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We present a method for the direct evaluation of the difference between the free energies of two crystalline structures, of different symmetry. The method rests on a Monte Carlo procedure which allows one to sample along a path, through atomic-displacement-space, leading from one structure to the other by way of an intervening transformation that switches one set of lattice vectors for another. The configurations of both structures can thus be sampled within a single Monte Carlo process, and the difference between their free energies evaluated directly from the ratio of the measured probabilities of each. The method is used to determine the difference between the free energies of the fcc and hcp crystalline phases of a system of hard spheres.

Three successive pressure-induced structural transformations of stishovite ͑rutile-structure SiO 2) to denser phases are predicted by the first-principles pseudopotential method within the local-density approximation. The transition from the rutile to the orthorhombic CaCl 2 phase occurs at 47 GPa, the transition from the CaCl 2 to the Pnc 2 structure at 98 GPa, and finally the Pnc 2 phase transforms to the pyrite phase at 226 GPa. It is clearly illustrated that the first transition is associated with an elastic instability which arises from the strong coupling between elastic constants and the softening rutile B 1g mode. The fully optimized structures of the four polymorphs of silica are obtained as a function of pressure. In addition, all zone-center transverse optic modes of the rutile and CaCl 2 phases are determined as a function of pressure. The results are in excellent agreement with available experimental data. ͓S0163-1829͑97͒02506-X͔