LongRangeCorrections

class LongRangeCorrections : public feasst::VisitModel

These are the long range corrections assuming a 12-6 Lennard-Jones potential and that the radial distribution function is one beyond the cutoff.

See Allen and Tildesley or Frenkel and Smit.

\( U_{LRC} = \sum_i \sum_j U_{LRC}^{ij} = \sum_i \sum_j C_{ij} n_i n_j\)

where \(i\) and \(j\) are particle types i and j, \(n\) are the number of sites of the given type, and the constant, C_{ij} is given by

\(C_{ij} = \frac{8\epsilon_{ij}\pi\sigma_{ij}^3}{3 V}\left[\frac{1}{3}\left(\frac{\sigma_{ij}}{r^c_{ij}}\right)^9 - \left(\frac{\sigma_{ij}}{r^c_{ij}}\right)^3\right]\)

which depends only on the LennardJones parameters.

Only trials that change the number of sites of a type contribute to changes in energy.

If a selection of sites are to be deleted, the energetic contribution of the selection may be computed as:

\(U_{with} - U_{without} \propto n_i n_j - (n_i-n_i^s)(n_j-n_j^2)\)

\(U_{with} - U_{without} \propto n_i^s n_j + n_i n_j^s - n_i^s n_j^s\)

where the superscript s refers to the number of sites in the selection.

If a selection is to be added, the energetic contribution of the selection may be computed as:

\(U_{with} - U_{without} \propto (n_i+n_i^s)(n_j+n_j^s) - n_i n_j\)

\(U_{with} - U_{without} \propto n_i^s n_j + n_i n_j^s + n_i^s n_j^s\)

For the Mie potential,

\(C_{ij} = \left(\frac{n}{n-m}\right)\left(\frac{n}{m}\right)^{m/(n-m)} \frac{2\epsilon_{ij}\pi\sigma_{ij}^3}{(\lambda_a-3)V}\left[\frac{\lambda_a-3}{\lambda_r-3}\left(\frac{\sigma_{ij}}{r^c_{ij}}\right)^{\lambda_r-3} - \left(\frac{\sigma_{ij}}{r^c_{ij}}\right)^{\lambda_a-3}\right]\)