Electronic structure calculations at macroscopic scales

Electronic structure 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 small cell-sizes (computational domains) consisting of few atoms (about 200 atoms). But material properties are influenced by defects---vacancies, dopants, dislocations, cracks, free surfaces---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 thesis presents the development of a seamless multi-scale scheme, Quasi-Continuum Orbital-Free Density-Functional Theory (QC-OFDFT) to address this significant issue. 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. The real-space formulation and the finite-element discretization gives freedom from periodicity, which is important in the study of defects in materials. More importantly, the real-space formulation and its finite-element discretization support unstructured coarse-graining of the basis functions, which is exploited to advantage in developing the QC-OFDFT method. This method has enabled for the first time a calculation of the electronic structure of samples with millions of atoms subjected to arbitrary boundary conditions. Importantly, the method is completely seamless, does not require any ad hoc assumptions, uses orbital-free density-functional theory as its only input, and enables convergence studies of its accuracy. From the viewpoint of mathematical analysis, the convergence of the finite-element approximation is established rigorously using Gamma-convergence, thus adding strength and validity to the formulation. The accuracy of the proposed multi-scale method under modest computational cost, and the physical insights it offers into properties of materials with defects, have been demonstrated by the study of vacancies in aluminum. One of the important results of this study is the strong cell-size effect observed on the formation energies of vacancies, where cells as large as tens of thousands of atoms were required to obtain convergence. This indicates the prevalence of long-range physics in materials with defects, and the need to calculate the electronic structure of materials at macroscopic scales, thus underscoring the importance of QC-OFDFT. Finally, QC-OFDFT was used to study a problem of great practical importance: the embrittlement of metals subjected to radiation. The brittle nature of metals exposed to radiation is associated with the formation of prismatic dislocation loops---dislocation loops whose Burgers vector has a component normal to their plane. QC-OFDFT provides an insight into the mechanism of prismatic dislocation loop nucleation, which has remained unclear to date. This study, for the first time using electronic structure calculations, establishes vacancy clustering as an energetically favorable process. Also, from direct numerical simulations, it is demonstrated that vacancy clusters collapse to form stable prismatic dislocation loops. This establishes vacancy clustering and collapse of these clusters as a possible mechanism for prismatic dislocation loop nucleation. The study also suggests that prismatic loops as small as those formed from a 7-vacancy cluster are stable, thus shedding new light on the nucleation size of these defects which was hitherto unknown.