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Short-range repulsion associated with the antisymmetry requirement for overlapping electronic states, related to the Pauli exclusion principle. It competes with attractive electrostatic and London dispersion forces at small intermolecular separations.

A physicist associated with the London dispersion force and London equations.

A force between molecules, including effects of electrostatics, polarizability and London dispersion forces. Hydrogen bonds describe a directional interaction pattern rather than an additional fundamental force to add independently to all those components.

Named after John Lennard-Jones, this is a pair potential energy combining steep short-range repulsion with nonretarded London dispersion force attraction. Its minimum is $U=-ϵ$ at $r=2^{1/6}\sigma$, found by differentiation. The power twelve is a convenient model for repulsion, rather than an exact consequence of the Pauli exclusion principle. The power six in the attractive energy follows from second-order coupling of atomic electric dipoles, before retardation matters.

CO and CO${}_2$ are discrete molecules, whereas the other oxides gain stability from extended bonding or ionic lattices. carbon's compact 2p orbitals overlap effectively with oxygen 2p orbitals, allowing strong multiple bonds that satisfy the valence requirements within small molecules. Intermolecular forces between CO or CO${}_2$ molecules are much weaker than those internal bonds, so low condensation temperatures and gaseous states at room conditions are unsurprising. CO${}_2$ is nonpolar overall but still has London dispersion forces and interactions between electric quadrupoles; “nonpolar” does not mean no attraction.

For larger Group 14 atoms, overlap with oxygen 2p orbitals is less effective for strong localized multiple bonding. SiO${}_2$ instead forms an extended network of SiO${}_4$ tetrahedra with each oxygen bridging two silicon atoms. Breaking or melting the network involves much more than separating intact SiO${}_2$ molecules. The Si–O electronegativity difference is $3.61-1.92=1.69$, so these are strongly polar covalent bonds rather than evidence for a molecular solid.

For Pb–O the difference is $3.61-1.85=1.76$, and ionic Pb${}^{2+}$/O${}^{2-}$ or Pb${}^{4+}$/O${}^{2-}$ descriptions provide useful starting points, with appreciable covalent contributions. PbO has layered structures with asymmetric oxygen coordination consistent with a stereochemically active 6s${}^2$ pair. A common PbO${}_2$ structure is rutile-like, with PbO${}_6$ octahedra linked through oxygen. Extended lattice interactions account for their solid states.^[ref] The similar Si–O and Pb–O electronegativity differences show why a single difference threshold cannot determine the entire bonding type, coordination number or structure.

The minimum is at $r=1.878\mathring{\mathrm{A}}$ with depth $-0.250E_h$; the model is not a smooth physical potential at the cage centre. Put $s=R-r$ and $x=(B/s{)}^6$. Thenso $x=1/2$ is the minimum. ThereforewhileThe positive inverse-twelfth-power term mimics short-range overlap repulsion, while the negative inverse-sixth-power term mimics London dispersion force. These powers are used in the Lennard-Jones potential; they are not a microscopic derivation for an atom inside a carbon cage.

The relevant domain is $0\le r<R$. As the atom approaches the wall the model diverges positively, which is qualitatively sensible. However, the energy depends only on the nearest radial wall distance and omits all the other cage atoms. More decisively, $V^{\prime }(0)\ne 0$: a radial function of the Cartesian position must have zero first radial derivative at the centre to be differentiable there. The model produces a cusp and a nonvanishing directional force as the centre is approached. An off-centre minimum is not by itself forbidden, but this centre behaviour is inconsistent with a smooth symmetric cage potential.

There is no universal ranking from the information supplied. The depletion force scales with polymer osmotic pressure, hence with $\rho$ in the ideal model. London dispersion forces instead depend on the material-dependent Hamaker constant, the intervening solvent, the closest approach and the contact geometry.

Close face contact can give a strong dispersion contribution; the ideal continuum expression even becomes singular without a molecular-distance cutoff. Nevertheless, refractive-index matching can greatly reduce that contribution, and increasing polymer concentration can strengthen depletion force. A quantitative comparison requires those missing material and concentration scales. The two attractive contributions may act together.

Write $C_4=\alpha Q^2/[2(4\pi {ϵ}_0{ϵ}_r{)}^2]$. Differentiating the stipulated potential givesFor positive $\alpha$, the force points toward smaller separation for either sign of $Q$.

Its physical origin is charge-induced dipole attraction, rather than London dispersion forces. A point-charge field induces $\mathbf{p}=\alpha \mathbf{E}$ and the energy cost of inducing that moment gives $u=-\alpha E^2/2$. The charge changes the electric dipole moment; it does not, within this linear model, change the intrinsic coefficient $\alpha$. For a conducting sphere in a uniform medium, $\alpha =4\pi {ϵ}_0{ϵ}_r{a}^3$; other dielectric contrasts give another electric polarizability.

There is a distance inconsistency in the stated model: the point-charge field uses center-to-center separation $R$, whereas the question calls $r$ a surface gap. For a point ion and sphere, $R=a+r$. A controlled far-field dipole model therefore uses $u\simeq -C_4/(a+r{)}^4$, with $R\gg a$. The boxed derivative is the literal result for the given mathematical $u(r)$; it is not a justification for extending that dipole formula to contact.

A useful five-part classification is electrostatic double-layer, dispersion, steric, depletion force and solvent-mediated interactions. It is a practical classification rather than a unique exhaustive list.

Electrostatic double-layer forces arise from surface charge and mobile-ion redistribution. Like-charged surfaces often repel; opposite charges can attract. Their range changes with salt screening. London dispersion forces comes from correlated electronic fluctuations and is one contribution to Van der Waals force; its sign and strength across a medium depend on dielectric response.

steric stabilization occurs when adsorbed or grafted layers overlap, increasing chain confinement and/or local osmotic free energy. depletion force arises when smaller nonadsorbing solutes are excluded from a narrow gap: their greater accessible volume outside favors bringing the large particles together. Structured-solvent forces include short-range hydration force and hydrophobic attraction; their microscopic origin differs from the depletion force of an added solute.

The original sketches show double-layer overlap, a polymer-coated gap, depletion force exclusion and solvent structure, alongside a dispersion-potential curve. A realistic suspension can experience several simultaneously. Charge neutrality of a particle does not remove dispersion, steric or solvent-mediated interactions.

Cyclodextrins are truncated-cone-shaped hosts that form host–guest complexes. Their hydroxyl-rich exterior interacts favourably with water, while the internal cavity is relatively nonpolar and can accommodate a suitably sized organic guest. The six-, seven- and eight-unit rings have different cavity sizes, so shape and size selectivity matter. Binding can involve London dispersion forces, displacement of cavity water and changes in solvation; no covalent modification of the guest is necessary.

An odorous organic guest can be included in the cavity, decreasing its free concentration and hence its tendency to enter the vapour phase. The same inclusion chemistry can increase the apparent aqueous solubility of a poorly soluble compound, and reversible association allows release when conditions or concentrations change. In a separation, guests with different association constants partition differently between free and bound states; a suitably immobilized host can therefore act as a selective stationary phase.

The unifying mechanism is reversible molecular inclusion in a size-selective cavity. These are chemical explanations of binding and partitioning. Biological action, therapeutic efficacy and physiological delivery mechanisms are outside the present nonbiology scope.

The attractive part of the Lennard-Jones potential represents the London dispersion force between fluctuating atomic electric dipoles. The dipole-dipole interaction is proportional to $r^{-3}$; for nonpolar ground-state atoms its first-order expectation vanishes. In second-order perturbation theory, virtual excitations give a negative energy shift proportional to the square of that interaction, hence $-A/r^6$ with $A>0$. This applies at separations large compared with atomic size but small enough that electromagnetic retardation is negligible.

The repulsive term models the steep energy increase when closed electron shells overlap, involving the Pauli exclusion principle and increased electronic energy. The $r^{-12}$ power is an empirical convenient approximation, not a universal power derived from the exclusion principle; its being the square of $r^{-6}$ makes computation convenient. At still larger separations retardation changes the attractive asymptotics, outside this model.