A generalization of the virial theorem is presented for all components of the average stress tensor of arbitrary systems of interacting particles. Explicit expressions are given for local-density-functional calculations and the method is tested by ab initio pseudopotential calculations on silicon. Accurate determinations are made of lattice constant, bulk moduli, second-, third-, and fourth-order elastic constants, and the internal strain parameter $\ensuremath{\zeta}$. Agreement with experiment is very good, except for $\ensuremath{\zeta}$.