Lennard-Jones potential

This form is a simplified formulation that is used by some simulation software:

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

where, ${\displaystyle A=4\varepsilon \sigma ^{12}}$ and ${\displaystyle B=4\varepsilon \sigma ^{6}}$. Conversely, ${\displaystyle \sigma ={\sqrt[{6}]{\frac {A}{B}}}}$ and ${\displaystyle \varepsilon ={\frac {B^{2}}{4A}}}$. This is the form in which Lennard-Jones wrote the 12-6 potential.[16]

A more mathematically general form, which contains an extra variable n is

${\displaystyle V_{\text{LJ}}(r)=\varepsilon \left(\left({\frac {r_{0}}{r}}\right)^{2n}-2\left({\frac {r_{0}}{r}}\right)^{n}\right),}$

where ${\displaystyle \varepsilon }$ 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 ${\displaystyle r_{0}}$, where ${\displaystyle V=-\varepsilon }$) as

${\displaystyle k=2\varepsilon \left({\frac {n}{r_{0}}}\right)^{2},}$

from where n can be calculated if k is known. Normally the harmonic states are known, ${\displaystyle \Delta E=\hbar \omega }$, where ${\displaystyle \omega ={\sqrt {k/m}}}$. n can also be related to the group velocity in a crystal, ${\displaystyle v_{\text{g}}}$

${\displaystyle v_{\text{g}}={\frac {a\cdot n}{r_{0}}}{\sqrt {\frac {\varepsilon }{m}}},}$

where a is the lattice distance, and m is the mass of an atom.

To save computing time and satisfy the minimum image convention when using periodic boundary conditions, the Lennard-Jones potential is often truncated at a cut-off distance of r_c = 2.5σ, where

${\displaystyle \displaystyle V_{\text{LJ}}(r_{\text{c}})=V_{\text{LJ}}(2.5\sigma )=4\varepsilon \left[\left({\frac {\sigma }{2.5\sigma }}\right)^{12}-\left({\frac {\sigma }{2.5\sigma }}\right)^{6}\right]\approx -0.0163\varepsilon ,}$

(1)

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

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

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

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

(2)

Then the truncated Lennard-Jones potential ${\displaystyle V_{{\text{LJ}}_{\text{trunc}}}}$ is defined as follows[17]

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

(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.[18][19] The potential energy can often be corrected for this effect in a mean-field manner by adding so-called tail corrections.[20] The stability of crystal structure is extremely sensitive to the truncation: a wide variety of close packed stackings may have the lowest energy.[21]

Lennard-Jones potentials have been combined with bonded potentials to describe virtual electrons and image charges in metals.[22] The combined potential reproduces lattice parameters, surface energies, the complete image potentials, hydration energies, and mechanical properties without additional fit parameters in higher accuracy than density functional theory calculations.