# Lennard-Jones potential

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Lennard-Jones potential

The Lennard-Jones potential (also referred to as the L-J potential, 6-12 potential, or 12-6 potential) is a mathematically simple model that approximates the interaction between a pair of neutral atoms or molecules. A form of the potential was first proposed in 1924 by John Lennard-Jones. The most common expressions of the L-J potential are \begin{alignat}{3} V_{LJ}& = 4 \varepsilon &\left[ \left( \frac {\sigma} {r} \right)^{12} - \left( \frac {\sigma} {r} \right)^6 \right]\\ & = \varepsilon &\left[\left( \frac {r_m} {r} \right)^{12} - 2 \left( \frac {r_m} {r} \right)^6 \right] \end{alignat}

where ε is the depth of the potential well, σ is the finite distance at which the inter-particle potential is zero, r is the distance between the particles, and rm is the distance at which the potential reaches its minimum. At rm, the potential function has the value −ε. The distances are related as rm = 21/6σ. These parameters can be fitted to reproduce experimental data or accurate quantum chemistry calculations. Due to its computational simplicity, the Lennard-Jones potential is used extensively in computer simulations even though more accurate potentials exist.

## Explanation  A graph of strength versus distance for the 12-6 Lennard-Jones potential.

The r−12 term, which is the repulsive term, describes Pauli repulsion at short ranges due to overlapping electron orbitals and the r−6 term, which is the attractive long-range term, describes attraction at long ranges (van der Waals force, or dispersion force). Whereas the functional form of the attractive term has a clear physical justification, the repulsive term has no theoretical justification. It is used because it approximates the Pauli repulsion well, and is more convenient due to the relative computational efficiency of calculating r12 as the square of r6. The Lennard-Jones (12,6) potential is an approximation to the (exp-6) potential later proposed by R. A. Buckingham, in which the repulsive part is exponential: $V_B = \gamma \left[ e^{-r/r_0} - \left( \frac{r_0}{r} \right)^6 \right]$

The L-J potential is a relatively good approximation and due to its simplicity is often used to describe the properties of gases, and to model dispersion and overlap interactions in molecular models. It is particularly accurate for noble gas atoms and is a good approximation at long and short distances for neutral atoms and molecules.

The lowest energy arrangement of an infinite number of atoms described by a Lennard-Jones potential is a hexagonal close-packing. On raising temperature, the lowest free energy arrangement becomes cubic close packing and then liquid. Under pressure the lowest energy structure switches between cubic and hexagonal close packing. Real materials include BCC structures as well.

Other more recent methods, such as the Stockmayer potential, describe the interaction of molecules more accurately. Quantum chemistry methods, Møller–Plesset perturbation theory, coupled cluster method or full configuration interaction can give extremely accurate results, but require large computational cost.

## Alternative expressions

There are many different ways of formulating the Lennard-Jones potential The following are some common forms.

### AB form

This form is a simplified formulation that is used by some simulation software packages: $V_{LJ}(r) = \frac{A}{r^{12}} - \frac{B}{r^6},$

where A = 4εσ12 and B = 4εσ6. Conversely, $\sigma = \sqrt{\frac{A}{B}}$ and $\varepsilon = \frac{B^2}{4A}$. This is the form in which Lennard-Jones wrote the 12-6 potential.

### Truncated Lennard-Jones potential

To save computational time, the Lennard-Jones potential is often truncated at a cut-off distance of rc = 2.5σ, where $\displaystyle V_{LJ} ( r_c ) = V_{LJ} ( 2.5 \sigma ) = 4 \varepsilon \left[ \left( \frac {\sigma} {2.5 \sigma} \right)^{12} - \left( \frac {\sigma} {2.5 \sigma} \right)^6 \right] \approx -0.0163 \varepsilon$ (1)

i.e., at rc = 2.5σ, the Lennard-Jones potential, VLJ, is about 1/60th of its minimum value, ε (the depth of the potential well). Beyond $\displaystyle r_c$, the truncated potential is set to zero.

To avoid a jump discontinuity at $\displaystyle r_c$, the LJ potential must be shifted upward a little so that the truncated potential would be zero exactly at the cut-off distance, $\displaystyle r_c$.

For clarity, let $\displaystyle V_{LJ}$ denote the LJ potential as defined above, i.e., $\displaystyle V_{LJ} (r) = 4 \varepsilon \left[ \left( \frac {\sigma} {r} \right)^{12} - \left( \frac {\sigma} {r} \right)^6 \right]$ (2)

Then the truncated Lennard-Jones potential $\displaystyle V_{{LJ}_{trunc}}$ is defined as follows $\displaystyle V_{{LJ}_{trunc}} (r) := \begin{cases} V_{LJ} (r) - V_{LJ} (r_c) & \text{for } r \le r_c \\ 0 & \text{for } r > r_c \end{cases}$ (3)

It can be easily verified that VLJtrunc(rc) = 0, thus eliminating the jump discontinuity at r = rc. Although the value of the (unshifted) Lennard Jones potential at r = rc = 2.5σ is rather small, the effect of the truncation can be significant, for instance on the gas–liquid critical point. Fortunately, the potential energy can be corrected for this effect in a mean field manner by adding so-called tail corrections.

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