Chemical Sciences: A Manual for CSIR-UGC National Eligibility Test for Lectureship and JRF/Lennard-Jones potential

Lennard-Jones potential for argon dimer

The Lennard-Jones potential (also referred to as the L-J potential, 6-12 potential or, less commonly, 12-6 potential) is a mathematically simple model that describes the interaction between a pair of neutral atoms or molecules. A form of the potential was first proposed in 1924 by John Lennard-Jones.[1]

The most common expression of the L-J potential is

${\displaystyle V(r)=4\epsilon \left[\left({\frac {\sigma }{r}}\right)^{12}-\left({\frac {\sigma }{r}}\right)^{6}\right],}$

where ε is the depth of the potential well, σ is the (finite) distance at which the inter-particle potential is zero, and r is the distance between the particles. These parameters can be fitted to reproduce experimental data or accurate quantum chemistry calculations. The r−12 term describes Pauli repulsion at short ranges due to overlapping electron orbitals and the r−6 term describes attraction at long ranges (van der Waals force, or dispersion force).

The Lennard-Jones potential is an approximation. The form of the repulsion term has no theoretical justification; the repulsion force should depend exponentially on the distance, but the repulsion term of the L-J formula is more convenient due to the ease and efficiency of computing r12 as the square of r6. Its physical origin is related to the Pauli principle: when the electronic clouds surrounding the atoms start to overlap, the energy of the system increases abruptly. The exponent 12 was chosen exclusively because of ease of computation.

The attractive long-range potential, however, is derived from dispersion interactions. 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. On the graph, Lennard-Jones potential for argon dimer is shown. Small deviation from the accurate empirical potential due to incorrect short range part of the repulsion term can be seen.

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.[2]

Other more recent methods, such as the Stockmayer equation and the so-called multi equation, 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

The Lennard-Jones potential function is also often written as

${\displaystyle V(r)=\varepsilon \left[\left({\frac {r_{\min }}{r}}\right)^{12}-2\left({\frac {r_{\min }}{r}}\right)^{6}\right],}$

where rmin = 62σ is the distance at the minimum of the potential. At rmin, the potential function has the value −ε.

The simplest formulation, often used internally by simulation software, is

${\displaystyle V(r)={\frac {A}{r^{12}}}-{\frac {B}{r^{6}}},}$

where A = 4εσ12 and B = 4εσ6; conversely, σ = 6A/B and εB2/(4A).

Molecular dynamics simulation: Truncated potential

To save computational time, the Lennard-Jones (LJ) potential is often truncated at the cut-off distance of rc = 2.5σ, where

${\displaystyle \displaystyle V(r_{c})=V(2.5\sigma )=4\varepsilon \left[\left({\frac {\sigma }{2.5\sigma }}\right)^{12}-\left({\frac {\sigma }{2.5\sigma }}\right)^{6}\right]=-0.0163\varepsilon =-{\frac {1}{61.3}}\varepsilon }$

(1)

i.e., at rc = 2.5σ, the LJ potential V is about 1/60th of its minimum value ε (depth of potential well).

Beyond ${\displaystyle \displaystyle r_{c}}$, the computational potential is set to zero. On the other hand, to avoid a jump discontinuity at ${\displaystyle \displaystyle r_{c}}$, as shown in Eq.(1), the LJ potential is shifted upward a little so that the computational potential would be zero exactly at the cut-off distance ${\displaystyle \displaystyle r_{c}}$.

For clarity, let ${\displaystyle \displaystyle V_{LJ}}$ denote the LJ potential as defined above, i.e.,

 ${\displaystyle \displaystyle V_{LJ}(r)=4\varepsilon \left[\left({\frac {\sigma }{r}}\right)^{12}-\left({\frac {\sigma }{r}}\right)^{6}\right]}$ (2)

The computational potential ${\displaystyle \displaystyle V_{\text{comp}}}$ is defined as follows [3]

 ${\displaystyle \displaystyle V_{\text{comp}}(r):={\begin{cases}V_{LJ}(r)-V_{LJ}(r_{c})&{\text{for }}r\leq r_{c}\\0&{\text{for }}r>r_{c}\end{cases}}}$ (3)

It can be easily verified that Vcomp(rc) = 0, thus eliminating the jump discontinuity at r = rc. It should be noted that, 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.[4] Fortunately, the potential energy can be corrected for this effect in a mean field manner by adding so-called tail corrections.[5]

References

1. Lennard-Jones, J. E. (1924), "On the Determination of Molecular Fields", Proc. R. Soc. Lond. A 106 (738): 463–477, doi:10.1098/rspa.1924.0082 .
2. Barron, T. H. K.; Domb, C. (1955), "On the Cubic and Hexagonal Close-Packed Lattices", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 227 (1171): 447–465, doi:10.1098/rspa.1955.0023 .
3. Smit, B. (1992), "Phase diagrams of Lennard-Jones fluids", Journal of Chemical Physics 96 (11): 8639, doi:10.1063/1.462271 .
4. Frenkel, D. & Smit, B. (2002), Understanding Molecular Simulation (Second ed.), San Diego: Academic Press, ISBN 0122673514 .