A spectral scheme for Kohn-Sham density functional theory of clusters

Amartya S. Banerjee, Ryan S. Elliott, Richard D. James

Research output: Contribution to journalArticlepeer-review

23 Scopus citations


Starting from the observation that one of the most successful methods for solving the Kohn-Sham equations for periodic systems - the plane-wave method - is a spectral method based on eigenfunction expansion, we formulate a spectral method designed towards solving the Kohn-Sham equations for clusters. This allows for efficient calculation of the electronic structure of clusters (and molecules) with high accuracy and systematic convergence properties without the need for any artificial periodicity. The basis functions in this method form a complete orthonormal set and are expressible in terms of spherical harmonics and spherical Bessel functions. Computation of the occupied eigenstates of the discretized Kohn-Sham Hamiltonian is carried out using a combination of preconditioned block eigensolvers and Chebyshev polynomial filter accelerated subspace iterations. Several algorithmic and computational aspects of the method, including computation of the electrostatics terms and parallelization are discussed. We have implemented these methods and algorithms into an efficient and reliable package called ClusterES (Cluster Electronic Structure). A variety of benchmark calculations employing local and non-local pseudopotentials are carried out using our package and the results are compared to the literature. Convergence properties of the basis set are discussed through numerical examples. Computations involving large systems that contain thousands of electrons are demonstrated to highlight the efficacy of our methodology. The use of our method to study clusters with arbitrary point group symmetries is briefly discussed.

Original languageEnglish (US)
Pages (from-to)226-253
Number of pages28
JournalJournal of Computational Physics
StatePublished - Apr 5 2015

Bibliographical note

Funding Information:
This work was primarily supported by Russell Penrose. It also benefited from the support of NSF-PIRE Grant No. OISE-0967140 , ONR N00014-14-1-0714 and the MURI project FA9550-12-1-0458 (administered by AFOSR ). We would like to thank the Minnesota Supercomputing Institute for making the parallel computing resources used in this work available. We would like to thank Phanish Suryanarayana (Georgia Tech.) for his many insightful comments and suggestions at various stages of this work. We would like to thank Vikram Gavini and Phani Motamarri (U. Michigan) for stimulating discussions as well as for making available some of their Finite Element Method results which helped us in carrying out validation studies. We would also like to thank Gero Friesecke (TU Munich, Germany) and Michael Ortiz (Caltech) for informative discussions. We gratefully acknowledge comments from the anonymous reviewers which helped us in improving the presentation of our work. ASB and RDJ would like to acknowledge the hospitality of the Hausdorff Research Institute for Mathematics, Bonn, Germany where this work was partially carried out.

Publisher Copyright:
© 2015 Elsevier Inc.


  • Chebyshev filtering
  • Computational efficiency
  • Eigenvalue problem
  • Kohn-Sham density functional theory
  • Nano clusters
  • Parallel scaling performance
  • Spectral convergence
  • Spectral scheme
  • Spherical Bessel functions
  • Spherical harmonics
  • Super atoms


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