\section{Basic Equations}

The purpose of this formalism is to provide a simple and
reasonable interpolation scheme to model ionic materials.

The model has the following components:
\begin{enumerate}
\item Each ion has a position index $\bf{R}$ and a positive charge
$Q_{\bf{R}}$.  The charge $Q_{\bf{R}} \equiv Z_{\bf{R}} -
z^{core}_{\bf{R}}$ includes the nuclear charge $Z_{\bf{R}}$ and
the core electron $z^{core}_{\bf{R}}$ contributions. In the model,
it is represented by a spherical charge density shape function
$S_{\bf{R}}(r)$ (assumed to be normalized to unity such as a
Gaussian form $S_{\bf{R}}(r) =
e^{-(r/\sigma_{\bf{R}})^2}/(\pi^{3/2}\sigma^3)$) so the total
density associated with the ions of the material is represented by
\begin{equation}\label{ionchrg}
\rho^{ion}({\bf{r}})=\sum_{\bf{R}} Q_{\bf{R}}
S_{\bf{R}}(|{\bf{r-R}}|).
\end{equation}
\item In addition to its charge distribution, each ion has an associated spherically symmetric local
potential which is used to adjust the model electron affinities
and negativities described by an amplitude $\mathcal{V}_{\bf{R}}$
and shape function $\Phi_{\bf{R}}(r)$ so that the total local
potential of the material is given by
\begin{equation}\label{locpot}
  V^{loc}({\bf{r}})=\sum_{\bf{R}} \mathcal{V}_{\bf{R}}\Phi_{\bf{R}}(|{\bf{r-R}}|).
\end{equation}

\end{enumerate
