Extended Hubbard model with renormalized Wannier wave functions in the correlated state III: Statistically consistent Gutzwiller approximation and the metallization of atomic solid hydrogen

We extend our previous approach (Eur. Phys. J. B, $\mathbf{\text{74}}$, 63(2010)) to modeling correlated electronic states and the metal-insulator transition by applying the so-called \emph{statistically consistent Gutzwiller approximation} (SGA) to carry out self-consistent calculations of the renormalized single-particle Wannier functions in the correlated state. The transition to the Mott-Hubbard insulating state at temperature T=0 is of weak first order even if antiferromagnetism is disregarded. The magnitude of the introduced self-consistent magnetic correlation field is calculated and shown to lead to a small magnetic moment in the magnetically uniform state. Realistic value of the applied magnetic field has a minor influence on the metallic-state characteristics near the Mott-Hubbard lcalization threshold. The whole analysis has been carried out for an extended Hubbard model on a simple cubic (SC) lattice and the evolution of physical properties is analyzed as a function of the lattice parameter for the renormalized 1s-type Wannier functions. Quantum critical scaling of the selected physical properties is analyzed as a function of the lattice constant $R\to R_c=4.1a_0$, where $R_c$ is the critical value for metal-insulator transition and $a_0=0.53\mathring{A}$ is the Bohr radius. A critical pressure for metallization of solid atomic hydrogen is estimated and is $\sim {10}^{2}GPa$.