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Extending the Scale with Real-Space Methods for the Electronic Structure Problem

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Handbook of Materials Modeling

Abstract

In principle, the electronic structure of a material can be determined by a solution of the many-body Schrödinger equation. This was first noted by Dirac shortly after the invention of quantum mechanics in 1929. However, Dirac also noted that the solution of the many-body quantum mechanical equations was much too difficult to be solved. He challenged his colleagues to develop “practical methods of applying quantum mechanics,” which can lead to an explanation of the main features of complex atomic systems.” In this chapter, we explore concepts and algorithms targeting “Dirac’s challenge.” Two key physical concepts will be employed: pseudopotential theory and density functional theory. For many weakly correlated systems, this formalism works well for ground-state properties such as phase stability, structural properties, and vibrational modes. However, applying this approach to large systems, e.g., systems with thousands of atoms, remains a challenge even with contemporary computational platforms. The goal of this chapter is to show how new algorithms can be used to extend computations to these systems. The approach centers on solving the nonlinear Kohn-Sham equation by a nonlinear form of the subspace iteration technique. This approach results in a significant speedup, often by more than an order of magnitude with no loss of accuracy. Numerical results are presented for nanoscale systems with tens of thousands of atoms and propose new methods to extend our work to even larger systems.

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References

  • Alemany MMG, Jain M, Chelikowsky JR, Kronik L (2004) Real-space pseudopotential method for computing the electronic properties of periodic systems. Phys Rev B 69:075101

    Article  ADS  Google Scholar 

  • Alemany MG, Huang X, Tiago ML, Chelikowsky JR (2007) The role of quantum confinement in p-type doped indium phosphide nanowires. Nano Lett 7:1878

    Article  ADS  Google Scholar 

  • Andrade X, Aspuru-Guzik A (2013) Real-space density functional theory on graphical processing units: computational approach and comparison to gaussian basis set methods. J Chem Theor Comput 9:4360

    Article  Google Scholar 

  • Beck TL (2000) Real-space mesh techniques in density-functional theory. Rev Mod Phys 72:1041

    Article  ADS  Google Scholar 

  • Boffi NM, Jain M, Natan A (2016) Efficient computation of the Hartree-Fock exchange in real space with projection operators. J Chem Theory Comput 12:3614

    Article  Google Scholar 

  • Chan T-L, Chelikowsky JR (2010) Controlling lithium diffusion in semiconductor nanostructures by size and dimensionality. Nano Lett 10:821

    Article  ADS  Google Scholar 

  • Chan T-L, Tiago ML, Kaxiras E, Chelikowsky JR (2008) Size limits on doping phosphorus into silicon nanocrystals. Nano Lett 8:596

    Article  ADS  Google Scholar 

  • Chan T-L, Zayak AT, Dalpian GM, Chelikowsky JR (2009) Role of confinement on diffusion barriers in semiconductor nanocrystals. Phys Rev Lett 102:025901

    Article  ADS  Google Scholar 

  • Chan T-L, Lee AJ, Mok WK, Chelikowsky JR (2014) The interaction range of p-dopants in si [110] nanowires: determining the non-degenerate limit. Nano Lett 14:6306

    Article  ADS  Google Scholar 

  • Chelikowsky JR, Cohen ML (1992) Ab initio pseudopotentials for semiconductors. In: Landsberg PT (ed) Handbook on semiconductors, vol 1, 59. Elsevier, Amsterdam

    Google Scholar 

  • Chelikowsky JR, Troullier N, Saad Y (1994a) Finite-difference-pseudopotential method: electronic structure calculations without a basis. Phys Rev Lett 72:1240

    Article  ADS  Google Scholar 

  • Chelikowsky JR, Troullier N, Wu K, Saad Y (1994b) Higher-order finite-difference pseudopotential method: an application to diatomic molecules. Phys Rev B 50:11355

    Article  ADS  Google Scholar 

  • Chelikowsky JR, Chan T-L, Alemany MMG, Dalpian G (2011) Computational studies of doped nanostructures. Rep Prog Phys 74:046501

    Article  ADS  Google Scholar 

  • Dalpian G, Chelikowsky JR (2006) Self-purification in semiconductor nanocrystals. Phys Rev Lett 96:226802

    Article  ADS  Google Scholar 

  • Daniel J, Gragg WB, Kaufman L, Stewart GW (1976) Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization. Math Comput 30:772

    MathSciNet  MATH  Google Scholar 

  • Dirac PAM (1929) Quantum mechanics of many-electron systems. Proc R Soc A 123:714

    Article  ADS  Google Scholar 

  • Enkovaara J, Rostgaard C, Mortensen JJ, Chen J, Duak M, Ferrighi L, Gavnholt J, Glinsvad C, Haikola V, Hansen HA, Kristoffersen HH, Kuisma M, Larsen AH, Lehtovaara L, Ljungberg M, Lopez-Acevedo O, Moses PG, Ojanen J, Olsen T, Petzold V, Romero NA, Stausholm-Mller J, Strange M, Tritsaris GA, Vanin M, Walter M, Hammer B, Häkkinen H, Madsen11 GKH, Nieminen RM, Nørskov JK, Puska M, Rantala TT, Schiøtz J, Thygesen KS, Jacobsen KW (2010) Electronic structure calculations with GPAW: a real-space implementation of the projector augmented-wave method. J Phys Condens Matter 22:253202

    Google Scholar 

  • Fattebert J.-L., Bernholc J (2000) Towards grid-based o(n) density-functional theory methods: optimized nonorthogonal orbitals and multigrid acceleration. Phys Rev B 62:1713

    Article  ADS  Google Scholar 

  • Fornberg B, Sloan DM (1994) A review of pseudospectral methods for solving partial differential equations. Acta Numer 3:203

    Article  ADS  MathSciNet  Google Scholar 

  • Giannozzi P, Baroni S, Bonini N, Calandra M, Car R, Cavazzoni C, Ceresoli D, Chiarotti GL, Cococcioni M, Dabo I, Dal Corso A, de Gironcoli S, Fabris S, Fratesi G, Gebauer R, Gerstmann U, Gougoussis C, Kokalj A, Lazzeri M, Martin-Samos L, Marzari N, Mauri F, Mazzarello R, Paolini S, Pasquarello A, Paulatto L, Sbraccia C, Scandolo S, Sclauzero G, Seitsonen AP, Smogunov A, Umari P, Wentzcovitch RM (2009) Quantum espresso: a modular and open-source software project for quantum simulations of materials. J Phys Condens Matter 21:395502

    Article  Google Scholar 

  • Goedecker S (1999) Linear scaling electronic structure methods. Rev Mod Phys 71:1085

    Article  ADS  Google Scholar 

  • Hohenberg P, Kohn W (1964) Inhomogeneous electron gas. Phys Rev 136:B864

    Article  ADS  MathSciNet  Google Scholar 

  • Kleinmann L, Bylander DM (1982) Efficacious form for model pseudopotentials. Phys Rev Lett 48:1425

    Article  ADS  Google Scholar 

  • Koch W, Holthausen MC (2000) A chemist’s guide to density functional theory. Wiley-VCH, Weinheim

    Google Scholar 

  • Kohn W, Sham LJ (1965) Self-consistent equations including exchange and correlation effects. Phys Rev 140:A1133

    Article  ADS  MathSciNet  Google Scholar 

  • Kresse G, Furthmülle J (1996) Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys Rev B 54:11169

    Article  ADS  Google Scholar 

  • Kronik L, Makmal A, Tiago ML, Alemany MMG, Jain M, Huang X, Saad Y, Chelikowsky JR (2006) Parsec-the pseudopotential algorithm for real space electronic structure calculations: recent advances and novel applications to nano-structures. Phys Stat Sol (b) 243:1063

    Article  ADS  Google Scholar 

  • Lehoucq RB, Sorensen DC, Yang C (1998) ARPACK users guide: solution of large scale eigenvalue problems by implicitly restarted Arnoldi methods. SIAM, Philadelphia

    Book  Google Scholar 

  • Martin RM (2004) Electronic structure: basic theory and practical methods. Cambridge University Press, Cambridge/New York

    Book  Google Scholar 

  • Ono T, Hirose K (2005) Real-space electronic-structure calculations with a time-saving double-grid technique. Phys Rev B 72:085115

    Article  ADS  Google Scholar 

  • Otsuka T, Miyazaki T, Ohno T, Bowler DR, Gillan MJ (2008) Accuracy of order-n density-functional theory calculations on dna systems using conquest. J Phys Condens Matter 20:29401

    Article  Google Scholar 

  • Parlett BN (1998) The symmetric eigenvalue problem. SIAM, Philadelphia

    Book  Google Scholar 

  • Phillips JC (1958) Energy-band interpolation scheme based on a pseudopotential. Phys Rev 112:685

    Article  ADS  Google Scholar 

  • Phillips JC, Kleinman L (1959) New method for calculating wave functions in crystals and molecules. Phys Rev 116:287

    Article  ADS  Google Scholar 

  • Saad Y (1992) Numerical methods for large eigenvalue problems. Wiley, New York

    MATH  Google Scholar 

  • Saad Y, Chelikowsky JR, Shontz S (2010) Numerical methods for electronic structure calculations of materials. SIAM Rev 52:3

    Article  ADS  MathSciNet  Google Scholar 

  • Sakai Y, Lee AJ, Chelikowsky JR (2016) First-principles non-contact atomic force microscopy image simulations with frozen density embedding theory. Nano Lett 16:3242

    Article  ADS  Google Scholar 

  • Schofield G, Chelikowsky JR, Saad Y (2012a) A spectrum slicing method for the Kohn-Sham problem. Comput Phys Commun 183:497

    Article  ADS  MathSciNet  Google Scholar 

  • Schofield G, Chelikowsky JR, Saad Y (2012b) Using Chebyshev-filtered subspace iteration and windowing methods to solve the Kohn-Sham problem, chap. 6. In: Leszczynski J, Shulka MK (eds) Practical aspects of computational chemistry I: an overview of the last two decades and current trends, vol 167. Springer, Berlin

    Google Scholar 

  • Seitsonen P, Puska MJ, Nieminen RM (1995) Real-space electronic-structure calculations: combination of the finite-difference and conjugate-gradient methods. Phys Rev B 51:14057

    Article  ADS  Google Scholar 

  • Sham LJ, Schlüter M (1983) Density functional theory of the energy gap. Phys Rev B 51:1888

    ADS  Google Scholar 

  • Stathopoulos A, Öğüt S, Saad Y, Chelikowsky JR, Kim H (2000) Parallel methods and tools for predicting material properties. IEEE Comput Sci Eng 2:19

    Article  Google Scholar 

  • Teter MP, Allan DC, Arias TA, Joannopoulos JD (1992) Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients. Rev Mod Phys 64:1045

    Article  ADS  Google Scholar 

  • Tiago ML, Zhou Y, Alemany MMG, Saad Y, Chelikowsky JR (2006) The evolution of magnetism in iron from the atom to the bulk. Phys Rev Lett 97:147201

    Article  ADS  Google Scholar 

  • Troullier N, Martins JL (1991) Efficient pseudopotentials for plane-wave calculations. Phys Rev B 43:1993

    Article  ADS  Google Scholar 

  • Wu K, Simon H (2000) Thick-restart Lanczos method for large symmetric eigenvalue problems. SIAM J Matrix Anal 22:602

    Article  MathSciNet  Google Scholar 

  • Wu K, Canning A, Simon HD, Wang L-W (1999) Thick-restart Lanczos method for electronic structure calculations. J Comput Phys 154:156

    Article  ADS  MathSciNet  Google Scholar 

  • Zhao Y, Du M-H, Kim Y-H, Zhang SB (2004) First-principles prediction of icosahedral quantum dots for tetravalent semiconductors. Phys Rev Lett 93:015502

    Article  ADS  Google Scholar 

  • Zhou Y, Saad Y (2007) A Chebyshev–Davidson algorithm for large symmetric eigenproblems. SIAM J Matrix Anal Appl 29:954

    Article  MathSciNet  Google Scholar 

  • Zhou Y, Saad Y, Tiago M, Chelikowsky JR (2006a) Parallel self-consistent-field calculations via Chebyshev-filtered subspace acceleration. Phys Rev E 74:066704

    Article  ADS  Google Scholar 

  • Zhou Y, Tiago ML, Saad Y, Chelikowsky JR (2006b) Chebyshev-filtered subspace iteration method free of sparse diagonalization for solving the Kohn–Sham equation. J Comput Phys 219:172

    Article  ADS  Google Scholar 

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Acknowledgements

This work is supported by a subaward from the Center for Computational Study of Excited-State Phenomena in Energy Materials at the Lawrence Berkeley National Laboratory, which is funded by the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division under Contract No. DE-AC02-05CH11231, as part of the Computational Materials Sciences Program.

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Authors and Affiliations

  1. Center for Computational Materials, Institute for Computational Engineering and Sciences, Departments of Physics and Chemical Engineering, The University of Texas at Austin, Austin, TX, USA

    James R. Chelikowsky

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  1. James R. Chelikowsky
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Correspondence to James R. Chelikowsky .

Editor information

Editors and Affiliations

  1. Institute of Physics, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Wanda Andreoni

  2. Department of Nuclear Science & Engineering , Massachusetts Institute of Technology , Cambridge, Massachusetts, USA

    Sidney Yip

Section Editor information

  1. Department of Chemistry, Princeton University, Princeton, NJ, USA

    Roberto Car

  2. Princeton University, Princeton, New Jersey, USA

    Biswajit Santra

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Chelikowsky, J.R. (2018). Extending the Scale with Real-Space Methods for the Electronic Structure Problem. In: Andreoni, W., Yip, S. (eds) Handbook of Materials Modeling . Springer, Cham. https://doi.org/10.1007/978-3-319-42913-7_57-1

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  • DOI: https://doi.org/10.1007/978-3-319-42913-7_57-1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-42913-7

  • Online ISBN: 978-3-319-42913-7

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