Finite element methods in<i>ab initio</i>electronic structure calculations

Pask, John E.; Sterne, P. A.

doi:10.1088/0965-0393/13/3/r01

Cited by 223 publications

(160 citation statements)

References 79 publications

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“…onetep belongs to the category of methods that aim for high accuracy by optimising self-consistently the energy not only with respect to K but also with respect to the NGWFs 15,16,17,18,19,20 . In onetep the NGWFs are expanded in a basis set of periodic cardinal sine (psinc) functions 13,21 , also known as Dirichlet or Fourier Lagrange-mesh functions 22,23 .…”

Section: Overview Of Theorymentioning

confidence: 99%

Using ONETEP for accurate and efficient density functional calculations

Skylaris¹,

Haynes²,

Mostofi³

et al. 2005

J. Phys.: Condens. Matter

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We present a detailed comparison between onetep, our linear-scaling density functional method, and the conventional pseudopotential plane wave approach in order to demonstrate its high accuracy. Further comparison with all-electron calculations shows that only the largest available Gaussian basis sets can match the accuracy of routine onetep calculations. Results indicate that our minimisation procedure is not ill-conditioned and that convergence to self-consistency is achieved efficiently. Finally we present calculations with onetep, on systems of about 1000 atoms, of electronic, structural and chemical properties of a wide variety of materials such as metallic and semiconducting carbon nanotubes, crystalline silicon and a protein complex.

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Section: Overview Of Theorymentioning

confidence: 99%

Using ONETEP for accurate and efficient density functional calculations

Skylaris¹,

Haynes²,

Mostofi³

et al. 2005

J. Phys.: Condens. Matter

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“…31 Our MW results provide quasi-exact reference values that can be employed to quantify the accuracy of standard basis sets, such as GTO, NAO, and APW methods, as well as of novel approaches based for instance on finite element methods [32][33][34][35] or discontinuous Galerkin methods.…”

mentioning

confidence: 98%

The Elephant in the Room of Density Functional Theory Calculations

Jensen

Saha

Flores‐Livas

et al. 2017

J. Phys. Chem. Lett.

109

151

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Using multiwavelets, we have obtained total energies and corresponding atomization energies for the GGA-PBE and hybrid-PBE0 density functionals for a test set of 211 molecules with an unprecedented and guaranteed µHartree accuracy. These quasi-exact references allow us to quantify the accuracy of standard all-electron basis sets that are believed to be highly accurate for molecules, such as Gaussian-type orbitals (GTOs), all-electron numeric atom-centered orbitals (NAOs) and full-potential augmented plane wave (APW) methods. We show that NAOs are able to achieve the so-called chemical accuracy (1 kcal/mol) for the typical basis set sizes used in applications, for both total and atomization energies. For GTOs, a triple-zeta quality basis has mean errors of ∼10 kcal/mol in total energies, while chemical accuracy is almost reached for a quintuple-zeta basis. Due to systematic error cancellations, atomization energy errors are reduced by almost an order of magnitude, placing chemical accuracy within reach also for medium to large GTO bases, albeit with significant outliers. In order to check the accuracy of the computed densities, we have also investigated the dipole moments, where in general, only the largest NAO and GTO bases are able to yield errors below 0.01 Debye. The observed errors are similar across the different functionals considered here.

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“…Initial ρ For the numerical solution, finite element basis functions are chosen [3,4] consisting of low order polynomials defined on a simple geometry like a cube or a simplex, which is then mapped together with the basis function into the physical domain. The locality of the basis functions provides algorithmic advantages, as it yields sparse matrices, which are accessible to fast iterative solver methods and parallelization is directly possible via domain decomposition, as shown in Fig.2.…”

Section: Kohn-sham Equations and Poisson Problemmentioning

confidence: 99%

All‐electron calculations with finite elements

Schauer

Linder

2012

Proc Appl Math and Mech

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The Kohn-Sham equations resemble a nonlinear eigenvalue problem for the determination of the electronic structure of an atomic system, where the electrons are exposed to an effective potential, accounting for the Coulomb and quantum mechanical interactions between the particles. The effectiveness of the potential requires an iterative solution procedure, until selfconsistency is reached. This work illustrates the implementation of the self consistent field algorithm based on nested finite elements spaces and analyzes its properties in the case of all-electron calculations on atoms as large as the noble gas Xenon. All-electron calculations have maximal requirements onto the numerical basis, as it must be able to represent all the orthogonal electronic wavefunctions simultaneously together with the electrostatic potential, showing singularities at the positions of the atoms. Kohn-Sham equations and Poisson ProblemIn density functional theory [1], the energy of an atomic system, consisting of M positively charged nuclei at positions R A and N negatively charged electrons, is given in the form of an energy functional depending on the electron density ρ askinetic energywhich takes its minimum at the ground state electron density ρ 0 . The functional consists of the kinetic and electrostatic contributions plus an additional exchange correlation term E xc , that accounts for quantum mechanical corrections. The latter is not exactly known, though fairly good approximations exist. In this work the local density approximation from Perdew and Zunger [2] is applied. In order to attain an expression for the kinetic energy, Kohn and Sham introduced the electron density and the kinetic energy of a non-interacting quantum mechanical system, described by a single Slater determinant with orthogonal single electron orbitals ψ i (r). Under consideration of the orthogonality constraint, the variation of the energy functional with respect to these orbitals leads to the Kohn-Sham equationswhich represents a stationary Schrödinger equation for single electron orbitals ψ i (r), where the ground state electron density ρ 0 (r) is calculated from the orbitals as ρ 0 (r) = i |ψ i (r)| 2 . In the Kohn-Sham equations the effective potential V eff accounts for the interactions between the electrons only in an averaged way through the electrostatic potentialbuild by all electrons and nuclei. As V eff contains the electron density, the eigenvalue problem is nonlinear and must therefore be solved iteratively, which is done in the self-consisting field algorithm outlined in Fig.1.The electrostatic potential can be gained from the Poisson equation in (3), where the boundary values can be determined from a multipole decomposition. Initial ρPoisson-EQ: −∆φ n = 4π(ρ n + A δ A ) ⇒ V eff KS-EQ: − 1 2 ∆ + V eff (r) ψ i = ε i ψ i ⇒ ρ n+1

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Finite element methods inab initioelectronic structure calculations

Cited by 223 publications

References 79 publications

Using ONETEP for accurate and efficient density functional calculations

Using ONETEP for accurate and efficient density functional calculations

The Elephant in the Room of Density Functional Theory Calculations

All‐electron calculations with finite elements

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