An Introduction to Linear-Scaling Ab Initio Calculations

  • David Daniel O’Regan
Part of the Springer Theses book series (Springer Theses)


Atomistic modelling is a powerful tool that allows numerical experiments to be conducted which may be used to predict the properties of new materials, or known ones under novel user-defined conditions, and to test the validity of physical models against experiment.


Density Functional Theory Support Function Local Density Approximation Density Functional Theory Method Wannier Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    R.M. Martin, Electronic Structure: Basic Theory and Practical Methods (Cambridge University Press, New York, 2004), p. 624, (ISBN 0-521-78285-6)Google Scholar
  2. 2.
    R.W. Godby, P.G. González , in Density Functional Theories and Self-Energy Approaches, ed. by C. Fiolhais, F. Nogueira, M.A.L. Marques. A Primer in Density Functional Theory, vol. 620, Lecture Notes in Physics (Springer, Heidelberg, 2003)Google Scholar
  3. 3.
    E.K.U. Gross, E. Runge, O. Heinonen, Many-Particle Theory (Adam Hilger, Bristol, 1991)Google Scholar
  4. 4.
    L. Hedin, New method for calculating the one-particle green’s function with application to the electron-gas problem. Phys. Rev. 139, A796 (1965)ADSCrossRefGoogle Scholar
  5. 5.
    F. Aryasetiawan, O. Gunnarsson, The GW method. Rep. Prog. Phys. 61, 273 (1998)CrossRefGoogle Scholar
  6. 6.
    C. Friedrich, A. Schindlmayr, in Many-Body Perturbation Theory: The GW Approximation, ed. by J. Grotendorst, S. Blügel, D. Marx. Computational Nanoscience: Do It Yourself!, vol. 31, NIC Series (John von Neumann Institute for Computing, Jülich, 2006)Google Scholar
  7. 7.
    A. Georges, G. Kotliar, Hubbard model in infinite demensions. Phys. Rev. B 45, 6479 (1992)ADSCrossRefGoogle Scholar
  8. 8.
    G. Kotliar, S.Y. Savrasov, K. Haule, V.S. Oudovenko, O. Parcollet, C.A. Marianetti, Electronic structure calculations with dynamical mean-field theory. Rev. Mod. Phys. 78, 865 (2006)ADSCrossRefGoogle Scholar
  9. 9.
    K. Held, I.A. Nekrasov, G. Keller, V. Eyert, N. Blümer, A. McMahan, R. Scalettar, T. Pruschke, A.I. Anisimov, D. Vollhardt, Realistic investigations of correlated electron systems with LDA+DMFT. Psi-k Newsletter 56, 65 (2003)Google Scholar
  10. 10.
    G. Kotliar, D. Vollhardt, Strongly correlated materials: insights from dynamical mean-field theory. Phys. Today 57(3) (2004).Google Scholar
  11. 11.
    E. Runge, E.K.U. Gross, Density-functional theory for time-dependent systems. Phys. Rev. Lett. 52(12), 997 (1984)ADSCrossRefGoogle Scholar
  12. 12.
    P. Elliott, F. Furche, K. Burke, Excited states from time-dependent density functional theory, in Reviews in Computational Chemistry, eds. by K.B. Lipkowitz, T.R. Cundari, (Wiley, Hoboken, NJ, 2009), pp. 91–165Google Scholar
  13. 13.
    T.A. Arias, M.C. Payne, J.D. Joannopoulos, Ab initio molecular dynamics techniques extended to large length-scale systems. Phys. Rev. B 45(4), 1538 (1992)ADSCrossRefGoogle Scholar
  14. 14.
    K. Capelle, A bird’s-eye view of density-functional theory. Braz. J. Phys 36, 1318 (2006)CrossRefGoogle Scholar
  15. 15.
    A.B. Gaspar, V. Ksenofontov, S. Reiman, P. Gütlich, A.L. Thompson, A.E. Goeta, M.C. Muoz, J.A. Real, Mössbauer investigation of the photoexcited spin states and crystal structure analysis of the spin-crossover dinuclear complex \(\{{\rm Fe(bt)(NCS)}_2\}_2\)bpym. (bt=2,2-Bithiazoline, bpym=2,2-Bipyrimidine). Chem. Eur. J. 12(36), 9289 (2006)Google Scholar
  16. 16.
    S. Atwell, E. Meggers, G. Spraggon, P.G. Schultz, Structure of a copper-mediated base pair in DNA. J. Am. Chem. Soc. 123(49), 12364 (2001)CrossRefGoogle Scholar
  17. 17.
    D.D. O’Regan, N.D.M. Hine, M.C. Payne, A.A. Mostofi, Projector self-consistent DFT+U using nonorthogonal generalized Wannier functions. Phys. Rev. B 82(8), 081102 (2010)ADSCrossRefGoogle Scholar
  18. 18.
    D.D. O’Regan, M.C. Payne, A.A. Mostofi, Subspace representations in ab initio methods for strongly correlated systems. Phys. Rev. B 83(24), 245124 (2011)ADSCrossRefGoogle Scholar
  19. 19.
    E. Schrödinger, An undulatory theory of the mechanics of atoms and molecules. Phys. Rev. 28(6), 1049 (1926)ADSCrossRefGoogle Scholar
  20. 20.
    M. Born, R. Oppenheimer, Zur quantentheorie der molekeln. Ann. d. Physik 84(20), 457 (1927)ADSCrossRefGoogle Scholar
  21. 21.
    R. Car, M. Parrinello, Unified approach for molecular dynamics and density functional theory. Phys. Rev. Lett. 55(22), 2471 (1985)ADSCrossRefGoogle Scholar
  22. 22.
    V. Antonov, B. Harmon, A. Yaresko, Electronic Structure and Magneto-Optical Properties of Solids. (Kluwer Academic/ Dordrecht/ Boston/ London, 2004)Google Scholar
  23. 23.
    P. Hohenberg, W. Kohn, Inhomogeneous electron gas. Phys. Rev. 136(3B), B864 (1964)MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    W. Pauli, The connection between spin and statistics. Phys. Rev. 58, 716 (1940)ADSzbMATHCrossRefGoogle Scholar
  25. 25.
    M. Levy, Electron densities in search of Hamiltonians. Phys. Rev. A 26(3), 1200 (1982)ADSCrossRefGoogle Scholar
  26. 26.
    E.H. Lieb, Density functionals for Coulomb-systems. Int. J. Quantum Chem. 24(3), 243 (1983)CrossRefGoogle Scholar
  27. 27.
    J.P. Perdew, M. Levy, Extrema of the density functional for the energy: excited states from the ground-state theory. Phys. Rev. B 31(10), 6264 (1985)ADSCrossRefGoogle Scholar
  28. 28.
    R.M. Dreizler, E.K.U. Gross, Density Functional Theory, An Approach to the Quantum Many-Body Problem (Springer, New York, 1990).Google Scholar
  29. 29.
    W. Kohn, L.J. Sham, Self-consistent equations including exchange and correlation effects. Phys. Rev. 140(4A), A1133 (1965)MathSciNetADSCrossRefGoogle Scholar
  30. 30.
    D.R. Hartree, The wave mechanics of an atom with a non-Coulomb central field. Proc. Camb. Phil. Soc. 24(1), 89 (1928)ADSCrossRefGoogle Scholar
  31. 31.
    J.C. Slater, Note on Hartree’s method. Phys. Rev. 35(2), 210 (1930)ADSCrossRefGoogle Scholar
  32. 32.
    V. Fock, Näherungsmethode zur lösung des quantenmechanischen Mehrkörperproblems. Z. Phys. 61(1–2), 126 (1930)ADSGoogle Scholar
  33. 33.
    L.H. Thomas, The calculation of atomic fields. Proc. Camb. Phil. Soc. 23, 542 (1927)ADSzbMATHCrossRefGoogle Scholar
  34. 34.
    E. Fermi, Un metodo statistico per la determinazione di alcune proprietà dell’atome. Rend. Accad. Naz. Lincei 6, 602 (1927)Google Scholar
  35. 35.
    E. Teller, On the stability of molecules in the Thomas-Fermi theory. Rev. Mod. Phys. 34(4), 627 (1962)ADSzbMATHCrossRefGoogle Scholar
  36. 36.
    W. Kohn, A. Becke, R. Parr, Density functional theory of electronic structure. J. Phys. Chem. 100, 12974 (1996)CrossRefGoogle Scholar
  37. 37.
    D.C. Langreth, J.P. Perdew, The exchange-correlation energy of a metallic surface. Solid State Commun. 17(1), 1425 (1975)ADSCrossRefGoogle Scholar
  38. 38.
    O. Gunnarsson, B.I. Lundqvist, Exchange and correlation in atoms, molecules, and solids by the spin-density-functional formalism. Phys. Rev. B 13(10), 4274 (1976)ADSCrossRefGoogle Scholar
  39. 39.
    R.O. Jones, O. Gunnarsson, The density functional formalism, its applications and prospects. Rev. Mod. Phys. 61(3), 689 (1989)ADSCrossRefGoogle Scholar
  40. 40.
    J.P. Perdew, A. Zunger, Self-interaction correction to density-functional approximations for many-electron systems. Phys. Rev. B 23(10), 5048 (1981)ADSCrossRefGoogle Scholar
  41. 41.
    D.M. Ceperley, B.J. Alder, Ground state of the electron gas by a stochastic method. Phys. Rev. Lett. 45(7), 566 (1980)ADSCrossRefGoogle Scholar
  42. 42.
    M. Gell-Mann, K.A. Brueckner, Correlation energy of an electron gas at high density. Phys. Rev. 106(2), 364 (1957)MathSciNetADSzbMATHCrossRefGoogle Scholar
  43. 43.
    J.B. Krieger, Y. Li, G.J. Iafrate, Construction and application of an accurate local spin-polarized Kohn–Sham potential with integer discontinuity: exchange-only theory. Phys. Rev. A 45, 101 (1992)ADSCrossRefGoogle Scholar
  44. 44.
    O. Eriksson, J. M. Wills, M. Colarieti-Tosti, S. Lebgue, A. Grechnev. Many-body projector orbitals for electronic structure theory of strongly correlated electrons. Int. J. Quantum Chem. 105 (2) (2005)Google Scholar
  45. 45.
    V.I. Anisimov, J. Zaanen, O.K. Andersen, Band theory and Mott insulators: Hubbard U instead of Stoner I. Phys. Rev. B 44(3), 943 (1991)ADSCrossRefGoogle Scholar
  46. 46.
    V.I. Anisimov, I.V. Solovyev, M.A. Korotin, M.T. Czy zyk, G.A. Sawatzky, Density-functional theory and NiO photoemission spectra. Phys. Rev. B 48(23), 16929 (1993)ADSCrossRefGoogle Scholar
  47. 47.
    J.P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple. Phys. Rev. Lett. 77(18), 3865 (1996)ADSCrossRefGoogle Scholar
  48. 48.
    J. Kubler, K.H. Hock, J. Sticht, A.R. Williams, Density functional theory of non-collinear magnetism. J. Phys. F Metal Phys. 18(3), 469 (1988)ADSCrossRefGoogle Scholar
  49. 49.
    von U. Barth, L. Hedin, A local exchange-correlation potential for the spin polarized case. i. J. Phys. C Solid State Phys. 5(13), 1629 (1972)ADSCrossRefGoogle Scholar
  50. 50.
    G.L. Oliver, J.P. Perdew, Spin-density gradient expansion for the kinetic energy. Phys. Rev. A 20(2), 397 (1979)ADSCrossRefGoogle Scholar
  51. 51.
    V. Heine, The pseudopotential concept, vol. 24, Solid State Physics (Academic Press, New York, 1970), p. 1ffGoogle Scholar
  52. 52.
    J.C. Phillips, Energy-band interpolation scheme based on a pseudopotential. Phys. Rev. 112(3), 685 (1958)ADSCrossRefGoogle Scholar
  53. 53.
    J.C. Phillips, L. Kleinman, New method for calculating wave functions in crystals and molecules. Phys. Rev. 116(2), 287 (1959)ADSzbMATHCrossRefGoogle Scholar
  54. 54.
    C. Herring, A new method for calculating wave functions in crystals. Phys. Rev. 57(12), 1169 (1940)ADSzbMATHCrossRefGoogle Scholar
  55. 55.
    A.M. Rappe, K.M. Rabe, E. Kaxiras, J.D. Joannopoulos, Optimized pseudopotentials. Phys. Rev. B 41(2), 1227 (1990)ADSCrossRefGoogle Scholar
  56. 56.
    M. Fuchs, M. Scheffler, Ab initio pseudopotentials for electronic structure calculations of poly-atomic systems using density-functional theory. Comput. Phys. Commun. 119(1), 67 (1999)ADSzbMATHCrossRefGoogle Scholar
  57. 57.
    D.R. Hamann, M. Schlüter, C. Chiang, Norm-conserving pseudopotentials. Phys. Rev. Lett 43(20), 1494 (1979)ADSCrossRefGoogle Scholar
  58. 58.
    G.P. Kerker, Non-singular atomic pseudopotentials for solid-state applications. J. Phys. C 13(9), L189 (1980)ADSCrossRefGoogle Scholar
  59. 59.
    D.R. Hamann, Generalized norm-conserving pseudopotentials. Phys. Rev. B 41(2), 2980 (1989)ADSCrossRefGoogle Scholar
  60. 60.
    S.G. Louie, S. Froyen, M.L. Cohen, Nonlinear ionic pseudopotentials in spin-density-functional calculations. Phys. Rev. B 26(4), 1738 (1982)ADSCrossRefGoogle Scholar
  61. 61.
    C.-K. Skylaris, P.D. Haynes, A.A. Mostofi, M.C. Payne, Introducing ONETEP: linear-scaling density functional simulations on parallel computers. J. Chem. Phys. 122, 084119 (2005)ADSCrossRefGoogle Scholar
  62. 62.
    P.D. Haynes, C.-K. Skylaris, A.A. Mostofi, M.C. Payne, Elimination of basis set superposition error in linear-scaling density-functional calculations with local orbitals optimised in situ. Chem. Phys. Lett. 422, 345 (2006)ADSCrossRefGoogle Scholar
  63. 63.
    N.W. Ashcroft, N.D. Mermin, Solid State Physics (Harcourt Brace College Publishers, Fort Worth, 1976)Google Scholar
  64. 64.
    C. Kittel, Introduction to Solid State Physics (Wiley, New York, 2005)Google Scholar
  65. 65.
    L.P. Bouckaert, R. Smoluchowski, E. Wigner, Theory of Brillouin zones and symmetry properties of wave functions in crystals. Phys. Rev. 50(1), 58 (1936)ADSzbMATHCrossRefGoogle Scholar
  66. 66.
    D.J. Chadi, M.L. Cohen, Special points in the Brillouin zone. Phys. Rev. B 8(12), 5747 (1973)MathSciNetADSCrossRefGoogle Scholar
  67. 67.
    H.J. Monkhorst, J.D. Pack, Special points for Brillouin-zone integrations. Phys. Rev. B 13(12), 5188 (1976)MathSciNetADSCrossRefGoogle Scholar
  68. 68.
    G. Makov, M.C. Payne, Periodic boundary conditions in ab initio calculations. Phys. Rev. B 51(7), 4014 (1995)ADSCrossRefGoogle Scholar
  69. 69.
    E. Hernández, M.J. Gillan, Self-consistent first-principles technique with linear scaling. Phys. Rev. B 51(15), 10157 (1995)ADSCrossRefGoogle Scholar
  70. 70.
    A.A. Mostofi, P.D. Haynes, C.-K. Skylaris, M.C. Payne, Preconditioned interative minimisation for linear-scaling electronic structure calculations. J. Chem. Phys. 119, 8842 (2003)ADSCrossRefGoogle Scholar
  71. 71.
    D. Baye, P.-H. Heenen, Generalised meshes for quantum mechanical problems. J. Phys. A Math. Gen. 19, 2041 (1986)MathSciNetADSzbMATHCrossRefGoogle Scholar
  72. 72.
    S. Goedecker, Linear scaling electronic structure methods. Rev. Mod. Phys. 71(4), 1085 (1999)ADSCrossRefGoogle Scholar
  73. 73.
    G. Galli, Linear scaling methods for electronic structure calculations and quantum molecular dynamics simulations. Curr. Opin. Solid State Mater. Sci. 1(6), 864 (1996)ADSCrossRefGoogle Scholar
  74. 74.
    P.D. Haynes, C.-K. Skylaris, A.A. Mostofi, M.C. Payne, ONETEP: linear-scaling density-functional theory with local orbitals and plane waves. Phys. Stat. Solidi (b) 243(11), 2489 (2006)ADSCrossRefGoogle Scholar
  75. 75.
    J.M. Soler, E. Artacho, J.D. Gale, A. Garcia, J. Junquera, P. Ordejon, D. Sanchez-Portal, The SIESTA method for ab initio order-N materials simulation. J. Phys. Condens. Matter 14, 2745 (2002)ADSCrossRefGoogle Scholar
  76. 76.
    D.R. Bowler, T. Miyazaki, M.J. Gillan, Recent progress in linear scaling ab initio electronic structure techniques. J. Phys. Condens. Matter 14(11), 2781 (2002)ADSCrossRefGoogle Scholar
  77. 77.
    M.J. Han, T. Ozaki, J. Yu, O(N) LDA+U electronic structure calculation method based on the nonorthogonal pseudoatomic orbital basis. Phys. Rev. B 73(4), 045110 (2006)ADSCrossRefGoogle Scholar
  78. 78.
    F. Mauri, G. Galli, Electronic-structure calculations and molecular-dynamics simulations with linear system-size scaling. Phys. Rev. B 50(7), 4316 (1994)ADSCrossRefGoogle Scholar
  79. 79.
    W. Kohn, Density functional and density matrix method scaling linearly with the number of atoms. Phys. Rev. Lett. 76(17), 3168 (1996)ADSCrossRefGoogle Scholar
  80. 80.
    R. McWeeny, Some recent advances in density matrix theory. Rev. Mod. Phys. 32(2), 335 (1960)MathSciNetADSCrossRefGoogle Scholar
  81. 81.
    P.D. Haynes, C.-K. Skylaris, A.A. Mostofi, M.C. Payne, Density kernel optimization in the ONETEP code. J. Phys. Condens. Matter 20(29), 294207 (2008)CrossRefGoogle Scholar
  82. 82.
    X.-P. Li, R.W. Nunes, D. Vanderbilt, Density-matrix electronic-structure method with linear system-size scaling. Phys. Rev. B 47(16), 10891 (1993)ADSCrossRefGoogle Scholar
  83. 83.
    R.W. Nunes, D. Vanderbilt, Generalization of the density-matrix method to a nonorthogonal basis. Phys. Rev. B 50(23), 17611 (1994)ADSCrossRefGoogle Scholar
  84. 84.
    M.S. Daw, Model for energetics of solids based on the density matrix. Phys. Rev. B 47(16), 10895 (1993)ADSCrossRefGoogle Scholar
  85. 85.
    J.F. Janak, Proof that \({\frac{\partial e} {\partial n_{i} }}=\epsilon_{i}\) in density-functional theory. Phys. Rev. B 18(12), 7165 (1978)Google Scholar
  86. 86.
    G.H. Wannier, The structure of electronic excitation levels in insulating crystals. Phys. Rev. 52(3), 191 (1937)ADSzbMATHCrossRefGoogle Scholar
  87. 87.
    J. des Cloizeaux, Energy bands and projection operators in a crystal: analytic and asymptotic properties. Phys. Rev. 135(3A), A685 (1964)MathSciNetCrossRefGoogle Scholar
  88. 88.
    L. He, D. Vanderbilt, Exponential decay properties of Wannier functions and related quantities. Phys. Rev. Lett. 86, 5341 (2001)ADSCrossRefGoogle Scholar
  89. 89.
    C. Brouder, G. Panati, M. Calandra, C. Mourougane, N. Marzari, Exponential localization of Wannier functions in insulators. Physical Review Letters 98(4), 046402 (2007)ADSCrossRefGoogle Scholar
  90. 90.
    G. Galli, M. Parrinello, Large scale electronic structure calculations. Phys. Rev. Lett. 69(24), 3547 (1992)ADSCrossRefGoogle Scholar
  91. 91.
    N. Hine, P. Haynes, A. Mostofi, C.-K. Skylaris, M. Payne, Linear-scaling density-functional theory with tens of thousands of atoms: expanding the scope and scale of calculations with onetep. Comput. Phys. Commun. 180(7), 1041 (2009)ADSzbMATHCrossRefGoogle Scholar
  92. 92.
    A. Einstein, Die grundlage der allgemeinen relativitätstheorie. Annalen der Physik 354, 769 (1916)ADSCrossRefGoogle Scholar
  93. 93.
    N. Marzari, D. Vanderbilt, M.C. Payne, Ensemble density-functional theory for ab initio molecular dynamics of metals and finite-temperature insulators. Phys. Rev. Lett. 79(7), 1337 (1997)ADSCrossRefGoogle Scholar
  94. 94.
    F. Mauri, G. Galli, R. Car, Orbital formulation for electronic-structure calculations with linear system-size scaling. Phys. Rev. B 47(15), 9973 (1993)ADSCrossRefGoogle Scholar
  95. 95.
    J. Kim, F. Mauri, G. Galli, Total-energy global optimizations using nonorthogonal localized orbitals. Phys. Rev. B 52(3), 1640 (1995)ADSCrossRefGoogle Scholar
  96. 96.
    P. Ordejón, D.A. Drabold, R.M. Martin, M.P. Grumbach, Linear system-size scaling methods for electronic-structure calculations. Phys. Rev. B 51(3), 1456 (1995)ADSCrossRefGoogle Scholar
  97. 97.
    C.-K. Skylaris, P.D. Haynes, A.A. Mostofi, M.C. Payne, Recent progress in linear-scaling density functional calculations with plane waves and pseudopotentials: the ONETEP code. J. Phys. Condens. Matter 20, 064209 (2008)ADSCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1. Cavendish LaboratoryTCM Group, University of CambridgeCambridgeUK

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