The basic idea of my research is quite simple, use a correlated wave function, \(|\psi>= e^{\frac{1}{2}\sum_{i,j} u(r_i,r_j)} |\phi_0>\), and optimize it. Here \(|\phi_0>\) is a Slater determinant, which means it takes care of Fermi statistics, but has no correlation built in, this means e.g. that two electrons with opposite spin can occupy the same state without any penalty. The correlation function \(u\) prevents electrons to come close together, thus brings in correlation to the wave function. This wave function is an excellent approximation for several classes of systems and is the starting point of my research.

Ground state

So far we haven’t specified the explicit form of \(u\) and \(|\phi_0>\). We determine this by minimizing the expectation value of the energy \(E= <\psi |H|\psi>\) with respect to \(u\) and \(|\phi_0>\). For the homogeneous system \(|\phi_0>\) is a Slater determinate of plane waves and we can concentrate on the correlation function \(u\). More details on the groundstate theory of the homogeneous system can be found in the Introduction.

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^1,2. With this basis set we can either improve the description of the ground state or do excitations (similar to time dependent perturbation theory). Results are quite good for the HEG³ and other quantum fluids⁴.