  Silicon undergoes a phase transition from the semiconducting diamond phase to
the metallic beta-Sn phase under pressure. We use quantum Monte Carlo
calculations to predict the transformation pressure and compare the results to
density functional calculations employing the LDA, PBE, PW91, WC, AM05, PBEsol
and HSE06 exchange-correlation functionals. Diffusion Monte Carlo predicts a
transition pressure of 14.0 +- 1.0 GPa slightly above the experimentally
observed transition pressure range of 11.3 to 12.6 GPa. The HSE06 hybrid
functional predicts a transition pressure of 12.4 GPa in excellent agreement
with experiments. Exchange-correlation functionals using the local-density
approximation and generalized-gradient approximations result in transition
pressures ranging from 3.5 to 10.0 GPa, well below the experimental values. The
transition pressure is sensitive to stress anisotropy. Anisotropy in the stress
along any of the cubic axes of the diamond phase of silicon lowers the
equilibrium transition pressure and may explain the discrepancy between the
various experimental values as well as the small overestimate of the quantum
Monte Carlo transition pressure.
