Chemical Sciences: A Manual for CSIRUGC National Eligibility Test for Lectureship and JRF/LennardJones potential
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The LennardJones potential (also referred to as the LJ potential, 612 potential or, less commonly, 126 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 LennardJones.^{[1]}
The most common expression of the LJ potential is
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\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 interparticle 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 LennardJones 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 LJ formula is more convenient due to the ease and efficiency of computing r^{12} as the square of r^{6}. 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 longrange potential, however, is derived from dispersion interactions. 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. On the graph, LennardJones 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 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.^{[2]}
Other more recent methods, such as the Stockmayer equation and the socalled multi equation, describe the interaction of molecules more accurately. Quantum chemistry methods, MøllerPlesset perturbation theory, coupled cluster method or full configuration interaction can give extremely accurate results, but require large computational cost.
Alternative expressions[edit]
The LennardJones potential function is also often written as
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle V(r) = \varepsilon \left[ \left(\frac{r_\min}{r}\right)^{12}  2\left(\frac{r_\min}{r}\right)^{6} \right],}
where r_{min} = ^{6}√2σ is the distance at the minimum of the potential. At r_{min}, the potential function has the value −ε.
The simplest formulation, often used internally by simulation software, is
 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V(r)={\frac {A}{r^{12}}}{\frac {B}{r^{6}}},}
where A = 4εσ^{12} and B = 4εσ^{6}; conversely, σ = ^{6}√A/B and ε = B^{2}/(4A).
Molecular dynamics simulation: Truncated potential[edit]
To save computational time, the LennardJones (LJ) potential is often truncated at the cutoff distance of r_{c} = 2.5σ, where

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 r_{c} = 2.5σ, the LJ potential V is about 1/60th of its minimum value ε (depth of potential well).
Beyond , the computational potential is set to zero. On the other hand, to avoid a jump discontinuity at , as shown in Eq.(1), the LJ potential is shifted upward a little so that the computational potential would be zero exactly at the cutoff distance .
For clarity, let denote the LJ potential as defined above, i.e.,

(2)
The computational potential is defined as follows ^{[3]}

(3)
It can be easily verified that V_{comp}(r_{c}) = 0, thus eliminating the jump discontinuity at r = r_{c}. It should be noted that, 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.^{[4]} Fortunately, the potential energy can be corrected for this effect in a mean field manner by adding socalled tail corrections.^{[5]}
References[edit]
 ↑ LennardJones, J. E. (1924), "On the Determination of Molecular Fields", Proc. R. Soc. Lond. A 106 (738): 463–477, doi:10.1098/rspa.1924.0082.
 ↑ 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, doi:10.1098/rspa.1955.0023.
 ↑ softmatter:LennardJones Potential, Soft matter, Materials Digital Library Pathway
 ↑ Smit, B. (1992), "Phase diagrams of LennardJones fluids", Journal of Chemical Physics 96 (11): 8639, doi:10.1063/1.462271.
 ↑ Frenkel, D. & Smit, B. (2002), Understanding Molecular Simulation (Second ed.), San Diego: Academic Press, ISBN 0122673514.