Multigrid Methods in Electronic Structure Calculations

摘要

We describe a set of techniques for performing large scale ab initiocalculations using multigrid accelerations and a real-space grid as a basis.The multigrid methods provide effective convergence acceleration andpreconditioning on all length scales, thereby permitting efficient calculationsfor ill-conditioned systems with long length scales or high energy cut-offs. Wediscuss specific implementations of multigrid and real-space algorithms forelectronic structure calculations, including an efficient multigrid-acceleratedsolver for Kohn-Sham equations, compact yet accurate discretization schemes forthe Kohn-Sham and Poisson equations, optimized pseudo\-potentials forreal-space calculations, efficacious computation of ionic forces, and acomplex-wavefunction implementation for arbitrary sampling of the Brilliounzone. A particular strength of a real-space multigrid approach is its readyadaptability to massively parallel computer architectures, and we present animplementation for the Cray-T3D with essentially linear scaling of theexecution time with the number of processors. The method has been applied to avariety of periodic and non-periodic systems, including disordered Si, a Nimpurity in diamond, AlN in the wurtzite structure, and bulk Al. The highaccuracy 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 totalenergy was conserved within 27 microeV per atom.