  We describe a set of techniques for performing large scale ab initio
calculations using multigrid accelerations and a real-space grid as a basis.
The multigrid methods provide effective convergence acceleration and
preconditioning on all length scales, thereby permitting efficient calculations
for ill-conditioned systems with long length scales or high energy cut-offs. We
discuss specific implementations of multigrid and real-space algorithms for
electronic structure calculations, including an efficient multigrid-accelerated
solver for Kohn-Sham equations, compact yet accurate discretization schemes for
the Kohn-Sham and Poisson equations, optimized pseudo\-potentials for
real-space calculations, efficacious computation of ionic forces, and a
complex-wavefunction implementation for arbitrary sampling of the Brillioun
zone. A particular strength of a real-space multigrid approach is its ready
adaptability to massively parallel computer architectures, and we present an
implementation for the Cray-T3D with essentially linear scaling of the
execution time with the number of processors. The method has been applied to a
variety of periodic and non-periodic systems, including disordered Si, a N
impurity in diamond, AlN in the wurtzite structure, and bulk Al. The high
accuracy of the atomic forces allows for large step molecular dynamics; e.g.,
in a 1 ps simulation of Si at 1100 K with an ionic step of 80 a.u., the total
energy was conserved within 27 microeV per atom.
