Coulomb Sum with Distance Dependent Dielectric

This potential attempts to address the difficulties of applying the direct Coulomb sum, without the brutal truncation of the previous case. It hinges on the assumption that the electrostatic forces are effectively `screened' in real systems - an effect which is approximated by introducing a dielectic term that increases with distance.

The interatomic potential for two charged ions is

$$U(r_{ij})=\frac{1}{4\pi\epsilon_{0}\epsilon(r_{ij})}\frac{q_{i}q_{j}}{r_{ij}}$$ (2.126)

with $q_{\ell}$ the charge on an atom labelled $\ell$, and $r_{ij}$ the magnitude of the separation vector $\mbox{$\underline{r}$}_{ij}=\mbox{$\underline{r}$}_{j}-\mbox{$\underline{r}$}_{i}$. $\epsilon(r)$ is the distance dependent dielectric function. In DL_POLY_2 it is assumed that this function has the form

$$\epsilon(r)=\epsilon r$$ (2.127)

where $\epsilon$ is a constant. Inclusion of this term effectively accelerates the rate of convergence of the Coulomb sum.

The force on an atom $j$ derived from this potential is

$$\mbox{$\underline{f}$}_{j}=\frac{1}{2\pi\epsilon_{0}\epsilon}\frac{q_{i}q_{j}}{r_{ij}^{4}}\mbox{$\underline{r}$}_{ij}$$ (2.128)

with the force on atom $i$ the negative of this.

The contribution to the atomic virial is

$${\cal W}=-\mbox{$\underline{r}$}_{ij}\cdot\mbox{$\underline{f}$}_{j}$$ (2.129)

which is $-2$ times the potential term.

The contribution to be added to the atomic stress tensor is given by

$$\sigma^{\alpha \beta}=r_{ij}^{\alpha}f_{j}^{\beta},$$ (2.130)

where $\alpha,\beta$ are $x,y,z$ components. The atomic stress tensor is symmetric.

In DL_POLY_2 these forces are handled by the routines COUL2 and COUL2NEU.