Lennard–Jones potential
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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. It was proposed in 1924 by John Lennard-Jones.[1]
The L-J potential is of the form
![V(r) = 4\epsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6} \right]](http://upload.wikimedia.org/math/e/a/c/eac9407d601624e0d5123ae78ccc1127.png)
,
where 
is the depth of the potential well and 
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 
term describes Pauli repulsion at short ranges due to overlapping electron orbitals and the 
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.
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[edit] Alternative expressions
The Lennard–Jones potential function is also often written as
![V(r) = \epsilon \left[ \left(\frac{r_{min}}{r}\right)^{12} - 2\left(\frac{r_{min}}{r}\right)^{6} \right]](http://upload.wikimedia.org/math/d/9/2/d9283be3ed70576130d56caa6710c081.png)
where
= 
is the distance at the minimum of the potential.
The simplest formulation, often used internally by simulation software, is:
where
and
.
[edit] Molecular dynamics simulation: Truncated potential
To save computational time, the Lennard-Jones (LJ) potential is often truncated at the cut-off distance of 
where
-
![\displaystyle
V ( r_c )
=
V ( 2.5 \sigma )
=
4 \epsilon
\left[
\left(
\frac
{\sigma}
{2.5 \sigma}
\right)^{12}
-
\left(
\frac
{\sigma}
{2.5 \sigma}
\right)^6
\right]
=
-0.0163 \epsilon
=
-
\frac
{1}
{61.3}
\epsilon](http://upload.wikimedia.org/math/b/e/f/bef09563e5795f7567db0ac31a3378f7.png)
(1)
i.e., at , the LJ potential 
is about 1/60th of its minimum value 
(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 cut-off distance .
For clarity, let 
denote the LJ potential as defined above, i.e.,
-
![\displaystyle
V_{LJ}
(r)
=
4 \epsilon
\left[
\left(
\frac
{\sigma}
{r}
\right)^{12}
-
\left(
\frac
{\sigma}
{r}
\right)^6
\right]](http://upload.wikimedia.org/math/7/7/7/777dac5a547800612f7ea372599eaff5.png)
(2)
The computational potential 
is defined as follows [3]
-
(3)
It can be easily verified that 
, thus eliminating the jump discontinuity at 
. 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].
[edit] See also
- Molecular mechanics
- Embedded atom model
- Morse potential
- Force field (chemistry)
- Lennard-Jones model on SklogWiki.
[edit] References
- ^ Lennard-Jones, J. E. — Proc. Roy. Soc., 1924, v. A 106, p. 463.
- ^ Barron, T. H. K., Domb, C. On the Cubic and Hexagonal Close-Packed Lattices. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 1955, 227, 447-465.
- ^ softmatter:Lennard-Jones Potential, Soft matter, Materials Digital Library Pathway
- ^ Smit, B., Phase diagrams of Lennard-Jones fluids. J. Chem. Phys., 1992 vol. 96, p. 8639
- ^ Frenkel, D. and Smit, B., Understanding Molecular Simulation, second edition, Academic Press , San Diago, California, USA, 2002.
