Real-space implementation of nonlocal pseudopotentials for first-principles total-energy calculations

We present a real-space method for performing the operations that involve the nonlocal parts of the Kohn-Sham Hamiltonian in a first-principles plane-wave total-energy calculation. In contrast to the conventional reciprocal-space formulation, where the number of operations required to compute the nonlocal contributions to the energies, forces, and stresses scales as the cube of the system size, the numerical work to compute these quantities with our real-space algorithm scales as the square of the number of atoms in the unit cell. The scheme, which can be applied to any potential expressible as a sum of separable terms, uses an approximate method to project the nonlocal potential on the core region of each atom. Errors introduced in the projection step are extremely well controlled and will not be a cause of problems in practical calculations. We have implemented the method in a conjugate-gradient total-energy program and, for illustrative purposes, demonstrate that the method produces excellent results on a two-atom cell of silicon.