Electronic Structure Calculations on a Real-Space Mesh with Multigrid Acceleration
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Mat. Res. Soc. Symp. Proc. Vol. 408 0 1996 Materials Research Society
number of iterations needed to converge the electronic wavefunctions. Furthermore, the real-space multigrid formulation does not involve long-range operations and is particularly suitable for parallelization and O(N) algorithms [14-17], because every operation can be partitioned into hierarchical real-space domains. There have been a number of previous real-space grid-based electronic-structure calculations. The finite-element method was applied by White et al. [18] to one-electron systems. They used both conjugate-gradient and multigrid acceleration to find the groundstate wavefunction. Two of the present authors [12] used nonuniform grids with locally enhanced regions in conjunction with multigrid acceleration to calculate the electronic properties of atomic and diatomic systems with nearly singular pseudopotentials. They verified that the preconditioning afforded by multigrid was effective in multi-length-scale systems. Chelikowsky et al. [19] have used high-order finite-difference methods and soft
non-local pseudopotentials on uniform grids to calculate the electronic structure, geometry, and short-time dynamics of small Si clusters. The present authors have developed a real-space grid method with multigrid methods and tested it extensively against planewave results [20]. Other recent real-space work includes finite-differencing on warped grids [21, 22], wavelet bases [23, 241, hierarchical non-linear grids applied to diatomics [25], and
multigrid calculations on first and second row atoms [26]. REAL-SPACE MESH DESIGN Several issues that are absent from plane-wave or orbital-based methods arise when using a real-space grid approach. In the former case the wavefunctions, potentials, and the electronic density are representable in explicit basis functions, and thus are known everywhere. In a real-space grid implementation, these quantities are known only at a discrete set of grid points, which can introduce a spurious dependence of the Kohn-Sham eigenvalues, the total energy, and the ionic forces on the positions of the ions with respect to the real-space grid. We have developed a set of techniques that can overcome these difficulties and can be used to compute accurate, static and dynamical properties of large physical systems. In our formalism the wavefunctions, density, and potentials are directly represented on a three-dimensional real-space grid with linear spacing hgrid and number of mesh points Ngrid. The ions are represented by soft-core norm-conserving pseudopotentials [27]. Exchange and correlation effects are treated using the local density approximation (LDA) of density functional theory. Discretization and Pseudopotentials For reasons of accuracy and computational efficiency, we discretize the Kohn-Sham equations in a generalized eigenvalue formulation: A[O.] + B[Vef/)f
]
cnB[•b].
(1)
A and B are the components of the Mehrstellen discretization [28], which is based on Hermite's generalization of Taylor's theorem. It uses a
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