deletions | additions       diff --git a/background.tex b/background.tex  index a373b90..54d1ae7 100644  --- a/background.tex  +++ b/background.tex 
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\begin{equation} \label{eqn:coupling_exact}  J_{ij} = \int \int \mathrm{d}\mathbf{r} \mathrm{d}\mathbf{r}' \frac{\rho(\mathbf{r})_{A,i}^T \rho(\mathbf{r}')_{B,j}^T}{|\mathbf{r}-\mathbf{r}'|}.  \end{equation}  Rather than using the transition densities directly, it is possible to represent them using a set of atomic point charges, $\{q^T\}$, that have been fitted to reproduce the electrostatic potential of the transition densities, $\rho(\mathbf{r})^T$. $\rho(\mathbf{r})^T$, i.e. \textbf{TrESP}.  This is turns redefines the coupling (eq~\ref{eqn:coupling_exact}) \begin{equation}  J_{ij} \approx \sum_{a\in A} \sum_{b\in B} \frac{q_{a,i}^T q_{b,j}^T}{|\mathbf{R}_a - \mathbf{R}_b|}.  \end{equation}