Elastic constants of beryllium: a first-principles investigation

We apply several recently introduced projector-augmented wave, ultrasoft, and norm-conserving pseudopotentials (PPs) to the calculation of the elastic constants of beryllium and compare the results with previous theory and experiments. We discuss how the elastic constants depend on the Brillouin zone integration, the PP type, and the exchange and correlation functional. We find that although in percentage terms the elastic constants of beryllium depend on the PPs more than the crystal parameters or the bulk moduli, the differences between the local density approximation (LDA) and the Perdew, Burke, and Ernzerhof (PBE) generalized-gradient approximation are larger than the PP differences. The LDA overestimates compared to experiments, while the PBE values are higher than those of experiments but show a much better agreement. The PBEsol functional gives values that are slightly higher than those from PBE, with differences comparable to the PP uncertainty. We propose a simple formula to rationalize the internal relaxations in hexagonal close-packed crystals and show that Be relaxations are in reasonable agreement with this formula. The effects of internal relaxations on the values of C11 and C12 amount to a few per cent of C11, but up to 50% of C12.