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for Partition-of-unity finite elements for quantum-mechanical calculations in condensed matter: Have planewaves finally met their match?

===Natarajan Sukumar, Ph.D.===

==Partition-of-unity finite elements for quantum-mechanical calculations in condensed matter: Have planewaves finally met their match?==

Over the past few decades, the planewave pseudopotential (PW) method as implemented in codes such as ABINIT, VASP and QBOX, has established itself as the method of choice for large, accurate quantum-mechanical calculations in condensed matter. However, due to its global Fourier basis, the PW method suffers from substantial inefficiencies in parallelization and in problems involving localized states, e.g., first-row, transition metal, actinide, or
other atoms at extreme conditions. Modern real-space methods such as finite-differences, finite elements (FE), and wavelets, have addressed these problems but have until now required a much larger number of basis functions to attain the required accuracy.  In this talk, I will discuss the formulation and application of a new real-space finite element method for large-scale ab initio electronic-structure calculations—with the aim to push back the current limits on such calculations, while retaining both generality (applicable to metals and insulators, all atomic species and configurations) and systematic improvability. In a C0 finite element method, the quantum-mechanical problem consists of the solution of coupled three-dimensional Schr¨odinger and Poisson equations within a parallelepiped unit cell. We employ partition-of-unity enrichment techniques to build the known atomic physics into the FE basis, thereby substantially reducing the degrees of freedom required. Particular issues that arise in the formulation will be discussed—imposition of Bloch boundary conditions, treatment of non-local pseudopotential operators, and construction of enrichment functions. The
enrichment functions are pseudoatomic wavefunctions, the product of solutions of the radial Schr¨odinger equation and spherical harmonics. Local and nonlocal ionic pseudopotentials, with arbitrary k-point sampling, are considered. Initial results show order-of-magnitude reductions in degrees of freedom vis-`a-vis current state-of-the-art PW and adaptive-mesh FE methods for model and ab initio pseudopotential problems with localized states such as those involving first-row, and d- and f-electron elements.

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