Electronic structure calculations (first-principle, quantum-mechanical calculations), especially those using density-functional theory have provided many insights into various materials properties in the recent decade. However, the computational complexity associated with electronic structure calculations has restricted these investigations to periodic geometries with cell-sizes consisting of few atoms (? 200 atoms). But, material properties are influenced by defects in small concentrations (parts per million). A complete description of such defects must include both the electronic structure of the core at the fine (sub-nanometer) scale and also elastic and electrostatic interactions at the coarse (micrometer and beyond) scale. This in turn requires electronic structure calculations at macroscopic scales, involving millions of atoms, well beyond the current capability.
This talk presents the development of a seamless multi-scale scheme, quasi-continuum orbital-free density-functional theory (QC-OFDFT) to perform electronic structure calculations at macroscopic scales. This multi-scale scheme has enabled for the first time a calculation of the electronic structure of multi-million atom systems using orbital-free density-functional theory, thus, paving the way to an accurate electronic structure study of defects in materials. The key ideas in the development of QC-OFDFT are (i) a real-space variational formulation of orbital-free density-functional-theory, (ii) a nested finite-element discretization of the formulation, and (iii) a systematic means of adaptive coarse-graining retaining full resolution where necessary, and coarsening elsewhere with no patches, assumptions or structure. Rigorous proofs of convergence of the finite-element approximation using the variational notion of Gamma-convergence will be presented. The accuracy of QC-OFDFT scheme and the physical insights it offers into the behavior of defects in materials are highlighted by the study of vacancies in aluminum.
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