Ricardo Ruíz-Coronado

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Intermolecular potentials

October 2, 2026

Intermolecular potentials are many-body interaction potentials, which are physical at best. As of now I devote this entry to analytical (theoretical?) pairwise major model name potentials only —yes, I know!—, whose modeling is described below. (These differ with interatomic potentials insofar as they relate various molecules and not just atoms. Examples of interatomic potentials are the Axilrod–Teller–Muto, the Hooke, the Morse, and the Stillinger–Weber potentials.) I will succintly refer to these as name potentials.

Among the analytical potentials there are some exponential (exp), some potential —inverse power potentials (em, en, em-en) abound—, and some other. These tend to be concave, for so they reproduce the behaviour of known interactions. We will overlook numerical potentials (Gō, Dymond–Alder) for now.

Name potentials

Below is a non-authorative, non-exhaustive list of name potentials.

Beware their functional form: It may depend on the system of units, whether international (SI) or atomic (au). (Recall that $\hbar = e = m_\textrm{e} = 4\pi \varepsilon_0 \overset{\textrm{au}}{=} 1$ . Read ‘atomic units’ (Hartree’s) for au.)

Let $A$ and $B$ be any two molecules, so that $r_{AB}=|\textbf{r}_{AB}|=|\textbf{r}_A-\textbf{r}_B|$ is their separation distance (from center of mass to center of mass). The name potentials follow.

Coulomb potential

This is an (unscreened, attractive-only) one potential,

$V_\textrm{Coulomb}(\textbf{r}_{AB})=-g^2r_{AB}^{-1}$

where

$g^2\overset{\textrm{SI}}{=}e(4\pi \varepsilon_0)^{-1}$

$g^2\overset{\textrm{au}}{=}1$

is a physical constant. (We assume the form in vacuo.)

Yukawa potential

This is an (screened, attractive-only) one potential,

$V_\textrm{Yukawa}(\textbf{r}_{AB})=-g^2 \exp{(-\alpha mr)}r_{AB}^{-1}$

where both $g$ and $\alpha$ are scaling constants. Do note that for $m=0$ a Yukawa potential reduces to a Coulomb potential.

Lennard-Jones potential

This is a (attractive-repulsive) twelve-six potential,

$V_\textrm{Lennard-Jones}(\textbf{r}_{AB})=4\epsilon [(\sigma_{AB}/r_{AB})^{12}-(\sigma_{AB}/r_{AB})^6]$

where $\epsilon = -V_\textrm{Lennard-Jones}(r_\textrm{AB,min})$ is the minimal potential energy and $\sigma = 2^{-1/6}r_\textrm{AB,min}$ is a scaled minimal distance.

Mie potential

This is a (attractive-repulsive) en-em potential,

$V_\textrm{Mie}(\textbf{r}_{AB})=\lambda\epsilon [(\sigma_{AB}/r_{AB})^n-(\sigma_{AB}/r_{AB})^m]$

where

$\lambda=n/(n-m)(n/m)^{m/(n-m)}$

with special care that $n$ be greater than $m$ (that is, $n > m$ ). Do note that for $n =12$ and $m=6$ a Mie potential reduces to a Lennard-Jones one.

Other, minor potentials are the Barker–Fisher–Watts, the Derjaguin–Landau–Verwey–Overbeek, the Girifalco, and the Kihara potentials.

Some notes

Note 1: There are model potentials (Coulomb, Lennard-Jones) which, in time, have been shown to reduce formally from much general others. So they might as well be called redux potentials. It is righteous from the mathematical point of view; from the historical point of view, however, it is dubious since the genre potentials would come out as generalised forms.

Note 2: Particles or substances that are fit to any one potential are named alike (e.g. a Mie particle for a Mie potential, and so on). The unique exception is the Leonard-Jonessium.

Note 3: It is ambiguous (to me) whether the listed name potentials are empirical or theoretical.

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Intermolecular potentials

Name potentials

Coulomb potential

Yukawa potential

Lennard-Jones potential

Mie potential

Some notes

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