The solid molecular hydrogen under hydrostatic pressure varying from 0 up to 2.2 GPa at low temperatures is studied by path-integral Monte Carlo simulations in the isothermal-isobaric ensemble. The quantum contribution to the vibrational energy lies between 40% and 90% of their total values. The zero-point energy increases with increasing pressure and the anharmonicity is more pronounced at low temperature and pressure. A satisfactory agreement between our simulation results and available experimental measurements has been achieved. At very low temperatures, the quantum effects are responsible for softening the molecular crystal in about 1 GPa in the studied range of pressure. (C) 2017 Elsevier B.V. All rights reserved.