Higher-order adaptive finite-element methods for orbital-free density functional theory

Motamarri, Phani; Iyer, Mrinal; Knap, Jaroslaw; Gavini, Vikram

doi:10.1016/j.jcp.2012.04.036

Cited by 30 publications

(66 citation statements)

References 53 publications

(123 reference statements)

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“…(9)). Here we adopt the recently developed local real-space reformulation of the kernel energy 30,31 , and recall the key ideas and local reformulation for the sake of completeness. We present the local reformulation of K 0 and the local reformulations for other kernels (K 1 , K 11 , K 12 ) follows along similar lines.…”

Section: A Local Real-space Formulationmentioning

confidence: 99%

“…However in electronic structure calculations, where the desired accuracy is commensurate with chemical accuracy, linear finite elements are computationally inefficient requiring of the order of hundred thousand basis functions per atom to achieve chemical accuracy. A recent study 31 has demonstrated the significant computational savings-of the order of 1000-fold compared to linear finite-elements-that can be realized by using higher-order finite-element discretizations. Thus, in the present work we use higher-order hexahedral finite elements, where the basis functions are constructed as a tensor product of basis functions in one-dimension 46 .…”

Section: A Finite-element Basismentioning

confidence: 99%

“…The extrapolation procedure proposed in Motamarri et. al 31 allows us to estimate the ground-state energy and hydrostatic stress in the limit as h → 0, denoted by E 0 and σ 0 . To this end, the energy and hydrostatic stress computed from the sequence of meshes using HEX64 finite-elements are fitted to expressions of the form…”

Section: B Convergence Of Finite-element Discretizationmentioning

confidence: 99%

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Real-space formulation of orbital-free density functional theory using finite-element discretization: The case for Al, Mg, and Al-Mg intermetallics

2015

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We propose a local real-space formulation for orbital-free DFT with density dependent kinetic energy functionals and a unified variational framework for computing the configurational forces associated with geometry optimization of both internal atomic positions as well as the cell geometry. The proposed real-space formulation, which involves a reformulation of the extended interactions in electrostatic and kinetic energy functionals as local variational problems in auxiliary potential fields, also readily extends to all-electron orbital-free DFT calculations that are employed in warm dense matter calculations. We use the local real-space formulation in conjunction with higher-order finite-element discretization to demonstrate the accuracy of orbital-free DFT and the proposed formalism for the Al-Mg materials system, where we obtain good agreement with Kohn-Sham DFT calculations on a wide range of properties and benchmark calculations. Finally, we investigate the cell-size effects in the electronic structure of point defects, in particular a mono-vacancy in Al. We unambiguously demonstrate that the cell-size effects observed from vacancy formation energies computed using periodic boundary conditions underestimate the extent of the electronic structure perturbations created by the defect. On the contrary, the bulk Dirichlet boundary conditions, accessible only through the proposed real-space formulation, which correspond to an isolated defect embedded in the bulk, show cell-size effects in the defect formation energy that are commensurate with the perturbations in the electronic structure. Our studies suggest that even for a simple defect like a vacancy in Al, we require cell-sizes of ∼ 10 3 atoms for convergence in the electronic structure.

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Section: A Local Real-space Formulationmentioning

confidence: 99%

Section: A Finite-element Basismentioning

confidence: 99%

Section: B Convergence Of Finite-element Discretizationmentioning

confidence: 99%

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Real-space formulation of orbital-free density functional theory using finite-element discretization: The case for Al, Mg, and Al-Mg intermetallics

2015

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“…5b, we define b(x, R) = − Na I Z I δ(x, R I ), where δ(x, R I ) is a Dirac distribution centered at R I . We refer to previous works on finite element based DFT calculations 46,48,50,53,60 for a comprehensive treatment of the local reformulation of electrostatic potentials into Poisson problems.…”

Section: Real-space Dft Formulationmentioning

confidence: 99%

Large-scale all-electron density functional theory calculations using an enriched finite-element basis

Kanungo

Gavini

2017

Phys. Rev. B

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We present a computationally efficient approach to perform large-scale all-electron density functional theory calculations by enriching the classical finite element basis with compactly supported atom-centered numerical basis functions that are constructed from the solution of the Kohn-Sham (KS) problem for single atoms. We term these numerical basis functions as enrichment functions, and the resultant basis as the enriched finite element basis. The compact support for the enrichment functions is obtained by using smooth cutoff functions, which enhances the conditioning and maintains the locality of the enriched finite element basis. The integrals involved in the evaluation of the discrete KS Hamiltonian and overlap matrix in the enriched finite element basis are computed using an adaptive quadrature grid that is constructed based on the characteristics of enrichment functions. Further, we propose an efficient scheme to invert the overlap matrix by using a block-wise matrix inversion in conjunction with special reduced-order quadrature rules, which is required to transform the discrete Kohn-Sham problem to a standard eigenvalue problem. Finally, we solve the resulting standard eigenvalue problem, in each self-consistent field iteration, by using a Chebyshev polynomial based filtering technique to compute the relevant eigenspectrum. We demonstrate the accuracy, efficiency and parallel scalability of the proposed method on semiconducting and heavymetallic systems of various sizes, with the largest system containing 8694 electrons. We obtain accuracies in the ground-state energies that are ∼ 1 mHa with reference ground-state energies employing classical finite element as well as gaussian basis sets. Using the proposed formulation based on enriched finite element basis, for accuracies commensurate with chemical accuracy, we observe a staggering 50−300 fold reduction in the overall computational time when compared to classical finite element basis. Further, we find a significant outperformance by the enriched finite element basis when compared to the gaussian basis for the modest system sizes where we obtained convergence with gaussian basis. We also observe good parallel scalability of the numerical implementation up to 384 processors for a representative benchmark system comprising of 280-atom silicon nano-cluster.

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“…It computes E[ (r)] without computing i , resulting in orders of magnitude faster calculations and significant memory savings [4][5][6]. The scaling of OF-DFT is near linear with system size.…”

Section: Introductionmentioning

confidence: 99%

Highly accurate local pseudopotentials of Li, Na, and Mg for orbital free density functional theory

Legrain

Manzhos

2015

Chemical Physics Letters

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a b s t r a c tWe present a method to make highly accurate pseudopotentials for use with orbital-free density functional theory (OF-DFT) with given exchange-correlation and kinetic energy functionals, which avoids the compounding of errors of Kohn-Sham DFT and OF-DFT. The pseudopotentials are fitted to reference (experimental or highly accurate quantum chemistry) values of interaction energies, geometries, and mechanical properties, using a genetic algorithm. This can enable routine large-scale ab initio simulations of many practically relevant materials. Pseudopotentials for Li, Na, and Mg resulting in accurate geometries and energies of different phases as well as of vacancy formation and bulk moduli are presented as examples.

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Higher-order adaptive finite-element methods for orbital-free density functional theory

Cited by 30 publications

References 53 publications

Real-space formulation of orbital-free density functional theory using finite-element discretization: The case for Al, Mg, and Al-Mg intermetallics

Real-space formulation of orbital-free density functional theory using finite-element discretization: The case for Al, Mg, and Al-Mg intermetallics

Large-scale all-electron density functional theory calculations using an enriched finite-element basis

Highly accurate local pseudopotentials of Li, Na, and Mg for orbital free density functional theory

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