Finite-element methods in electronic-structure theory Original Research Article

We discuss the application of the finite-element (FE) method to ab initio solid-state electronic-structure calculations. In this method, the basis functions are strictly local, piecewise polynomials. Because the basis is composed 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 space; produces sparse, structured matrices, enabling the effective use of iterative 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 for large, accurate ab initio calculations.We discuss the construction and properties of the required FE bases and develop in detail their use in the solution of the Schrödinger and Poisson equations subject to boundary conditions appropriate for a periodic solid. We present results for the Schrödinger equation illustrating the rapid, variational convergence of the method in electronic band-structure calculations. We present results for the Poisson equation illustrating the rapid convergence of the method, both pointwise and in the L2 norm, and its linear scaling 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 and 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 beyond 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.