The high-pressure properties of carbon in eight different structures are calculated using an ab initio pseudopotential local-orbital method. In particular, the structural properties of hexagonal diamond and variation of its fundamental band gap with pressure are calculated for the first time. The variation of the fundamental gap in hexagonal diamond is found to have the opposite sign to that in cubic diamond, although the cubic and hexagonal forms have almost identical structural properties. Among the structures examined, diamond is found to transform under hydrostatic pressure first to the fourfold coordinated bc-8 (or Si-III) structure. The bc-8 form is favored at pressures greater than 11.1 Mbar. This is a slightly lower transformation pressure than that recently calculated using the pseudopotential plane-wave method. The sixfold coordinated structures, simple cubic and β-tin, are found at low pressure to be kinetically unstable, transforming spontaneously, without an energy barrier, into the cubic diamond structure. This result suggests that sixfold coordination of liquid carbon will be unlikely to occur at moderate pressure and temperature. Similarly, sixfold coordination is not expected in carbon clusters. A method of fully utilizing crystal symmetry to reduce the amount of computation in evaluating two- and three-center integrals needed in the local-orbital method is developed for the present calculations.