightly held electrons (α′ = 2.6 ų) and cannot form hydrogen bonds, so it boils at −78.4 °C. Replacing the hydrogen atoms with large, diffuse 17-electron chlorine atoms (α′ ≈ 2.2 ų per Cl atom) increases total molecular polarizability to α′ = 8.5 ų in polar CHCl₃ (T_b = +61.2 °C) and 10.5 ų in symmetric, nonpolar CCl₄. Even though CCl₄ has zero permanent dipole moment (μ = 0 D), the fourth chlorine atom adds more London dispersion energy (V ∝ (α′)²) than the lost Keesom dipole interaction of CHCl₃, raising the boiling point of CCl₄ to +76.7 °C.

Why does the attractive London dispersion potential scale specifically with the inverse sixth power of distance (−1/r⁶)?

The −1/r⁶ scaling follows directly from applying classical electrostatics twice in succession:

1. First, the electric field E₁ created at distance r by an instantaneous electric dipole μ₁ on atom 1 decays with the inverse cube of distance: E₁(r) ∝ μ₁ / r³.
2. Second, this field induces a dipole on atom 2 proportional to the field strength (μ_2,ind = α₂ E₁ ∝ α₂ μ₁ / r³).
3. Third, the potential energy of an induced dipole μ_2,ind sitting in the electric field E₁ is V = −½ μ_2,ind E₁ = −½ α₂ (E₁)². Squaring the 1/r³ electric field dependence yields (1/r³)² = 1/r⁶.

1. London, F. (1930). "Zur Theorie und Systematik der Molekularkräfte." Zeitschrift für Physik, 63(3–4), 245–279. https://doi.org/10.1007/BF01421741
2. Israelachvili, J. N. (2011). Intermolecular and Surface Forces (3rd ed.). Academic Press. ISBN 978-0123751829
3. Autumn, K., Sitti, M., Liang, Y. A., Peattie, A. M., Hansen, W. R., Sponberg, S., Kenny, T. W., Fearing, R., Israelachvili, J. N., & Full, R. J. (2002). "Evidence for van der Waals adhesion in gecko setae." Proceedings of the National Academy of Sciences, 99(19), 12252–12256. https://doi.org/10.1073/pnas.192252799
4. Wagner, J. P., & Schreiner, P. R. (2015). "London dispersion in molecular chemistry—Reconsidering steric effects." Angewandte Chemie International Edition, 54(42), 12274–12296. https://doi.org/10.1002/anie.201503476