Electronic Structure Methods: Augmented Waves, Pseudopotentials and The Projector Augmented Wave Method

  • Peter E. Blöchl
  • Johannes Kästner
  • Clemens J. Först

Abstract

The main goal of electronic structure methods is to solve the Schrödinger equation for the electrons in a molecule or solid, to evaluate the resulting total energies, forces, response functions and other quantities of interest. In this paper we describe the basic ideas behind the main electronic structure methods such as the pseudopotential and the augmented wave methods and provide selected pointers to contributions that are relevant for a beginner. We give particular emphasis to the projector augmented wave (PAW) method developed by one of us, an electronic structure method for ab initio molecular dynamics with full wavefunctions. We feel that it allows best to show the common conceptional basis of the most widespread electronic structure methods in materials science.

Keywords

Lithium Boron Catalysis Chlorine Assure 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    P. Hohenberg and W. Kohn, “Inhomogeneous electron gas,” Phys. Rev., 136, B864, 1964.CrossRefMathSciNetADSGoogle Scholar
  2. [2]
    W. Kohn and L.J. Sham, “Self-consistent equations including exchange and correlation effects,” Phys. Rev., 140, A1133, 1965.CrossRefMathSciNetADSGoogle Scholar
  3. [3]
    R.G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules, Oxford University Press, Oxford, 1989.Google Scholar
  4. [4]
    P.E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, 50, 17953, 1994.CrossRefADSGoogle Scholar
  5. [5]
    J.C. Slater, “Wave functions in a periodic potential,” Phys. Rev., 51, 846, 1937.MATHCrossRefADSGoogle Scholar
  6. [6]
    J. Korringa, “On the calculation of the energy of a Bloch wave in a metal,” Physica (Utrecht), 13, 392, 1947.CrossRefMathSciNetADSGoogle Scholar
  7. [7]
    W. Kohn and J. Rostocker, “Solution of the schrödinger equation in periodic lattices with an application to metallic lithium,” Phys. Rev., 94, 1111, 1954.MATHCrossRefADSGoogle Scholar
  8. [8]
    O.K. Andersen, “Linear methods in band theory,” Phys. Rev. B, 12, 3060, 1975.CrossRefADSGoogle Scholar
  9. [9]
    H. Krakauer, M. Posternak, and A.J. Freeman, “Linearized augmented plane-wave method for the electronic band structure of thin films,” Phys. Rev. B, 19, 1706, 1979.CrossRefADSGoogle Scholar
  10. [10]
    S. Singh, Planewaves, Pseudopotentials and the LAPW method, Kluwer Academic, Dordrecht, 1994.Google Scholar
  11. [11]
    J.M. Soler and A.R. Williams, “Simple formula for the atomic forces in the augmented-plane-wave method,” Phys. Rev. B, 40, 1560, 1989.CrossRefADSGoogle Scholar
  12. [12]
    D. Singh, “Ground-state properties of lanthanum: treatment of extended-core states,” Phys. Rev. B, 43, 6388, 1991.CrossRefADSGoogle Scholar
  13. [13]
    E. Sjöstedt, L. Nordström, and DJ. Singh, “An alternative way of linearizing the augmented plane-wave method,” Solid State Commun., 114, 15, 2000.CrossRefADSGoogle Scholar
  14. [14]
    G.K.H. Madsen, P. Blaha, K. Schwarz, E. Sjöstedt, and L. Nordström, “Efficient linearization of the augmented plane-wave method,” Phys. Rev. B, 64, 195134, 2001.CrossRefADSGoogle Scholar
  15. [15]
    H.L. Skriver, The LMTO Method, Springer, New York, 1984.Google Scholar
  16. [16]
    O.K. Andersen and O. Jepsen, “Explicit, first-principles tight-binding theory,” Phys. Rev. Lett., 53, 2571, 1984.CrossRefADSGoogle Scholar
  17. [17]
    O.K. Andersen, T. Saha-Dasgupta, and S. Ezhof, “Third-generation muffin-tin orbitals,” Bull. Mater. Sci., 26, 19, 2003.CrossRefGoogle Scholar
  18. [18]
    K. Held, I.A. Nekrasov, G. Keller, V. Eyert, N. Blümer, A.K. McMahan, R.T. Scalettar, T. Pruschke, V.I. Anisimov, and D. Vollhardt, “The LDA+DMFT approach to materials with strong electronic correlations,” In: J. Grotendorst, D. Marx, and A. Muramatsu (eds.) Quantum Simulations of Complex Many-Body Systems: From Theory to Algorithms, Lecture Notes, vol. 10 NIC Series. John von Neumann Institute for Computing, Jülich, p. 175, 2002.Google Scholar
  19. [19]
    C. Herring, “A new method for calculating wave functions in crystals,” Phys. Rev., 57, 1169, 1940.MATHCrossRefADSGoogle Scholar
  20. [20]
    J.C. Phillips and L. Kleinman, “New method for calculating wave functions in crystals and molecules,” Phys. Rev, 116, 287, 1959.MATHCrossRefADSGoogle Scholar
  21. [21]
    E. Antoncik, “Approximate formulation of the orthogonalized plane-wave method,” J. Phys. Chem. Solids, 10, 314, 1959.CrossRefADSGoogle Scholar
  22. [22]
    D.R. Hamann, M. Schlüter, and C. Chiang, “Norm-conserving pseudopotentials,” Phys. Rev. Lett., 43, 1494, 1979.CrossRefADSGoogle Scholar
  23. [23]
    A. Zunger and M. Cohen, “First-principles nonlocal-pseudopotential approach in the density-functional formalism: development and application to atoms,” Phys. Rev. B, 18, 5449, 1978.CrossRefADSGoogle Scholar
  24. [24]
    G.P. Kerker, “Non-singular atomic pseudopotentials for solid state applications,” J. Phys. C, 13, L189, 1980.CrossRefADSGoogle Scholar
  25. [25]
    G.B. Bachelet, D.R. Hamann, and M. Schlüter, “Pseudopotentials that work: from H to Pu,” Phys. Rev. B, 26, 4199, 1982.CrossRefADSGoogle Scholar
  26. [26]
    N. Troullier and J.L. Martins, “Efficient pseudopotentials for plane-wave calculations,” Phys. Rev. B, 43, 1993, 1991.CrossRefADSGoogle Scholar
  27. [27]
    J.S. Lin, A. Qteish, M.C. Payne, and V. Heine, “Optimized and transferable nonlocal separable ab initio pseudopotentials,” Phys. Rev. B, 47, 4174, 1993.CrossRefADSGoogle Scholar
  28. [28]
    M. Fuchs and M. Scheffler, “Ab initio pseudopotentials for electronic structure calculations of poly-atomic systems using density-functional theory,” Comput. Phys. Commun., 119, 67, 1999.MATHCrossRefADSGoogle Scholar
  29. [29]
    L. Kleinman and D.M. Bylander, “Efficacious form for model pseudopotentials,” Phys. Rev. Lett., 48, 1425, 1982.CrossRefADSGoogle Scholar
  30. [30]
    P.E. Blöchl, “Generalized separable potentials for electronic structure calculations,” Phys. Rev. B, 41, 5414, 1990.CrossRefADSGoogle Scholar
  31. [31]
    D. Vanderbilt, “Soft self-consistent pseudopotentials in a generalized eigenvalue formalism,” Phys. Rev. B, 41, 17892, 1990.CrossRefADSGoogle Scholar
  32. [32]
    S.G. Louie, S. Froyen, and M.L. Cohen, “Nonlinear ionic pseudopotentials in spindensity-functional calculations,” Phys. Rev. B, 26, 1738, 1982.CrossRefADSGoogle Scholar
  33. [33]
    D.R. Hamann, “Generalized norm-conserving pseudopotentials,” Phys. Rev. B, 40, 2980, 1989.CrossRefADSGoogle Scholar
  34. [34]
    K. Laasonen, A. Pasquarello, R. Car, C. Lee, and D. Vanderbilt, “Implementation of ultrasoft pseudopotentials in ab initio molecular dynamics,” Phys. Rev. B, 47, 110142, 1993.CrossRefGoogle Scholar
  35. [35]
    X. Gonze, R. Stumpf, and M. Scheffler, “Analysis of separable potentials,” Phys. Rev. B, 44, 8503, 1991.CrossRefADSGoogle Scholar
  36. [36]
    C.G. Van de Walle and P.E. Blöchl, “First-principles calculations of hyperfine parameters,” Phys. Rev. B, 47, 4244, 1993.CrossRefADSGoogle Scholar
  37. [37]
    M.C. Payne, M.P. Teter, D.C. Allan, T.A. Arias, and J.D. Joannopoulos, “Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate-gradients,” Rev. Mod. Phys., 64, 11045, 1992.CrossRefADSGoogle Scholar
  38. [38]
    R. Car and M. Parrinello, “Unified approach for molecular dynamics and density-functional theory,” Phys. Rev. Lett., 55, 2471, 1985.CrossRefADSGoogle Scholar
  39. [39]
    S. Nosé, “A unified formulation of the constant temperature molecular-dynamics methods,” Mol. Phys., 52, 255, 1984.CrossRefADSGoogle Scholar
  40. [40]
    Hoover, “Canonical dynamics: equilibrium phase-space distributions,” Phys. Rev. A, 31, 1695, 1985.Google Scholar
  41. [41]
    P.E. Blöchl and M. Parrinello, “Adiabaticity in first-principles molecular dynamics,” Phys. Rev. B, 45, 9413, 1992.CrossRefADSGoogle Scholar
  42. [42]
    P.E. Blöchl, “Second generation wave function thermostat for ab initio molecular dynamics,” Phys. Rev. B, 65, 1104303, 2002.CrossRefGoogle Scholar
  43. [43]
    S.C. Watson and E.A. Carter, “Spin-dependent pseudopotentials,” Phys. Rev. B, 58, R13309, 1998.CrossRefADSGoogle Scholar
  44. [44]
    G. Kresse and J. Joubert, “From ultrasoft pseudopotentials to the projector augmented-wave method,” Phys. Rev. B, 59, 1758, 1999.CrossRefADSGoogle Scholar
  45. [45]
    N.A.W. Holzwarth, G.E. Mathews, R.B. Dunning, A.R. Tackett, and Y. Zheng, “Comparison of the projector augmented-wave, pseudopotential, and linearized augmented-plane-wave formalisms for density-functional calculations of solids,” Phys. Rev. B, 55, 2005, 1997.CrossRefADSGoogle Scholar
  46. [46]
    A.R. Tackett, N.A.W. Holzwarth, and G.E. Matthews, “A projector augmented wave (PAW) code for electronic structure calculations. Part I: atompaw for generating atom-centered functions. A projector augmented wave (PAW) code for electronic structure calculations. Part II: pwpaw for periodic solids in a plane wave basis,” Comput. Phys. Commun., 135, 329–347, 2001. See also pp. 348–376.MATHCrossRefADSGoogle Scholar
  47. [47]
    M. Valiev and J.H. Weare, “The projector-augmented plane wave method applied to molecular bonding,” J. Phys. Chem. A, 103, 10588, 1999.CrossRefGoogle Scholar
  48. [48]
    P.E. Blöchl, “Electrostatic decoupling of periodic images of plane-wave-expanded densities and derived atomic point charges,” J. Chem. Phys., 103, 7422, 1995.CrossRefADSGoogle Scholar
  49. [49]
    T.K. Woo, P.M. Margl, P.E. Blöchl, and T. Ziegler, “A combined Car-Parrinello QM/MM implementation for ab initio molecular dynamics simulations of extended systems: application to transition metal catalysis,” J. Phys. Chem. B, 101, 7877, 1997.CrossRefGoogle Scholar
  50. [50]
    O. Bengone, M. Alouani, P.E. Blöchl, and J. Hugel, “Implementation of the projector augmented-wave LDA+U method: application to the electronic structure of NiO,” Phys. Rev. B, 62, 16392, 2000.CrossRefADSGoogle Scholar
  51. [51]
    B. Arnaud and M. Alouani, “All-electron projector-augmented-wave GW approximation: application to the electronic properties of semiconductors,” Phys. Rev. B., 62, 4464, 2000.CrossRefADSGoogle Scholar
  52. [52]
    D. Hobbs, G. Kresse, and J. Hafner, “Fully unconstrained noncollinear magnetism within the projector augmented-wave method,” Phys. Rev. B, 62, 11556, 2000.CrossRefADSGoogle Scholar
  53. [53]
    H.M. Petrilli, P.E. Blöchl, P. Blaha, and K. Schwarz, “Electric-field-gradient calculations using the projector augmented wave method,” Phys. Rev. B, 57, 14690, 1998.CrossRefADSGoogle Scholar
  54. [54]
    P.E. Blöchl, “First-principles calculations of defects in oxygen-deficient silica exposed to hydrogen,” Phys. Rev. B, 62, 6158, 2000.CrossRefADSGoogle Scholar
  55. [55]
    C.J. Pickard and F. Mauri, “All-electron magnetic response with pseudopotentials: NMR chemical shifts,” Phys. Rev. B., 63, 245101, 2001.CrossRefADSGoogle Scholar
  56. [56]
    F. Mauri, B.G. Pfrommer, and S.G. Louie, “Ab initio theory of NMR chemical shifts in solids and liquids,” Phys. Rev. Lett., 77, 5300, 1996.CrossRefADSGoogle Scholar
  57. [57]
    D.N. Jayawardane, CJ. Pickard, L.M. Brown, and M.C. Payne, “Cubic boron nitride: experimental and theoretical energy-loss near-edge structure,” Phys. Rev. B, 64, 115107, 2001.CrossRefADSGoogle Scholar
  58. [58]
    H. Kageshima and K. Shiraishi, “Momentum-matrix-element calculation using pseudopotentials,” Phys. Rev. B, 56, 14985, 1997.CrossRefADSGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Peter E. Blöchl
    • 1
  • Johannes Kästner
    • 1
  • Clemens J. Först
    • 1
  1. 1.Institute for Theoretical PhysicsClausthal University of TechnologyClausthal-ZellerfeldGermany

Personalised recommendations