| Hydrogen is the first element in the periodic table of the elements and one of the most abundant in the universe. The properties of solid hydrogen under pressure have been the focus of a great deal of experimental and theoretical researches dating back at least to the pioneering work of Wigner and Huntington in 1935. They predicted that hydrogen will undergo a molecular to atomic transition as the density is increased. In this paper, pressure effect on the ground state molecular solid has been simulated through a molecular confinement model with various volumes, both VMC and DMC methods are employed to calculate the properties of a hydrogen molecule confined in a spheroidal box. Different from the previous clamped nuclei model, we treat with the degrees of freedom both electrons and protons simultaneously in the quantum frame work, the Born- Oppenheimer approximation is also avoided and the zero point motion (ZPM) of the protons is considered fully. The ground state properties of the solid molecular hydrogen have been studied under high pressure by quantum Monte Carlo method through this model.We have calculated the equation of state of solid hydrogen using two methods and obtained the pressure that is approaching to the result of LeSar et al. Because the hard-box model exaggerates the compression effect, it brings higher pressure than the result at the same volume in the experiment and other theoretical work. The ground state energy increases and the bond length decreases as the confinement becoming stronger. The ground state energy obtained is higher and the equilibrium bond length becomes longer than the results of clamped nuclei approximation due to the quantum zero point motion of the protons.We also studied the variety of spheroidal shape with the increase of density. It is found that the ratio of a/b has increscent trend with the increase of pressure. The two abrupt changes of the ratio of a/b appear at the density of 0.27～0.29 mol/cm3 and 0.35～0.37 mol/cm3 with the increase of pressure, which are likely to be related to the two phase transitions of solid hydrogen induced by pressure at low temperature, have been found.A degree of anisotropyδhas been defined for understanding the change of the confined shape under different pressure. Our results show thatδdecrease with the increase of pressure and two obvious changes ofδare present where a/b changes abruptly. Comparingδof the confined ellipse with that of the W-S cell of the actual crystal, we find that its trend of change for phaseⅡis similar to that of like-Pa3 structure. This result is very important to confirm the structure of phaseⅡin much debate previously. <θ> and <θ2>1/2 (θis the angle between the molecular axis and the major axis of the spheroid) are investigated in order to further understand the nature of the abrupt changes of a/b. The H2 rotate freely at low densities and <θ>1/2 approximates to 56°which is a little bigger than the actual value (52°) of <θ2 >1/2 for free rotor. With the increase of pressure, <θ2>1/2 becomes smaller and <θ> approximates to zero. Two changes of <θ2>1/2 occur, indicating that the two abrupt changes of a/b correspond to two orientational phase transitions induced by pressure in solid hydrogen.The quantum effect of protonsΔRp/Rp is also investigated. As the density increases,ΔRp/Rp decreases. The fluctuations of the quantum effect of protonΔRp/Rp near the two densities are strong. This testifies that the ZPM of protons causes the two abrupt changes of a/b. The densities increase, which shortens the distance of protons, so potential energy increases. Two potential gains are found which ensure that energy is minimum at the two densities where the abrupt changes of a/b happen. So the ZPM of protons is important and the quantum effect of protons cannot be ignored in the study of hydrogen system.According to the above discussion, we conclude that the results we have obtained are significant in studying the properties of the H2 molecular solid. Additionally, the molecular confinement model of spheroid provides an effective way to investigate the dense H2 system.
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