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. Since the system sizes studied in ab-initio calculations are relatively modest, the extra computation in the adopted procedure is negligible.Algorithm 1: Pseudocharge density generation and self energy calculationis interpolated on to the finite-difference grid in the overlap region Ω r b J ∩ Ω p = ∅ (and an additional 4n 0 points in each direction) using cubic-splines, 13 from which b (i,j,k) J is calculated using Eqn. 24. Subsequently, f h,p J,loc -contribution of the p th processor to the local component of the force-is calculated using Eqn. 41. Finally, the contributions from all processors are summed to simultaneously obtain f h J,loc for all the atoms. Algorithm 3: Calculation of the local component of the atomic force. Input: R, φ (i,j,k) , V J , and r b J for J ∈ P p r b J do Determine starting and ending indices i s , i e , j s , j e , k s , k e for Ω r b JAgain, the integral in Eqn. 17 has been approximated using the integration rule in Eqn. 22. The calculation of f h J,nloc in SPARC is summarized in Algorithm 4. We use P p r c J to denote the set of all atoms whose Ω r c Jcube with side of length 2r c J centered on the J th atom-has overlap with the processor domain Ω p . The value of r c J corresponds to the maximum cutoff radius amongst the non-local components of the pseudopotential for the J th atom. We have chosen a cube rather than a sphere due to its simplicity and efficiency within the Euclidean finite-difference discretization. While describing Algorithm 4, we use the subscripts s and e to denote the starting and ending indices of Ω r c J ∩ Ω p = ∅, respectively. In th...

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...

We present a symmetry-adapted real-space formulation of Kohn-Sham density functional theory for cylindrical geometries and apply it to the study of large X (X=C, Si, Ge, Sn) nanotubes. Specifically, starting from the Kohn-Sham equations posed on all of space, we reduce the problem to the fundamental domain by incorporating cyclic and periodic symmetries present in the angular and axial directions of the cylinder, respectively. We develop a high-order finite-difference parallel implementation of this formulation, and verify its accuracy against established planewave and realspace codes. Using this implementation, we study the band structure and bending properties of X nanotubes and Xene sheets, respectively. Specifically, we first show that zigzag and armchair X nanotubes with radii in the range 1 to 5 nm are semiconducting, other than the armchair and zigzag type III carbon variants, for which we find a vanishingly small bandgap, indicative of metallic behavior. In particular, we find an inverse linear dependence of the bandgap with respect to the radius for all nanotubes, other than the armchair and zigzag type III carbon variants, for which we find an inverse quadratic dependence. Next, we exploit the connection between cyclic symmetry and uniform bending deformations to calculate the bending moduli of Xene sheets in both zigzag and armchair directions, while considering radii of curvature up to 5 nm. We find Kirchhoff-Love type bending behavior for all sheets, with graphene and stanene possessing the largest and smallest moduli, respectively. In addition, other than graphene, the sheets demonstrate significant anisotropy, with larger bending moduli along the armchair direction. Finally, we demonstrate that the proposed approach has very good parallel scaling and is highly efficient, enabling ab initio simulations of unprecedented size for systems with a high degree of cyclic symmetry. In particular, we show that even micron-sized nanotubes can be simulated with modest computational effort. Overall, the current work opens an avenue for the efficient ab-initio study of 1D nanostructures with large radii as well as 1D/2D nanostructures under uniform bending.

We present a real-space formulation and higher-order finite-difference implementation of periodic Orbitalfree 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 as well as 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 Anderson's 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. Geometry optimization: forces on nucleiConsider the minimization problem in Eqn. 33 for determining the equilibrium configuration of the atoms. During this geometry optimization, the forces on the nuclei can be calculated using the relationwhere f J denotes the force on the J th nucleus and the summation over J ′ signifies the J th atom and its periodic images. Additionally, φ * (x) is the solution of the Poisson equation in Eqn. 36 for u(x) = u * (x) and∂R J corrects for the error in forces due to overlapping charge density of nuclei. The expression for this correction has been derived in Appendix B. The second equality in Eqn. 46 is obtained by using the

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