Restarted Pulay mixing for efficient and robust acceleration of fixed-point iterations

Chemical Physics Letters

Pask

2016

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Pulay's 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 Pulay's method significantly improves its robustness and efficiency.

Section: Periodic Pulay Methodsmentioning

confidence: 99%

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

Banerjee

Chemical Physics Letters

Pask

2016

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“…Specifically, the non-linear eigenvalue problem described in Eqn. 14 is solved using a fixed-point iterationaccelerated using mixing/extrapolation schemes [70,71,72,73]-with respect to the potential V ef f = V xc + φ. In each iteration of the SCF method, the electron density is calculated by solving for the eigenfunctions of the linearized Hamiltonian, and the effective potential is evaluated by solving the Poisson equation for the electrostatic potential.…”

Section: Formulation and Implementationmentioning

confidence: 99%

SPARC: Accurate and efficient finite-difference formulation and parallel implementation of Density Functional Theory: Isolated clusters

Ghosh

Computer Physics Communications

2017

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101

128

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...

“…The superposition of isolated-atom electron densities is used as the initial guess for the first SCF iteration in the simulation, whereas for every subsequent atomic configuration encountered, extrapolation based on previous configurations' solutions is employed. 89 The SCF method is accelerated using the restarted variant 90 of Periodic Pulay mixing 91 with real-space preconditioning. 92 For spin polarized calculations, mixing is performed simultaneously on both components.…”

Section: Real-space Formulation and Implementation A M-sparc Code Basementioning

confidence: 99%

Implementation of Perdew–Zunger self-interaction correction in real space using Fermi–Löwdin orbitals

Diaz

The Journal of Chemical Physics

et al. 2021

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Most widely used density functional approximations suffer from self-interaction (SI) error, which can be corrected using the Perdew-Zunger (PZ) self-interaction correction (SIC). We implement the recently proposed size-extensive formulation of PZ-SIC using Fermi-Löwdin Orbitals (FLOs) in real space, which is amenable to systematic convergence and large-scale parallelization. We verify the new formulation within the generalized Slater scheme by computing atomization energies and ionization potentials of selected molecules and comparing to those obtained by existing FLOSIC implementations in Gaussian based codes. The results show good agreement between the two formulations, with new real-space results somewhat closer to experiment on average for the systems considered. We also obtain the ionization potentials and atomization energies by scaling down the Slater statistical average of SIC potentials. The results show that scaling down the average SIC potential improves both atomization energies and ionization potentials, bringing them closer to experiment. Finally, we verify the present formulation by calculating the barrier heights of chemical reactions in the BH6 dataset, where significant improvements are obtained relative to Gaussian based FLOSIC results.