2 Methods

To search for the densest structures, evolutionary structure prediction was performed using the USPEX code4; 79 in conjunction with ab initio structure relaxations using DFT within the PBE-GGA functional, as implemented in the VASP code64. We used the all-electron PAW with [1s$^2$] core and plane wave basis set with the 600 eV cutoff. Such calculations provide an excellent description of the density of tetrahedral phases of carbon; e.g., the computed densities are 3.504 g/cm$^3$ for diamond (3.52 g/cm$^3$ from experiment) and 3.496 g/cm$^3$ for lonsdaleite (3.52 g/cm$^3$ from experiment). The most significant feature of USPEX is the capability of searching for the global minimum according to the fitness function, given only the chemical composition. Here we used the density as the fitness function, and all structures are fully relaxed at constant pressure. To ensure that the obtained superdense structures are dynamically stable, we calculated phonon frequencies across the Brillouin zone using the finite-displacement approach as implemented in the Phononpy code67.