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Journal of Chemical Physics / Volume 139 / Issue 5 / ARTICLES / Theoretical Methods and Algorithms

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J. Chem. Phys. 139, 054107 (2013); http://dx.doi.org/10.1063/1.4817001 (8 pages)

A variational method for density functional theory calculations on metallic systems with thousands of atoms

Álvaro Ruiz-Serrano and Chris-Kriton Skylaris

School of Chemistry, University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom

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(Received 8 May 2013; accepted 16 July 2013; published online 2 August 2013)

A new method for finite-temperature density functional theory calculations which significantly increases the number of atoms that can be simulated in metallic systems is presented. A self-consistent, direct minimization technique is used to obtain the Helmholtz free energy of the electronic system, described in terms of a set of non-orthogonal, localized functions which are optimized in situ using a periodic-sinc basis set, equivalent to plane waves. Most parts of the calculation, including the demanding operation of building the Hamiltonian matrix, have a computational cost that scales linearly with the number of atoms in the system. Also, this approach ensures that the Hamiltonian matrix has a minimal size, which reduces the computational overhead due to diagonalization, a cubic-scaling operation that is still required. Large basis set accuracy is retained via the optimization of the localized functions. This method allows accurate simulations of entire metallic nanostructures, demonstrated with calculations on a supercell of bulk copper with 500 atoms and on gold nanoparticles with up to 2057 atoms.

© 2013 AIP Publishing LLC

Article Outline

  1. INTRODUCTION
  2. FINITE-TEMPERATURE KS-DFT
  3. IMPLEMENTATION IN ONETEP
  4. DIRECT FREE ENERGY MINIMIZATION
    1. Inner loop
    2. Outer loop
  5. RESULTS AND DISCUSSION
    1. Scaling with the system size
    2. Validation tests
  6. CONCLUSIONS

KEYWORDS, PACS, and IPC

Keywords

copper, density functional theory, electronic density of states, free energy, gold, nanoparticles, variational techniques

PACS

  • 73.22.-f

    Electronic structure of nanoscale materials and related systems

  • 65.40.G-

    Other thermodynamical quantities

  • 65.80.-g

    Thermal properties of small particles, nanocrystals, nanotubes, and other related systems

  • 71.15.Ap

    Basis sets (LCAO, plane-wave, APW, etc.) and related methodology (scattering methods, ASA, linearized methods, etc.)

  • 71.15.Mb

    Density functional theory, local density approximation, gradient and other corrections

  • 71.20.Be

    Transition metals and alloys

International Patent Classification (IPC)

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.4817001

PUBLICATION DATA

ISSN

0021-9606 (print)  
1089-7690 (online)

Publisher

$abstract.society.value is a Member of CrossRef AIP Publishing LLC

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Figures (4) Tables (2)

Figures (click on thumbnails to view enlargements)

FIG.1
Time taken to complete one outer loop iteration with five inner loop iterations, in calculations on Au nanoparticles of increasing size. The plot shows the time taken by different parts of the algorithm. “Hamiltonian DD” and “Hamiltonian DI” must be interpreted as the density-dependent and density-independent terms of the Hamiltonian, respectively.

FIG.1 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.2
Density of states of Pt13 obtained with Onetep and Castep. Agreement is achieved for NGWF radii of 4.0 Å and above.

FIG.2 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.3
Lattice parameter stretching of bulk Cu. There are 4 atoms in the Castep simulation cell and 500 atoms in the Onetep simulation cell, forming a 5 × 5 × 5 supercell.

FIG.3 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.4
Convergence of the Helmholtz free energy functional with the number of outer loop (NGWF optimization) iterations, for a set of Au cuboctahedral nanoparticles of increasing sizes. The structures of Au13 and Au2057 are also shown in the plot.

FIG.4 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

Tables

Table I. BFGS geometry optimization of Pt13 with Onetep and Castep. The table shows the optimized value of the distance to the nearest-neighbour Pt atom.

View Table
Table II. Bulk modulus, B, and equilibrium lattice parameter, L0, of bulk Cu, calculated with Castep and Onetep. The value of χ2 corresponding to the fitting of the results to the third-order Birch-Murnaghan equation is also shown.

View Table

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