High order forces and nonlocal operators in a Kohn–Sham Hamiltonian

Computer Physics Communications

Pask

2016

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.

Section: Lithium Hydridementioning

confidence: 81%

Section: Introductionmentioning

confidence: 99%

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

Pratapa

Computer Physics Communications

Pask

2016

“…These results demonstrate that SPARC is able to obtain high convergence rates in both the energy and forces, which contributes to its accuracy and efficiency. Moreover, the energies and forces in SPARC converge at comparable rates, without the need for additional measures such as double-grid [64] or high-order integration [43] techniques. 5 Mesh size (Bohr) Figure 2: Convergence of the energy and atomic forces with respect to mesh size to reference planewave result for the lithium hydride, silicon, and gold systems.…”

Section: Convergence With Discretizationmentioning

confidence: 99%

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

Ghosh

Computer Physics Communications

2017

132

120

As the second 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 extended systems. Specifically, employing a local formulation of the electrostatics, the Chebyshev polynomial filtered self-consistent field iteration, and a reformulation of the non-local force component, we develop a finite-difference framework wherein both the energy and atomic forces can be efficiently calculated to within desired accuracies in DFT. We demonstrate using a wide variety of materials systems that SPARC achieves high convergence rates in energy and forces with respect to spatial discretization to reference plane-wave result; exponential convergence in energies and forces with respect to vacuum size for slabs and wires; energies and forces that are consistent and display negligible 'egg-box' effect; accurate properties of crystals, slabs, and wires; and negligible drift in molecular dynamics simulations. We also demonstrate that the weak and strong scaling behavior of SPARC is similar to well-established and optimized plane-wave implementations for systems consisting up to thousands of electrons, but with a significantly reduced prefactor. Overall, SPARC represents an attractive alternative to plane-wave codes for performing DFT simulations of extended systems.where the summation index J runs over all atoms in Ω, and the summation index J ′ runs over the J th atom and its periodic images. In addition, the coefficients γ Jl and projection functions χ Jlm can be expressed as(6) Above, u J ′ lm are the isolated atom pseudowavefunctions, V J ′ l are the angular momentum dependent pseudopotentials, and V J ′ are the local components of the pseudopotentials, with l and m signifying the azimuthal and magnetic quantum numbers, respectively.Above, the summation index J runs over all atoms in Ω, and the summation index J ′ runs over the J th atom and its periodic images. The electronic ground-state is determined by solving the non-linear eigenvalue problem in Eqn. 16 using the Self-Consistent Field (SCF) method [52]. Specifically, a fixed-point iteration is performed with respect to the potential V ef f = V xc + φ, which is further accelerated using mixing/extrapolation schemes [53,54,55,56]. In each iteration of the SCF method, the electrostatic potential is calculated by solving the Poisson equation, and the electron density is determined by computing the eigenfunctions of the linearized Hamiltonian. The orthogonality requirement amongst the Kohn-Sham orbitals makes such a procedure scale asymptotically as O(N 3 ) with respect to the number of atoms, which severely limits the size of systems that can be studied. To overcome this restrictive scaling, O(N ) approaches [20,21,57,58] will be subsequently developed and implemented into SPARC. are summed to obtain f h J,loc . Algorithm 2: Calculation of the local component of the atomic force. Input: R, φ (i,j,k) , V J , and r b J f h,p J...

“…This choice of coordinate system presents its own challenges for the finite-difference method. For example, even in the Cartersian coordinate system where the mesh is uniformly spaced, calculation of accurate atomic forces is challenging [100,101]. This and a number of such issues have been overcome in the formulation and implementation of Cyclic DFT so as to enable the accurate and efficient evaluation of energies and forces, as verified in Section 3.2.…”

Section: Symmetry-adapted Finite-difference Discretizationmentioning

confidence: 99%

Cyclic density functional theory: A route to the first principles simulation of bending in nanostructures

Banerjee

Journal of the Mechanics and Physics of Solids

2016

We formulate and implement Cyclic Density Functional Theory (Cyclic DFT) -a selfconsistent first principles simulation method for nanostructures with cyclic symmetries. Using arguments based on Group Representation Theory, we rigorously demonstrate that the Kohn-Sham eigenvalue problem for such systems can be reduced to a fundamental domain (or cyclic unit cell) augmented with cyclic-Bloch boundary conditions. Analogously, the equations of electrostatics appearing in Kohn-Sham theory can be reduced to the fundamental domain augmented with cyclic boundary conditions. By making use of this symmetry cell reduction, we show that the electronic ground-state energy and the Hellmann-Feynman forces on the atoms can be calculated using quantities defined over the fundamental domain. We develop a symmetry-adapted finite-difference discretization scheme to obtain a fully functional numerical realization of the proposed approach. We verify that our formulation and implementation of Cyclic DFT is both accurate and efficient through selected examples.The connection of cyclic symmetries with uniform bending deformations provides an elegant route to the ab-initio study of bending in nanostructures using Cyclic DFT. As a demonstration of this capability, we simulate the uniform bending of a silicene nanoribbon and obtain its energy-curvature relationship from first principles. A selfconsistent ab-initio simulation of this nature is unprecedented and well outside the scope of any other systematic first principles method in existence. Our simulations reveal that the bending stiffness of the silicene nanoribbon is intermediate between that of graphene and molybdenum disulphide -a trend which can be ascribed to the variation in effective thickness of these materials. We describe several future avenues and applications of Cyclic DFT, including its extension to the study of non-uniform bending deformations and its possible use in the study of the nanoscale flexoelectric effect.