Phanish Suryanarayana

Coarse-graining Kohn-Sham Density Functional Theory

We present a real-space formulation for coarse-graining Kohn–Sham Density Functional Theory that significantly speeds up the analysis of material defects without appreciable loss of accuracy. The approximation scheme consists of two steps. First, we develop a linear-scaling method that enables the direct evaluation of the electron density without the need to evaluate individual orbitals. We achieve this by performing Gauss quadrature over the spectrum of the linearized Hamiltonian operator appearing in each iteration of the self-consistent field method. Building on the linear-scaling method, we introduce a spatial approximation scheme resulting in a coarse-grained Density Functional Theory. The spatial approximation is adapted so as to furnish fine resolution where necessary and to coarsen elsewhere. This coarse-graining step enables the analysis of defects at a fraction of the original computational cost, without any significant loss of accuracy. Furthermore, we show that the coarse-grained solutions are convergent with respect to the spatial approximation. We illustrate the scope, versatility, efficiency and accuracy of the scheme by means of selected examples.

A mesh-free convex approximation scheme for Kohn-Sham density functional theory

Density functional theory developed by Hohenberg, Kohn and Sham is a widely accepted, reliable ab initio method. We present a non-periodic, real space, mesh-free convex approximation scheme for Kohn-Sham density functional theory. We rewrite the original variational problem as a saddle point problem and discretize it using basis functions which form the Pareto optimum between competing objectives of maximizing entropy and minimizing the total width of the approximation scheme. We show the utility of the approximation scheme in performing both all-electron and pseudopotential calculations, the results of which are in good agreement with literature.

Anderson acceleration of the Jacobi iterative method

We employ Anderson extrapolation to accelerate the classical Jacobi iterative method for large, sparse linear systems. Specifically, we utilize extrapolation at periodic intervals within the Jacobi iteration to develop the Alternating Anderson-Jacobi (AAJ) method. We verify the accuracy and efficacy of AAJ in a range of test cases, including nonsymmetric systems of equations. We demonstrate that AAJ possesses a favorable scaling with system size that is accompanied by a small prefactor, even in the absence of a preconditioner. In particular, we show that AAJ is able to accelerate the classical Jacobi iteration by over four orders of magnitude, with speed-ups that increase as the system gets larger. Moreover, we find that AAJ significantly outperforms the Generalized Minimal Residual (GMRES) method in the range of problems considered here, with the relative performance again improving with size of the system. Overall, the proposed method represents a simple yet efficient technique that is particularly attractive for large-scale parallel solutions of linear systems of equations.

Evolution of polarization and space charges in semiconducting ferroelectrics

Ferroelectric perovskites and polymers that are used in a variety of electronic, ultrasonic, and optical applications are often wide-band-gap semiconductors. We present a time-dependent and thermodynamically consistent theory that describes the evolution of polarization and space charges in such materials. We then use it to show that the semiconducting nature of ferroelectrics can have a profound effect on polarization domain switching, hysteresis, and leakage currents. Further, we show how hysteresis and leakage are affected by doping, film thickness, electrode work function, ambient temperature, and loading frequency.

Augmented Lagrangian formulation of orbital-free density functional theory

We present an Augmented Lagrangian formulation and its real-space implementation for non-periodic Orbital-Free Density Functional Theory (OF-DFT) calculations. In particular, we rewrite the constrained minimization problem of OF-DFT as a sequence of minimization problems without any constraint, thereby making it amenable to powerful unconstrained optimization algorithms. Further, we develop a parallel implementation of this approach for the Thomas-Fermi-von Weizsacker (TFW) kinetic energy functional in the framework of higher-order finite-differences and the conjugate gradient method. With this implementation, we establish that the Augmented Lagrangian approach is highly competitive compared to the penalty and Lagrange multiplier methods. Additionally, we show that higher-order finite-differences represent a computationally efficient discretization for performing OF-DFT simulations. Overall, we demonstrate that the proposed formulation and implementation are both efficient and robust by studying selected examples, including systems consisting of thousands of atoms. We validate the accuracy of the computed energies and forces by comparing them with those obtained by existing plane-wave methods.

SPARC: Accurate and efficient finite-difference formulation and parallel implementation of Density Functional Theory: Extended systems

Abstract As the first component of SPARC (Simulation Package for Ab-initio Real-space Calculations), we present an accurate and efficient finite-difference formulation and parallel implementation of Density Functional Theory (DFT) for isolated clusters. Specifically, utilizing a local reformulation of the electrostatics, the Chebyshev polynomial filtered self-consistent field iteration, and a reformulation of the non-local component of the force, we develop a framework using the finite-difference representation that enables the efficient evaluation of energies and atomic forces to within the desired accuracies in DFT. Through selected examples consisting of a variety of elements, we demonstrate that SPARC obtains exponential convergence in energy and forces with domain size; systematic convergence in the energy and forces with mesh-size to reference plane-wave result at comparably high rates; forces that are consistent with the energy, both free from any noticeable ‘egg-box’ effect; and accurate ground-state properties including equilibrium geometries and vibrational spectra. In addition, for systems consisting up to thousands of electrons, SPARC displays weak and strong parallel scaling behavior that is similar to well-established and optimized plane-wave implementations, but with a significantly reduced prefactor. Overall, SPARC represents an attractive alternative to plane-wave codes for practical DFT simulations of isolated clusters. Program summary Program title: SPARC Catalogue identifier: AFBL_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AFBL_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: GNU GPL v3 No. of lines in distributed program, including test data, etc.: 47525 No. of bytes in distributed program, including test data, etc.: 826436 Distribution format: tar.gz Programming language: C/C++. Computer: Any system with C/C++ compiler. Operating system: Linux. RAM: Problem dependent. Ranges from 80 GB to 800 GB for a system with 2500 electrons. Classification: 7.3. External routines: PETSc 3.5.3 ( http://www.mcs.anl.gov/petsc ), MKL 11.2 ( https://software.intel.com/en-us/intel-mkl ), and MVAPICH2 2.1 ( http://mvapich.cse.ohio-state.edu/ ). Nature of problem: Calculation of the electronic and structural ground-states for isolated clusters in the framework of Kohn–Sham Density Functional Theory (DFT). Solution method: High-order finite-difference discretization. Local reformulation of the electrostatics in terms of the electrostatic potential and pseudocharge densities. Calculation of the electronic ground-state using the Chebyshev polynomial filtered Self-Consistent Field (SCF) iteration in conjunction with Anderson extrapolation/mixing. Evaluation of boundary conditions for the electrostatic potential through a truncated multipole expansion. Reformulation of the non-local component of the force. Geometry optimization using the Polak–Ribiere variant of non-linear conjugate gradients with secant line search. Restrictions: System size less than ∼4000 electrons. Local Density Approximation (LDA). Troullier–Martins pseudopotentials without relativistic or non-linear core corrections. Running time: Problem dependent. Timing results for selected examples provided in the paper.

Periodic Pulay method for robust and efficient convergence acceleration of self-consistent field iterations

Abstract Pulays Direct Inversion in the Iterative Subspace (DIIS) method is one of the most widely used mixing schemes for accelerating the self-consistent solution of electronic structure problems. In this work, we propose a simple generalization of DIIS in which Pulay extrapolation is performed at periodic intervals rather than on every self-consistent field iteration, and linear mixing is performed on all other iterations. We demonstrate through numerical tests on a wide variety of materials systems in the framework of density functional theory that the proposed generalization of Pulays method significantly improves its robustness and efficiency.

Ab initio strain engineering of graphene: opening bandgaps up to 1 eV

We employ electronic structure calculations based on Density Functional Theory (DFT) to strain engineer graphenes bandgap. Specifically, working in the finite deformation setting, we traverse the three-dimensional in-plane strain space to determine states capable of opening significant bandgaps in graphene. We find that biaxial strains comprising of tension in the zigzag direction and compression in the armchair direction are particularly effective at tuning graphenes electronic properties, with resulting bandgaps of up to 1 eV. Notably, we ascertain that a 11% strain in the zigzag direction in combination with −20% in the armchair direction produces a bandgap of approximately 1 eV. We also establish that uniaxial and isotropic biaxial strains of up to ±20% are incapable of opening bandgaps, while shear strains of ±20% can introduce bandgaps of around 0.4 eV.

Higher-order finite-difference formulation of periodic Orbital-free Density Functional Theory

We present a real-space formulation and higher-order finite-difference implementation of periodic Orbital-free Density Functional Theory (OF-DFT). Specifically, utilizing a local reformulation of the electrostatic and kernel terms, we develop a generalized framework for performing OF-DFT simulations with different variants of the electronic kinetic energy. In particular, we propose a self-consistent field (SCF) type fixed-point method for calculations involving linear-response kinetic energy functionals. In this framework, evaluation of both the electronic ground-state and forces on the nuclei are amenable to computations that scale linearly with the number of atoms. We develop a parallel implementation of this formulation using the finite-difference discretization. We demonstrate that higher-order finite-differences can achieve relatively large convergence rates with respect to mesh-size in both the energies and forces. Additionally, we establish that the fixed-point iteration converges rapidly, and that it can be further accelerated using extrapolation techniques like Andersons mixing. We validate the accuracy of the results by comparing the energies and forces with plane-wave methods for selected examples, including the vacancy formation energy in Aluminum. Overall, the suitability of the proposed formulation for scalable high performance computing makes it an attractive choice for large-scale OF-DFT calculations consisting of thousands of atoms.

Spectral Quadrature method for accurate O(N) electronic structure calculations of metals and insulators

Abstract We present the Clenshaw–Curtis Spectral Quadrature (SQ) method for real-space O ( N ) Density Functional Theory (DFT) calculations. In this approach, all quantities of interest are expressed as bilinear forms or sums over bilinear forms, which are then approximated by spatially localized Clenshaw–Curtis quadrature rules. This technique is identically applicable to both insulating and metallic systems, and in conjunction with local reformulation of the electrostatics, enables the O ( N ) evaluation of the electronic density, energy, and atomic forces. The SQ approach also permits infinite-cell calculations without recourse to Brillouin zone integration or large supercells. We employ a finite difference representation in order to exploit the locality of electronic interactions in real space, enable systematic convergence, and facilitate large-scale parallel implementation. In particular, we derive expressions for the electronic density, total energy, and atomic forces that can be evaluated in O ( N ) operations. We demonstrate the systematic convergence of energies and forces with respect to quadrature order as well as truncation radius to the exact diagonalization result. In addition, we show convergence with respect to mesh size to established O ( N 3 ) planewave results. Finally, we establish the efficiency of the proposed approach for high temperature calculations and discuss its particular suitability for large-scale parallel computation.