The combination of these techniques with the density functional theory (DFT) has become a widely accepted and powerful

In this thesis, we present a new method that treats the electrons within the many-body QMC and perform Molecular Dynamic ''on the fly'' on the ions. This method provides improved dynamical trajectories and significantly more accurate total energies.

In the past two different approaches were proposed to couple Quantum Monte Carlo with ionic Molecular Dynamic. The first, called Coupled Electronic-Ionic Monte Carlo (CEIMC) (8), is based on a generalized Metropolis algorithm that takes into account the statistical noise present in the QMC evaluation of the Bohr-Oppenheimer surface energy. In the second approach, called Continuous Diffusion Monte Carlo (CDMC) (64), the Molecular Dynamics trajectories are generated with some empirical models or by CPMD-DFT, and then the CDMC technique is used to efficiently evaluate energy along the trajectories. Both methods present some drawbacks. In the second method even if all the properties are evaluated using the Diffusion Monte Carlo, the trajectories are generated using empirical models without the accuracy given by the QMC for the structural properties, as radial distribution, bonding lengths and so on. Instead, in the first one the QMC energies are used to perform the Monte Carlo sampling leading to accurate static properties. In order to have a reasonable acceptance rate within this scheme simulations have to be carryed out with a statistical error on the energy of the order of Furthermore, in order to have a fixed acceptance rate the amplitude of the ionic move has to be decreased with the size of the system.

The method we present here, allows to solve two major drawbacks of the previous two techniques. Following the idea of Car and Parrinello (54) we will show that it is possible to perform a feasible

- The Born-Oppenheimer approximation
- Dealing with Quantum Monte Carlo noise
- Canonical ensemble by Generalized Langevin Dynamics

- Practical implementation of the finite temperature dynamics