Optimized correlations

The homogeneous system

As the single particle orbitals are known, we only have to minimize the energy with respect to \(u\), which can be done by \(\delta  E[u]=0\). The resulting Euler-Lagrange equation determines the optimal \(u\). The equations have the familiar form of a zero energy Schrödinger equation, if we replace \(u \) by \(\sqrt{g}\), the square root of the pair distribution function

\begin{equation}
\bigg[-\frac{\hbar^2}{m}\nabla^2 + v(r) + w_{\mathrm I}(r) +
\frac{\hbar^2\nabla^2\sqrt{g_{\mathrm F}(r)}}{m\sqrt{g_{\mathrm F}(r)}}
\bigg] \sqrt{g(r)} = 0 \;,
\label{eq:realspaceeqFHNC}
\end{equation}

where \(v \) is the bare interaction, \(w_{\mathrm I}\) is called induced interaction (if we make connection with parquet theory of perturbation theory we identify it with the ring diagram contribution to the  particle particle irreducible interaction ).

Excited states

As shown above, the correlated wave functions yields a good description of the ground state. Here we create excitations with the correlation function found in the ground state, this is called correlated basis functions in the literature