  We have used diffusion quantum Monte Carlo (DMC) calculations to study the
pressure-induced phase transition from the diamond to $\beta$-tin structure in
silicon. The calculations employ the pseudopotential technique and
systematically improvable B-spline basis sets. We show that in order to achieve
a precision of 1 GPa in the transition pressure the non-cancelling errors in
the energies of the two structures must be reduced to 30 meV/atom. Extensive
tests on system size errors, non-local pseudopotential errors, basis-set
incompleteness errors, and other sources of error, performed on periodically
repeated systems of up to 432 atoms, show that all these errors together can be
reduced to well below 30 meV/atom. The calculated DMC transition pressure is
about 3-4 GPa higher than the accepted experimental range of values, and we
argue that the discrepancy may be due to the fixed-node error inherent in DMC
techniques.
