A graph of strength versus distance for the 126 LennardJones potential.
The LennardJones potential (also referred to as the LJ potential, 612 potential, or 126 potential) is a mathematically simple model that approximates the interaction between a pair of neutral atoms or molecules. A form of this interatomic potential was first proposed in 1924 by John LennardJones.^{[1]} The most common expressions of the LJ potential are

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],
where ε is the depth of the potential well, σ is the finite distance at which the interparticle potential is zero, r is the distance between the particles, and r_{m} is the distance at which the potential reaches its minimum. At r_{m}, the potential function has the value −ε. The distances are related as r_{m} = 2^{1/6}σ ≈ 1.122σ. These parameters can be fitted to reproduce experimental data or accurate quantum chemistry calculations. Due to its computational simplicity, the LennardJones potential is used extensively in computer simulations even though more accurate potentials exist.
Contents

Explanation 1

Alternative expressions 2

AB form 2.1

Truncated and shifted LennardJones potential 2.2

Limitations 3

See also 4

References 5
Explanation
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 longrange term, describes attraction at long ranges (van der Waals force, or dispersion force).
Differentiating the LJ potential with respect to 'r' gives an expression for the net intermolecular force between 2 molecules. This intermolecular force may be attractive or repulsive, depending on the value of 'r'. When 'r' is very small, the 2 molecules repel each other.
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 r^{12} as the square of r^{6}.
The LennardJones (12,6) potential can be further approximated by the (exp6) potential later proposed by R. A. Buckingham, in which the repulsive part is exponential:^{[2]}

V_B = \gamma \left[ e^{r/r_0}  \left( \frac{r_0}{r} \right)^6 \right].
The LJ 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 LennardJones potential is a hexagonal closepacking. 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.^{[3]} Real materials include BCC structures as well.^{[4]}
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 LennardJones 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 = 4\varepsilon \sigma^{12} and B = 4\varepsilon \sigma^6. Conversely, \sigma = \sqrt[6]{\frac{A}{B}} and \varepsilon = \frac{B^2}{4A}. This is the form in which LennardJones wrote the 126 potential.^{[5]}
A more mathematical general form, which contains an extra variable, n, is:

V_{LJ}(r) = \varepsilon \left(\left(\frac{r_0}{r}\right)^{2n}  2\left(\frac{r_0}{r}\right)^{n}\right),
where \varepsilon is the bonding energy of the molecule (the energy required to separate the atoms). The exponent n could be related to the spring constant, k (at r_0, where V=\varepsilon) as:

k = \varepsilon \left(\frac{n}{r_0}\right)^2,
from where n can be calculated if k is known (normally the harmonic states are known \Delta E=\hbar\omega, where \omega=\sqrt{k/m}). Also to the group velocity in a crystal, v_g

v_g=\frac{a\cdot n}{r_0}\sqrt{\frac{\varepsilon}{m}},
where a is the lattice distance and m is the mass of an atom.
Truncated and shifted LennardJones potential
To save computational time and satisfy the minimum image convention, the LennardJones potential is often truncated at a cutoff distance of r_{c} = 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 r_{c} = 2.5σ, the LennardJones potential, V_{LJ}, 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 cutoff 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 LennardJones potential \displaystyle V_ is defined as follows^{[6]}

\displaystyle V_ (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 V_{LJtrunc}(r_{c}) = 0, thus eliminating the jump discontinuity at r = r_{c}. Although the value of the (unshifted) Lennard Jones potential at r = r_{c} = 2.5σ is rather small, the effect of the truncation can be significant, for instance on the gas–liquid critical point.^{[7]} Fortunately, the potential energy can be corrected for this effect in a mean field manner by adding socalled tail corrections.^{[8]}
Limitations

The LJ potential has only two parameters (A and B) which determine the length and energy scales and may, without loss of generality, be set to unity. The potential is therefore unique, and cannot be fitted to properties of any real material.

With the LJ potential, the number of atoms bonded to an atom does not affect the bond strength. The bond energy per atom therefore increases linearly with the number of bonds per atom. Experiments show instead that the bond energy per atom increases quadratically with the number of bonds ^{[9]}

Bonding has no directionality: the potential is spherically symmetric.

The sixthpower term models dipoledipole interactions due to electron dispersion in noble gases (London dispersion forces), but it does not represent other kinds of bonding well. The twelfthpower term appearing in the potential is chosen for its ease of calculation for simulations (by squaring the sixthpower term) and is not physically based.

The potential diverges when two atoms approach one another. This may create instabilities that require special treatment in molecular dynamics simulations.
See also
References

^ LennardJones, J. E. (1924), "On the Determination of Molecular Fields", Proc. R. Soc. Lond. A 106 (738): 463–477, .

^ Peter Atkins and Julio de Paula, "Atkins' Physical Chemistry" (8th edn, W. H. Freeman), p.637

^ Barron, T. H. K.; Domb, C. (1955), "On the Cubic and Hexagonal ClosePacked Lattices", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 227 (1171): 447–465, .

^ Calculation of the LennardJones n–m potential energy parameters for metals. Shu Zhen,G. J. Davies. physica status solidi (a)Volume 78, Issue 2, pages 595–605, 16 August 1983

^ LennardJones, J. E. (1931). "Cohesion". Proceedings of the Physical Society 43 (5): 461.

^ softmatter:LennardJones Potential, Soft matter, Materials Digital Library Pathway

^ Smit, B. (1992), "Phase diagrams of LennardJones fluids", .

^ Frenkel, D. & Smit, B. (2002), Understanding Molecular Simulation (Second ed.), San Diego: Academic Press, .

^ Stachurski, Zbigniew H., Fundamentals of Amorphous Solids: Structure and Properties, p. 15.
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