In the second part of the thesis we applied the JAGP wave-function to study high pressure hydrogen. The JAGP wave-function is a crucial ingredient to study correlation effects. In fact, as it is known from lattice models with electronic repulsion, it is not possible to obtain a superconducting ground state at the mean-field Hartree-Fock level. Instead as soon as a correlated Jastrow term is applied to the BCS wave function (equivalent to the AGP wave function in momentum space(33)), the stabilization of a d-wave superconducting order parameter is possible. Furthermore the presence of the Jastrow factor can qualitatively change the wave function especially at one electron per site filling, by converting a BCS superconductor to a Mott insulator with a finite charge gap(105). When the charge is locally conserved the phase of the BCS-AGP wave-function cannot have a definite value and phase coherence is correctly forbidden by the Jastrow factor.

In the second part of the thesis we studied the hydrogen close to the transition between the molecular solid to the atomic one, where it is expected a metal-insulator transition due to the closure of the band gap. We introduced a new technique to perform a Car-Parrinello like dynamics on ions by Quantum Monte Carlo noisy forces. This technique opens the possibility to use QMC to study finite temperature system with a reasonable computational effort. The combination of GLQ technique and JAGP wave-function has allowed us to study the electronic pairing structure during the nuclear motion. We have observed a non trivial behaviour on the eigenvalues of the matrix, see figure 6.6. This has led us to study the Off-Diagonal Long Range Order in this system. The study of the ODLRO evidences a non conventional superconductivity. Because of the classical nuclei, this superconductivity can be due only to correlation effects as in lattice models used to describe High Tc superconductor (see for instance (106)). Moreover our results showed that the dominant channel for superconductivity may be not be s-wave. Unfortunately the small size of the systems studied does not allow a conclusive answer. In fact as for lattice models, a finite size scaling is very difficult to perform (107). However motivated by recent results obtained on lattice models using renormalization group (102) we are planing to study larger systems to clarify our results. This implies the solution of some technical problems and the reduction of the size effects as discussed in the following.

The new advances in this thesis can be summarized in three points: a new highly correlated wave-function; a new technique to study finite temperature system with QMC; and the possibility, combining the two previous point,s to study exotic phases due to electronic correlation effects.