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John Lennard-Jones used gas-transport measurements to constrain models of intermolecular forces. His 1924 work connects the temperature dependence of viscosity with an assumed force law. The familiar modern Lennard-Jones potential represents short-range repulsion and longer-range attraction as $V(r)=4ϵ[(\sigma /r{)}^{12}-(\sigma /r{)}^6]$. Its minimum at $r=2^{1/6}\sigma$ has depth $ϵ$, making the two parameters physically interpretable. This 12–6 form is a simplified interaction model, rather than an exact force law for every atom pair.^[ref]

A physicist associated with the Lennard-Jones potential.

A nearest-neighbour chain with the Lennard-Jones potential $V(x)=4ϵ[(\sigma /x{)}^{12}-(\sigma /x{)}^6]$ has equilibrium spacing $a=2^{1/6}\sigma$ and spring constant $K=V^{\prime \prime }(a)=72ϵ/a^2$. The monatomic-chain dispersion relation gives speed of sound $v=a\sqrt{K/m}=\sqrt{72ϵ/m}$ and maximum angular frequency $2v/a$. The cancellation of $\sigma$ from the sound speed holds in this chain model; it does not remove the spacing from the cutoff frequency.

A noncovalent interaction is an attraction or repulsion between chemical groups that does not require forming a new ordinary covalent bond between them. In biochemistry, examples include hydrogen bonds, electrostatic interactions and Van der Waals forces, acting either between molecules or within one molecule. These interactions help stabilize biomolecular structures and enable molecular recognition.^[ref]

The Lennard-Jones potential gives a simple distance-dependent example:The steep positive term models short-range repulsion; the negative term supplies attraction. Here $r$ is separation, $ϵ$ sets the energy scale and $\sigma$ sets a length scale.^[ref]

Substitution gives $U(\sigma )=0$. At $r=2^{1/6}\sigma$, the sixth-power term is $1/2$ and the twelfth-power term is $1/4$, so $U=-ϵ$. A pair can therefore have a preferred separation without acquiring a new covalent bond. Bringing the groups still closer eventually raises the energy sharply.

This is an illustrative pair model, not a complete description of every hydrogen bond or protein-binding event. Real molecular association also depends on geometry, other contacts and the surrounding solvent. Its central lesson is that noncovalent does not mean energetically negligible or independent of distance.

Differentiate the Lennard-Jones potential. Its minimum satisfiesFor dilithium, the two wells consequently haveUnder an additional same-orbital, Hund-type exchange comparison, assign $(3.4,0.06)$ to the triplet and $(4.9,0.05)$ to the singlet: the lower-energy minimum is associated with the exchange-stabilized triplet state, as in part (a).

This assignment needs that extra orbital-parentage assumption. A scalar Lennard-Jones potential does not itself encode spin, and a common atomic dissociation limit does not specify a unique molecular electronic configuration or spatial symmetry. The two curves even cross at about $5.006$ bohr: neither lies below the other at every separation. Thus the minima and well depths are determined unambiguously; the spin labels are a conditional qualitative inference, rather than a unique consequence of the four supplied parameters. Neither “a triplet must always have a longer bond” nor a universal ordering of unrelated excited states is justified.

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.

With an attractive Lennard-Jones potential and $k_{B}T\ll ϵ$, a short run can remain trapped in an initial conformation or a metastable compact state. A distance trace may show a long apparently stationary segment, an initial collapse, or occasional jumps between local structures; its histogram depends on initialization and is not necessarily an equilibrium distribution. An extended start need not collapse on the available run length, and a compact start need not already be the global minimum.

Accepting only downhill moves would make this trapping worse and would not sample the specified finite-temperature canonical ensemble. The sketch should therefore label this trace as unequilibrated and avoid assigning its mean a thermodynamic interpretation.

For the Lennard-Jones potential,Its minimum and harmonic spring constant areUsing the monatomic-chain dispersion relation,The maximum is at $|k|=\pi /a$. Taking the argon atomic mass as $m\simeq 39.95\mathrm{u}=6.63\times 10^{-26}\mathrm{kg}$ and $ϵ=1.602\times 10^{-21}\mathrm{J}$ givesThese are the stipulated one-dimensional nearest-neighbour model estimates; a three-dimensional argon crystal has additional neighbours and phonon polarizations.

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.

Use the number-density virial expansion, following the density versus molar second virial convention, with $n=N/V$:Thus $B_2$ is the coefficient of the first density correction and has units of volume. With molar density instead, the molar coefficient is $N_A{B}_2$. For a classical pair potential $u(r)$,Attractive separations contribute negatively and short-range repulsion contributes positively. For inert-gas interactions the attractive contribution dominates at sufficiently low temperature, $B_2$ crosses zero at the Boyle temperature, and repulsion dominates above it.

The soft-core high-temperature second virial coefficient illustrates why the far-high-temperature tail depends on the model. A hard-core estimate $B_2\simeq b-a/(k_{B}T)$ tends to positive excluded volume $b$. A soft repulsive potential, such as the Lennard-Jones potential in the original diagram, allows a positive maximum followed by decay toward zero. The sign change is the main attraction-versus-repulsion feature; neither sketch should be interpreted as measured data for a specified gas. This classical, dilute-gas sketch omits low-temperature quantum-exchange corrections.^[ref]

A Van der Waals potential describes the separation-dependent potential energy associated with Van der Waals forces. For two neutral, nonpolar particles at separations where the nonretarded dispersion description applies, a familiar attractive contribution is $U(r)=-C_6/r^6$ with $C_6>0$. Its radial force is $-dU/dr=-6C_6/r^7$, directed toward smaller separation.

At sufficiently short separation, overlapping electronic states produce strong exchange repulsion; continuing the attractive power law to zero separation is not a physical model. A Lennard-Jones potential combines an attractive inverse-sixth-power term with a simple repulsive term to produce a finite-separation minimum. Geometry also changes the effective power law: summing pair interactions over macroscopic bodies gives different distance dependences, as in Hamaker summation. The pair potential and a measured force between extended surfaces must therefore be distinguished.