Finite-element methods in electronic-structure theory

We discuss the application of the finite-element (FE) method to ab initio s olid-state: electronic-structure calculations. In this method, the basis fu nctions are strictly local, piecewise polynomials. Because the basis is com posed of polynomials, the method is completely general and its convergence can be controlled systematically. Because the basis functions are strictly local in real space, the method allows for variable resolution in real spac e; produces sparse, structured matrices, enabling the effective use of iter ative solution methods; and is well suited to parallel implementation. The method thus combines the significant advantages of both real-space-grid and basis-oriented approaches and so promises to be particularly well suited f or large, accurate ab initio calculations. We discuss the construction and properties of the required FE bases and dev elop in detail their use in the solution of the Schrodinger and Poisson equ ations subject to boundary conditions appropriate for a periodic solid. We present results for the Schrodinger equation illustrating the rapid, variat ional convergence of the method in electronic band-structure calculations. We present results for the Poisson equation illustrating the rapid converge nce of the method, both pointwise and in the L-2 norm, and its linear scali ng with system size in the context of a model charge-density and Si pseudo- charge-density. Finally, we discuss the application of the method to large- scale ab initio positron distribution and lifetime calculations in solids a nd present results for a host of systems within the range of a conventional LMTO based approach for comparison, as well as results for systems well be yond the range of the conventional approach. The largest such calculation, involving a unit cell of 4092 atoms, was shown to be well within the range of the FE approach on existing computational platforms. (C) 2001 Elsevier S cience B.V. All rights reserved.