\begin{equation}
E^{av} = \frac{1}{N}\sum_K^N \sum_I^N\sum_J^N \left<\Phi_I\left|H\right|\Phi_J\right>c^*_{IK}c_{JK} = \frac{1}{N}\sum_K^N \left<\Phi_K\left|H\right|\Phi_K\right>
\end{equation}