Finite Element Method in Density Functional Theory Electronic Structure Calculations

Vackář, Jiří; Čertı́k, Ondřej; Cimrman, Robert; Novák, Matyáš; Šipr, O.; Plešek, Jiřı́

doi:10.1007/978-94-007-2076-3_12

Cited by 11 publications

(7 citation statements)

References 8 publications

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“…Finite element methods (FEMs) have been developed for the non-relativistic and relativistic calculation of atomic [117][118][119][120] and molecular [5,[120][121][122][123][124][125][126] wave functions. Another application of FEMs are non-relativistic [127][128][129] and relativistic DFT calculations [130,131] of molecules. The role of FEMs for DFT-based simulations of solids have been discussed by Pask et al [132][133][134], Gavini et al [135] and Masud et al [4].…”

Section: Appendix: Grid-based Vs Spectral Approaches In Electronic-smentioning

confidence: 99%

An efficient pseudo-spectral method for the description of atomic electronic wave functions – Application to the hydrogen atom in a uniform magnetic field

et al. 2018

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The mapping of an electronic state on a real-space support lattice may offer advantages over a basis set ansatz in cases where there are linear dependences due to basis set overcompleteness or when strong internal or external fields are present. Such discretization methods are also of interest because they allow for the convenient numerical integration of matrix elements of local operators. We have developed a pseudo-spectral approach to the numerical solution of the time-dependent and time-independent Schrödinger equations describing electronic motion in atoms and atomic ions in terms of a spherical coordinate system. A key feature of this scheme is the construction of a Variational Basis Representation (VBR) for the non-local component and of a Generalized Finite Basis Representation (GFBR) for the local component of the electronic Hamiltonian operator. Radial Hamiltonian eigenfunctions χ nl;β (r) of the H atom-like system and spherical harmonics form the basis set. Two special cases of this approach are explored: In one case, the functions of the field-free H atom are used as the elements of the basis set, and in the second case, each radial basis function has been obtained by variationally optimizing a shielding parameter β to yield a minimum energy for a particular eigenstate of the H atom in a uniform magnetic field. We derive a new quadrature rule of nearly Gaussian accuracy for the computation of matrix elements of local operators between radial basis functions and perform numerical evaluation of local operator matrix elements involving spherical harmonics. With this combination of radial and angular quadrature prescriptions we ensure to a good approximation the discrete orthogonality of Hamiltonian eigenfunctions of H atom-like systems for summation over the grid points. We further show that sets of χ nl;β (r) functions are linearly independent, whereas sets of the polarangle-dependent components of the spherical harmonics, i.e., the associated Legendre functions, are not and provide a physical interpretation of this mathematical observation. The pseudo-spectral approach presented here is applied to two model systems: the field-free H atom and the H atom in a uniform magnetic field. The results demonstrate the potential of this method for the description of challenging systems such as highly charged atomic ions.

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Section: Appendix: Grid-based Vs Spectral Approaches In Electronic-smentioning

confidence: 99%

An efficient pseudo-spectral method for the description of atomic electronic wave functions – Application to the hydrogen atom in a uniform magnetic field

et al. 2018

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“…Our research in this field is motivated by the interest in a detailed modelling of cracks in iron, where we wish to compute potentials for molecular dynamics simulations around a crack tip that would be as precise as possible. The potentials will be constructed with help of our ab-initio electronic calculations code (Vackář et al, 2011;Cimrman et al, 2018), that relies on the pseudopotential approach. This contribution is devoted to describing a new implementation of the approach developed in (Vackář, 1992) for generating and optimizing the so called environment-reflecting all-electron pseudopotentials.…”

Section: Introductionmentioning

confidence: 99%

A Software for Generating and Optimizing Pseudopotentials

Cimrman¹,

Vackář²,

Novák³

2019

Engineering Mechanics

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We wish to compute potentials for molecular dynamics simulations around a crack tip in iron that would be as precise as possible. The potentials will be constructed with help of our ab-initio electronic calculations code, that relies on the density functional theory and the pseudopotential approach. In this contribution we describe an approach and its new implementation for generating and optimizing the so called environmentreflecting all-electron pseudopotentials. A preliminary example is shown.

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“…The electronic structure calculations allow to predict and understand material properties such as stable atomic arrangements by minimizing the total internal energy of a system of atoms, as well as to determine derived properties such as elasticity, hardness, electric and magnetic properties, etc. 5 We are developing a real space code [29] for electronic structure calculations based on…”

Section: Introductionmentioning

confidence: 99%

Isogeometric analysis in electronic structure calculations

Cimrman

Novák

Kolman

et al. 2018

Mathematics and Computers in Simulation

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In electronic structure calculations, various material properties can be obtained by means of computing the total energy of a system as well as derivatives of the total energy w.r.t. atomic positions. The derivatives, also known as HellmanFeynman forces, require, because of practical computational reasons, the discretized charge density and wave functions having continuous second derivatives in the whole solution domain. We describe an application of isogeometric analysis (IGA), a spline modification of finite element method (FEM), to achieve the required continuity. The novelty of our approach is in employing the technique of Bézier extraction to add the IGA capabilities to our FEM based code for ab-initio calculations of electronic states of non-periodic systems within the density-functional framework, built upon the open source finite element package SfePy. We compare FEM and IGA in benchmark problems and several numerical results are presented.

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Finite Element Method in Density Functional Theory Electronic Structure Calculations

Cited by 11 publications

References 8 publications

An efficient pseudo-spectral method for the description of atomic electronic wave functions – Application to the hydrogen atom in a uniform magnetic field

An efficient pseudo-spectral method for the description of atomic electronic wave functions – Application to the hydrogen atom in a uniform magnetic field

A Software for Generating and Optimizing Pseudopotentials

Isogeometric analysis in electronic structure calculations

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