List of Figures

1. Example of the convergence of the SR method for the variational parameters as a function of the number of stochastic iterations. In the upper(lower) panel the Jastrow (geminal) parameters are shown. For each iteration, a variational Monte Carlo calculation is employed with a bin containing $15000$ samples of the energy, yielding at the equilibrium a standard deviation of $\simeq 0.0018 H$ . For the first 200 iteration $\Delta t= 0.00125 H^{-1}$ , for the further 200 iterations $\Delta t= 0.0025 H^{-1}$ , whereas for the remaining ones $\Delta t= 0.005 H^{-1}$ .
2. Calculation of the derivative of the energy with respect to the second $Z$ in the $2p$ orbital of the geminal function for the Be atom. The calculation of the force was obtained, at fixed variational parameters, by averaging over $10^7$ samples, allowing e.g. a statistical accuracy in the total energy of $0.07 mH$ . The variational parameters have been obtained by an SR minimization with fixed bin length shown in the x label. The parameter considered has the largest deviation from the Euler conditions.
3. Plot of the equilibrium distance of the $Li_2$ molecule as a function of the inverse bin length. The total energy and the binding energy are reported in Tables 3.3 and 3.2 respectively. For all simulations the initial wave-function is optimized at $Li-Li$ distance $6$ a.u.
4. Electron density (atomic units) projected on the plane of $C_6H_6$ . The surface plot shows the difference between the resonating valence bond wave function, with the correct $A1g$ symmetry of the molecule, and a non-resonating one, which has the symmetry of the Hartree Fock wave function.
5. Surface plot of the charge density projected onto the molecular plane. The difference between the non-resonating (indicated as HF) and resonating Kekulé 3-body Jastrow wave function densities is shown. Notice the corresponding change from a dimerized structure to a $C_6$ rotational invariant density profile.
6. Plot of the convergence toward the equilibrium geometry for the $^2 B_{2g}$ acute and the $^2 B_{3g}$ obtuse benzene cation. Notice that both the simulations start form the ground state neutral benzene geometry and relax with a change both in the $C-C$ bond lengths and in the angles. The symbols are the same of Tab. 3.5.
7. A simulation box with periodic boundary conditions.
8. Ionic dynamics of 54 hydrogen atoms using GLQ, with a time step $0.4 fs$ , starting from a BCC lattice. The trial wave-function contains 2920 variational parameters and we have optimized 300 of them at each step. In the inset the maximum deviation $F_i/\Delta F_i$ of the forces acting on the variational parameters is shown.
9. Energy per atom of 16 hydrogen atoms at Rs=1.31 calculated on configurations obtained by CEIMC with the method (8). The first 10 configurations are in the atomic liquid phase at 2000k while in the last ten the system is forming clusters at T=500.
10. Variance per atom of 16 hydrogen atoms at Rs=1.31 calculated on configuration obtained by CEIMC with the method (8). The first 10 configurations are in the atomic liquid phase at 2000k while in the last ten the system is forming clusters at T=500.
11. Proton-proton correlation function, g(r), at Rs=1.31. The GLQ and CEIMC have used a periodical simulation box with 32 atoms while Hohl et al. with 64 atoms. All the calculations were performed for a single $k$ point ($\Gamma$ ).
12. Comparison of the proton-proton correlation function, g(r), at Rs=2.1 and T=4350 obtained with different methods CEIMC (8) (7) and GLQ. All the simulations were performed with 32 atoms for a single $k$ point ($\Gamma$ ).
13. Variational and Diffusion Condensation Energy per atom
14. Eigenvalues of the $\lambda$ matrix for 16 hydrogen atoms at Rs=1.31 and 100K as function of the simulation time
15. Off-Diagonal Long Range Order for 16 hydrogen atoms at Rs=1.31 and 100K, in a box of size $L=5.3211$ , for $\theta =0$ as function of the distance $x$
16. Off-Diagonal Long Range Order for 54 hydrogen atoms at Rs=1.31 at 100K in a box of size $L=7.9817$ , for $\theta =0$ as function of the distance $x$