Long-range Correction (Energy and Virial)
To accelerate the simulation performance, the nonbonded potential is usually truncated at specific cut-off (Rcut) distance.
To compensate the missing potential energy and force, beyond the Rcut distance, the long-range correction (LRC) or tail correction to energy and virial must be
calculated and added to total energy and virial of the system, to account for infinite cutoff distance.
The VDW and EXP6 energy functions, evaluates the energy up to specified Rcut distance. In this section, the LRC equations for virial and energy term
for Van der Waals interaction are discussed in details.
VDW
This option calculates potential energy using standard Lennard Jones (12-6) or Mie (n-6) potentials, up to specific Rcut distance.
EnergyFor homogeneous system, the long-range correction energy can be analytically calculated:
\[E_{\texttt{LRC(VDW)}} = \frac{2\pi N^2}{V} \int_{r=r_{cut}}^{\infty} r^2 E_{\texttt{VDW}}(r) dr\]\[E_{\texttt{VDW}}(r) = C_{n} \epsilon \bigg[\bigg(\frac{\sigma}{r}\bigg)^{n} - \bigg(\frac{\sigma}{r}\bigg)^6\bigg]\]where \(N\), \(V\), \(r\), \(\epsilon\), and \(\sigma\) are the number of molecule, volume of the system, separation, minimum potential, and collision diameter, respectively. The constant \(C_n\) is a normalization factor such that the minimum of the potential remains at \(-\epsilon\) for all \(n\). In the 12-6 potential, \(C_n\) reduces to the familiar value of 4.
\[C_{n} = \bigg(\frac{n}{n - 6} \bigg)\bigg(\frac{n}{6} \bigg)^{6/(n - 6)}\]Substituting the general Lennard Jones energy equation into the integral, the long-range correction energy term is defined by:
\[E_{\texttt{LRC(VDW)}} = \frac{2\pi N^2}{V} C_{n} \epsilon {\sigma}^3 \bigg[\frac{1}{n-3}\bigg(\frac{\sigma}{r_{cut}}\bigg)^{(n-3)} - \frac{1}{3} \bigg(\frac{\sigma}{r_{cut}}\bigg)^3\bigg]\]VirialFor homogeneous system, the long-range correction virial can be analytically calculated:
\[W_{\texttt{LRC(VDW)}} = \frac{2\pi N^2}{V} \int_{r=r_{cut}}^{\infty} r^3 F_{\texttt{VDW}}(r) dr\]\[F_{\texttt{VDW}}(r) = \frac{6C_{n} \epsilon}{r} \bigg[\frac{n}{6} \times \bigg(\frac{\sigma}{r}\bigg)^{n} - \bigg(\frac{\sigma}{r}\bigg)^6\bigg]\]Substituting the general Lennard Jones force equation into the integral, the long-range correction virial term is defined by:
\[W_{\texttt{LRC(VDW)}} = \frac{2\pi N^2}{V} C_{n} \epsilon {\sigma}^3 \bigg[\frac{n}{n-3}\bigg(\frac{\sigma}{r_{cut}}\bigg)^{(n-3)} - 2 \bigg(\frac{\sigma}{r_{cut}}\bigg)^3\bigg]\]
EXP6
This option calculates potential energy using Buckingham potentials, up to specific Rcut distance.
EnergyFor homogeneous system, the long-range correction energy can be analytically calculated:
\[E_{\texttt{LRC(VDW)}} = \frac{2\pi N^2}{V} \int_{r=r_{cut}}^{\infty} r^2 E_{\texttt{VDW}}(r) dr\]\[\begin{split}E_{\texttt{VDW}}(r) = \begin{cases} \frac{\alpha\epsilon}{\alpha-6} \bigg[\frac{6}{\alpha} \exp\bigg(\alpha \bigg[1-\frac{r}{R_{min}} \bigg]\bigg) - {\bigg(\frac{R_{min}}{r}\bigg)}^6 \bigg] & r \geq R_{max} \\ \infty & r < R_{max} \end{cases}\end{split}\]where \(r\), \(\epsilon\), and \(R_{min}\) are, respectively, the separation, minimum potential, and minimum potential distance. The constant \(\alpha\) is an exponential-6 parameter. The cutoff distance \(R_{max}\) is the smallest positive value for which \(\frac{dE_{\texttt{VDW}}(r)}{dr}=0\).
Substituting the Buckingham potential into the integral, the long-range correction energy term is defined by:
\[E_{\texttt{LRC(VDW)}} = \frac{2\pi N^2}{V} \bigg[AB \exp\big(\frac{-r_{cut}}{B}\big) \bigg(2 B^2 + 2 B r_{cut} + {r_{cut}}^2 \bigg) - \frac{C}{3 {r_{cut}}^3} \bigg]\]\[A = \frac{6 \epsilon \exp(\alpha)}{\alpha - 6}\]\[B = \frac{R_{min}}{\alpha}\]\[C = \frac{\epsilon \alpha {R_{min}}^6}{\alpha - 6}\]VirialFor homogeneous system, the long-range correction virial can be analytically calculated:
\[W_{\texttt{LRC(VDW)}} = \frac{2\pi N^2}{V} \int_{r=r_{cut}}^{\infty} r^3 F_{\texttt{VDW}}(r) dr\]\[\begin{split}F_{\texttt{VDW}}(r) = \begin{cases} \frac{6 \alpha\epsilon}{r\big(\alpha-6\big)} \bigg[\frac{r}{R{min}} \exp\bigg(\alpha \bigg[1-\frac{r}{R_{min}} \bigg]\bigg) - {\bigg(\frac{R_{min}}{r}\bigg)}^6 \bigg] & r \geq R_{max} \\ \infty & r < R_{max} \end{cases}\end{split}\]Substituting the Buckingham potential into the integral, the long-range correction virial term is defined by:
\[W_{\texttt{LRC(VDW)}} = \frac{2\pi N^2}{V} \bigg[A \exp\big(\frac{-r_{cut}}{B}\big) \bigg(6 B^3 + 6 B^2 r_{cut} + 3 B {r_{cut}}^2 + {r_{cut}}^3 \bigg) - \frac{2C}{3 {r_{cut}}^3} \bigg]\]