• Quantum Monte Carlo
• electronic wave function
• strongly correlated systems

Our aim is to study the electronic wave function and the correlation energy of a low dimensional system. In particular we want to consider how the wave function changes as a function of the atomic positions and
the geometry configuration. We use Quantum Monte Carlo to describe accurately the correlation energy of the system by a variational approach based on an highly accurate many-body wave function, that can
be used also for very large systems with good computational scaling (at most the fourth power of the number of electrons). In our approach we have found useful to exploit the full variational freedom of two
ansatzes. The first one is obtained by applying a correlated Jastrow factor to the Antisymmetrized Geminal Power (JAGP).
The second one is a multi-reference Linear scaling Generalized Valence Bond with Jastrow factor (JLGVB). We want to compare the results of JAGP, that has a very good computational scaling, with JLGVB that has a
high flexibility because of its greater variational freedom. In this way we consider the systems with a very accurate description of their correlation energy, within the Born-Oppenheimer approximation.

In particular we study a molecule of four hydrogen atoms in a planar geometry and a linear chain containing an even number of hydrogen atoms.
In this thesis we want to study the ground state wave function through Quantum Monte Carlo methods and compare two different approaches: one that uses a single pairing function (JAGP) and the other that is based on a multi-determinantal approach (JLGVB).

We calculate the energy of our systems as a function of the increasing distance between the atoms and optimize our wave functions in order to obtain the optimal variational result.

Both the JAGP and JLGVB should play a crucial role in determining the energy and its relative error, and the comparison of the performances of these two wave functions for different physical systems is extremely important. Nowadays, this is possible using the most advanced Quantum Monte Carlo techniques and in particular through the Variational Quantum Monte Carlo (VMC) and the fixed node Diffusion Monte Carlo (DMC).

Our final goal is to apply the above methods in order to validate these particular JAGP and JLGVB ansatzes in these simple systems.
This is particularly important because these ansatzes can be extended to much larger number of atoms and can help us in determining the correlation energy of more realistic systems.

In the thesis, we compare explicitly the results obtained through our methods with the results that are present in literature. The agreement is good for both JAGP and JLGVB approaches.