Hydrogen, being the first element in the periodic table, has the simplest electronic structure of any atom, and the hydrogen molecule contains the simplest covalent chemical bond. Nevertheless, the phase diagram of hydrogen is poorly understood. Determining the stable structures of solid hydrogen is a tremendous experimental challenge 1-3 , because hydrogen atoms scatter X-rays only weakly, leading to low-resolution diffraction patterns. Theoretical studies encounter major difficulties owing to the small energy differences between structures and the importance of the zero-point motion of the protons. We have systematically investigated the zerotemperature phase diagram of solid hydrogen using firstprinciples density functional theory (DFT) electronic-structure methods 4 , including the proton zero-point motion at the harmonic level. Our study leads to a radical revision of the DFT phase diagram of hydrogen up to nearly 400 GPa. That the most stable phases remain insulating to very high pressures eliminates a major discrepancy between theory 5 and experiment 6 . One of our new phases is calculated to be stable over a wide range of pressures, and its vibrational properties agree with the available experimental data for phase III.The low-pressure phase I of solid hydrogen, which consists of freely rotating molecules on a hexagonal close-packed lattice 2 , transforms at pressures of about 110 GPa to the broken-symmetry phase II, in which the mean molecular orientations are ordered, and then to phase III at about 150 GPa (ref. 1). However, even the combination of X-ray and neutron scattering data and Raman and infrared vibrational data has not so far yielded the structures of phases II and III of hydrogen.The theoretical prediction of stable crystal structures is very difficult because of the need to search the very large space of possible structures, and the necessity of obtaining accurate energies for each of these structures. First-principles DFT methods have proved an efficient means of calculating quite accurate energies, and they have provided many insights into the properties of materials, including solid hydrogen 5,7 . At present, DFT offers the highest level of theoretical description at which we can carry out searches over many possible candidate structures.Our approach is to relax many random structures to minima in the enthalpy at fixed pressure 8 . This method does not rely on previous theoretical or experimental results, and it allows for the possibility of finding radically new structures. In some cases we used the intuition gained from the random searches to build other candidate structures. We then calculated the enthalpies of the most stable phases at a larger number of pressures, generating the data shown in Fig. 1. We refer to each structure by its short HermannMauguin space-group symbol, giving additional information where an ambiguity might occur.The lowest-enthalpy structures found around 100 GPa were those of space groups Pca2 1 and P2 1 /c, which were considered in previous studies 5,7 , and ...