The Lennard-Jones potential is a simple model for the interaction between two neutral atoms or molecules.
It combines a short-range repulsive term with a longer-range attractive term.
1. Lennard-Jones \(6\text{-}12\) potential
The potential energy between two atoms separated by a distance \(r\) is
\[
U(r)=4\varepsilon\left[
\left(\frac{\sigma}{r}\right)^{12}
-
\left(\frac{\sigma}{r}\right)^6
\right].
\]
Here:
| Symbol |
Name |
Meaning |
| \(r\) |
Separation distance |
Distance between the two atomic centers. |
| \(\sigma\) |
Size parameter |
Distance where \(U(r)=0\). It is related to the effective atomic size. |
| \(\varepsilon\) |
Well depth |
Magnitude of the minimum potential energy. |
| \(U(r)\) |
Potential energy |
Interaction energy at separation \(r\). |
2. Repulsive and attractive terms
The repulsive term is
\[
4\varepsilon\left(\frac{\sigma}{r}\right)^{12}.
\]
It becomes very large when atoms are too close. This represents strong short-range repulsion due to overlapping
electron clouds and the Pauli exclusion principle.
The attractive term is
\[
-4\varepsilon\left(\frac{\sigma}{r}\right)^6.
\]
This term represents longer-range attraction, such as dispersion forces.
3. Equilibrium distance
The equilibrium separation occurs at the minimum of the potential energy curve. At a minimum,
\[
\frac{dU}{dr}=0.
\]
Starting with
\[
U(r)=4\varepsilon\left(\sigma^{12}r^{-12}-\sigma^6r^{-6}\right),
\]
differentiate:
\[
\frac{dU}{dr}
=
4\varepsilon
\left(
-12\sigma^{12}r^{-13}
+
6\sigma^6r^{-7}
\right).
\]
Set the derivative equal to zero:
\[
-12\sigma^{12}r^{-13}+6\sigma^6r^{-7}=0.
\]
Rearranging gives
\[
2\sigma^6=r^6.
\]
Therefore,
\[
r_{\mathrm{eq}}=2^{1/6}\sigma.
\]
Since \(2^{1/6}\approx1.122\), the equilibrium distance is slightly larger than \(\sigma\).
4. Potential energy at equilibrium
At the equilibrium distance,
\[
r_{\mathrm{eq}}=2^{1/6}\sigma.
\]
Substituting this into the Lennard-Jones potential gives
\[
U(r_{\mathrm{eq}})=-\varepsilon.
\]
This means \(\varepsilon\) is the depth of the potential well.
5. Force from the potential
The radial force is related to the potential by
\[
F(r)=-\frac{dU}{dr}.
\]
For the Lennard-Jones potential,
\[
F(r)
=
\frac{24\varepsilon}{r}
\left[
2\left(\frac{\sigma}{r}\right)^{12}
-
\left(\frac{\sigma}{r}\right)^6
\right].
\]
The force changes sign at equilibrium:
| Region |
Potential curve |
Force meaning |
| \(r<r_{\mathrm{eq}}\) |
Steep repulsive wall |
The atoms repel each other strongly. |
| \(r=r_{\mathrm{eq}}\) |
Minimum of \(U(r)\) |
The net force is zero. |
| \(r>r_{\mathrm{eq}}\) |
Attractive tail |
The atoms attract each other weakly. |
6. Dimensionless graph
The calculator plots
\[
\frac{U}{\varepsilon}
\quad\text{versus}\quad
\frac{r}{\sigma}.
\]
This is useful because the shape of the Lennard-Jones curve is the same for all atom pairs after scaling.
The equilibrium point is always at
\[
\frac{r_{\mathrm{eq}}}{\sigma}=2^{1/6}\approx1.122,
\]
and the minimum is always
\[
\frac{U_{\min}}{\varepsilon}=-1.
\]
7. Worked example
Suppose
\[
\sigma=0.263\ \mathrm{nm},
\qquad
\varepsilon=1.51\times10^{-22}\ \mathrm{J}.
\]
The equilibrium distance is
\[
r_{\mathrm{eq}}=2^{1/6}\sigma.
\]
Since
\[
2^{1/6}\approx1.122,
\]
we get
\[
r_{\mathrm{eq}}\approx1.122(0.263\ \mathrm{nm})
\approx0.295\ \mathrm{nm}.
\]
The minimum potential energy is
\[
U_{\min}=-\varepsilon=-1.51\times10^{-22}\ \mathrm{J}.
\]
Therefore,
\[
\boxed{r_{\mathrm{eq}}\approx0.295\ \mathrm{nm}},
\qquad
\boxed{U_{\min}=-1.51\times10^{-22}\ \mathrm{J}}.
\]
Formula summary
| Concept |
Formula |
Use |
| Lennard-Jones potential |
\(U(r)=4\varepsilon[(\sigma/r)^{12}-(\sigma/r)^6]\) |
Potential energy at separation \(r\). |
| Equilibrium distance |
\(r_{\mathrm{eq}}=2^{1/6}\sigma\) |
Most stable atom separation in this model. |
| Minimum energy |
\(U_{\min}=-\varepsilon\) |
Potential energy at the well bottom. |
| Zero-potential distance |
\(r=\sigma\) |
Distance where attractive and repulsive terms cancel. |
| Radial force |
\(F(r)=\frac{24\varepsilon}{r}[2(\sigma/r)^{12}-(\sigma/r)^6]\) |
Force direction and magnitude from the potential curve. |
Key idea: \(\sigma\) is not the equilibrium distance. The stable separation is
\(r_{\mathrm{eq}}=2^{1/6}\sigma\), which is about \(1.122\sigma\).
