Quasi-continuum orbital-free density-functional theory : A route to multi-million atom electronic structure (DFT) calculation

I would like to share the research work I have been pursuing over the past four years. I believe, through this forum, I will be able to reach researchers with various backgrounds and expertise. Suggestions and comments from members will be very useful. I am also attaching links to preprints of manuscripts describing this work. Please follow these links:

http://www-personal.umich.edu/~vikramg/academic/Preprints/QC-OFDFT.pdf

http://www-personal.umich.edu/~vikramg/academic/Preprints/OFDFT-FE.pdf

Motivation: Most materials exhibit varying features and undergo various processes across different length and time scales. Moreover, these features change quantitatively as well as qualitatively across different materials. Therefore, understanding all aspects of materials behavior in a cohesive picture calls for the bridging of length and time scales, which is a key issue in computational materials science. Multi-scale modelling is a
paradigm to address this key issue. The success of a multi-scale model depends on the accuracy and transferability of the theory used to model the materials as well as the scheme through which information is transferred across scales.

Thesis work The central theme of my doctoral thesis, working with Prof. Michael Ortiz and Prof. Kaushik Bhattacharya, focuses on developing a seamless multi-scale scheme with density-functional theory as its sole input. Density-functional theory of Hohenberg, Kohn and Sham (KS-DFT), which is derived from quantum mechanics, is widely accepted as a reliable computational tool to compute a wide range of material properties. In metallic systems, it is common to use an approximate orbital-free density-functional theory (OFDFT) where the kinetic energy is modelled and fitted to finer calculations. Below, I describe, in brief, the various steps that constitute my thesis work.

Real-space formulation and analysis of OFDFT: Owing to the computational complexity of density-functional theory, traditional implementations of
KS-DFT/OFDFT, have for the most part, been based on the use of a plane-wave basis and periodic boundary conditions on samples consisting of few atoms (around 200 atoms). I developed a real-space formulation of OFDFT to overcome the serious limitation of periodicity which is not an appropriate assumption for various problems of interest in materials science, especially defects. An important step in developing this formulation was to reformulate the electrostatic interactions that are extended in real-space as a local variational principle. This results in a saddle-point variational problem (min-max problem) with a local functional in real space. Further, I established that this problem is mathematically well-posed by proving existence of solutions using the direct method in calculus of variations.

Finite-element discretization of OFDFT and Gamma-convergence: The local and variational structure of the real-space formulation of OFDFT motivated the use of a finite-element basis to discretize and compute the formulation. The use of finite-element basis enables consideration of complex geometries, general boundary conditions and locally-adapted grids. I proved the convergence of the finite-element approximation, including numerical quadratures, using the mathematical technique of Gamma-convergence. Gamma-convergence, which is a variational form of convergence, states in spirit that the solutions of the sequence of approximate functionals generated by finer and finer finite-element approximations converge to the solution of the exact functional. The key ideas used in these proofs include Sobolev embeddings, inverse inequalities and a novel approach to treat the Gamma-convergence of min-max problems.

Numerical implementation of OFDFT and examples: Numerical implementation of OFDFT using the finite-element method requires care since the
electron-density and the electrostatic potential are localized near the atomic cores and are convected as the atomic positions change. Consequently, a fixed spatial mesh would be extremely inefficient as we alternate between relaxing these electronic fields and atomic positions. I overcame this hurdle by designing an approach which convects the finite-element mesh with the atomic positions. This approach uses a nested-mesh scheme, where the finite-element mesh which describes the electronic fields is constructed as a sub-grid of the triangulation of atomic positions.

I demonstrated the approach on a host of examples, which included atoms, molecules and clusters of aluminum, and validated it by comparison with other numerical simulations and experiments. I performed simulations on varying sizes of aluminum clusters, including those as large as 3730 atoms, and these demonstrate the efficacy and advantages of the approach. Being clusters, they possess no natural periodicity and thus are not amenable to plane-wave basis. Second, since the boundaries of the clusters satisfy physically meaningful boundary conditions, it is possible to extract information regarding the scaling of the ground state energy density with size.

Countours of electron-density on the mid-plane of an aluminum cluster with 5x5x5 fcc unit cells (666 atoms).

Quasi-continuum orbital-free density-functional theory (QC-OFDFT): A route to multi-million atom non-periodic OFDFT calculation

Motivation : The real-space formulation of OFDFT and its computation using a finite-element basis, though very effective in addressing non-periodic systems, is restricted to samples consisting of a few thousands of atoms. However, many properties of materials 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-scale (sub-nanometer) as well as the elastic, electrostatic and other interactions at the coarse-scale (micrometer and beyond). This in turn requires calculations involving millions of atoms which are well beyond the current capability. This was the main motivation for the development of QC-OFDFT, which is a seamless scheme for systematic and adaptive coarse-graining of OFDFT in a manner that enables consideration of multi-million atom systems at no significant loss of accuracy and without the introduction of spurious physics or assumptions.

The key-idea: The method is developed in the spirit of the theory of quasi-continuum (QC), which is a computational technique for seamlessly bridging atomistic and continuum scales by the judicious introduction of kinematic constraints on the atomic degrees of freedom. However, QC-OFDFT differs from the earlier QC approaches in several notable respects. Apart from the displacement field, QC-OFDFT requires additional representation of the electron-density and the electrostatic potential which exhibit sub-lattice structure as well as lattice scale modulation. These electronic fields are decomposed into a local, oscillating solution and a non-local correction. The local, oscillating component of the electronic fields is represented by a sub-lattice finite-element interpolation (fine-mesh) in the entire domain, whereas the non-local correction is effectively represented by a finite-element interpolation (coarse-mesh) which is sub-lattice close to defects and coarse-grains away from defects . The local solution is computed by performing a periodic calculation and the non-local correction is determined from a variational principle.

Key-Idea: (a) atomistic-mesh (triangulation of the lattice sites) coarse-grains away from the defect (red dot); (b) coarse-mesh, which describes the non-local corrections to electronic fields; (c) fine-mesh, which describes the local, oscillating component of electronic fields; atomistic and coarse mesh coarse-grain away from vacancy, whereas fine-mesh is a uniform triangulation.

In order to avoid computational complexities of the order of the fine-mesh, I exploit the conceptual framework of the theory of homogenization of periodic media to define quadrature rules of a complexity commensurate with that of the coarse-mesh. Convergence of the OFDFT solution with increasing number of nodes in the coarse-mesh was investigated using numerical tests. The reduction in computational effort afforded by QC-OFDFT, at no significant loss of accuracy with respect to a full-atom calculation, is quite staggering. For instance, I have analyzed million-atom samples with modest computational resources, giving access to cell-sizes that have never been analyzed using OFDFT.

Some Results:

Contour of electron-density around a vacancy in an million atom aluminum cluster

Contour of electron-density around a di-vacancy complex in an million atom aluminum cluster