Self-Consistent Orthogonalized-PLane-Wave Calculations

  • R. N. Euwema
  • D. J. Stukel
  • T. C. Collins
Part of the The IBM Research Symposia Series book series (IRSS)


A natural way to describe mathematically a valence wave function in a periodic crystal is in terms of a Fourier series. However, convergence of such a plane-wave series is very poor because thousands of plane-wave terms are required to simulate the rapid oscillations of the wave function close to the atomic nuclei. To improve convergence, Herring1 proposed the Orthogonalized-Plane-Wave (OPW) method in which the plane-wave terms making up the Fourier series are orthogonalized to all the tightly-bound, core-wave functions. This orthogonalization vastly improves the convergence because the core functions present in the valence wave function expansion correctly simulate the behavior of the valence wave function in the core regions; while the plane-wave terms adequately describe the overall crystalline behavior of the function.


Wave Function Brillouin Zone Core State Atom Site Energy Band Structure 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    C. Herring, “A New Method for Calculating Wave Functions in Crystals”, Phys. Rev. 57, 1169 (1940)CrossRefGoogle Scholar
  2. 2.
    F. Herman, S. Skillman, “Theoretical Investigation of the Energy Band Structure of Semiconductors”, Proc. Int’l Conf. on Semiconductor Phys. Prague, 1960 ( Publishing House of the Czechoslovak Acad. Sci. )Google Scholar
  3. 3.
    F. Herman, R. L. Kortum, C. D. Kuglin, J. P. Van Dyke, and S. Skillman, “Electronic Structure of Tetrahedrally-Bonded Semiconductors: Empirically Adjusted OPW Energy Band Calculations”, Methods in Computational Physics. B. Alder, S. Fernbach, M. Rotenberg, eds. (Academic Press, New York, 1968), Vol. 8, pp 193.Google Scholar
  4. 4.
    R. N. Euwema, T. C. Collins, D. Shankland, and J. S. DeWitt, “Convergence Study of a Self-Consistent Orthogonalized-Plane-Wave Band Calculation for Hexagonal CdS”, Phys. Rev, 162, 710 (1967)CrossRefGoogle Scholar
  5. 5.
    D. J. Stukel, R. N, Euvema, “Self-Consistent Orthogonalized-Plane-Wave Energy-Band Study of Silicon”, Phys. Rev. B1, 1635 (1970)Google Scholar
  6. 6.
    D. J. Stukel, “Self-Consistent Energy Bands and Related Properties of Boron Phosphide”, Phys, Rev. B, 15 April (1970)Google Scholar
  7. 7.
    D. J. Stukel, “Electronic Structure and Optical Spectrum of Boron Arsenide”, Phys. Rev. B, 15 June (1970)Google Scholar
  8. 8.
    D. J. Stukel, R. N. Euvema, “Electronic Band Structure and Related Properties of Cubic MP”, Phys. Rev. 186, 154 (1969)CrossRefGoogle Scholar
  9. 9m.
    D. J. Stukel, R. N. Euwema, “Energy Band Structure of Aluminum Arsenide”, Phys. Rev. 188, 1193 (1969)CrossRefGoogle Scholar
  10. 10.
    T. C. Collins, D. J. Stukel, R. N. Euwema, “Self-Consistent Orthogonali zed-Plane-Wave Band Calculation on GaAs”, Phys. Rev. B1, 724 (1970)CrossRefGoogle Scholar
  11. 11.
    D. J. Stukel, “Energy Band Structure of BeS, BeSe and BeTe”, Phys. Rev. B (1970)Google Scholar
  12. 12.
    D. J. Stukel, R. N. Euwema, T. C. Collins, F. Herman, R. L. Kortum, “Self-Consistent Orthogonalized-Plane-Wave Energy- Band Models for Cubic ZnS, ZnSe, CdS, and CdSe”, Phys. Rev. 179, 740 (1969)CrossRefGoogle Scholar
  13. 13.
    R. A. Deegan, W. D. Twose, “Modifications to the Orthogenalized-Plane-Wave Method for Use in Transition Metals: Electronic Band Structure of Niobium”, Phys. Rev. 164 993 (1967) F. A. Butler, F. K. Bloom, E. Brown, “Modifications of the Orthogonali zed-Plane-Wave Method Applied to Copper”, Phys. Rev. 180, 744 (1969)CrossRefGoogle Scholar
  14. 14.
    T. O. Woodruff, “The Orthogonali zed Plane-Wave Method”, Solid State Physics 4 367, edited by F. Seitz and D. Turnbull (Academic Press, Inc., New York, 1957)Google Scholar
  15. 15.
    B. Wendroff, Theoretical Numerical Analysis, (Academic Press, 1966) J. Wilkinson, Algebraic Eigenvalue Problem, (Oxford University Press, 1966 )Google Scholar
  16. l6. J. H. Wilkinson, “Householder’s Method for the Solution of the Algebraic Eigenproblem”, Computer Journal 3, 23 (i960)Google Scholar
  17. 17.
    R. N. Euwema, D. J. Stukel, “OPW Convergence of Some Tetrahedral Semiconductors”, Phys. Rev. B2, 15 June (1970)Google Scholar
  18. 18.
    J, Callaway, “Orthogonalized Plane Wave Method”, Phys. Rev, 21. 933 (1955) V. Heine, “The Band Structure of Aluminum I, II, III”, Proc. Roy, Soc. A240 340, 354, 363 F. Herman, “Calculation of Energy Band Structure of Diamond and Ge Crystals by Method of Orthogonalized Plane Waves”,. Phys. Rev, 93, 1214 (1955)CrossRefGoogle Scholar
  19. 19.
    J. C. Slater, “A Simplification of the Hartree-Fock Method”, Phys. Rev. 8l, 385 (1951)CrossRefGoogle Scholar
  20. 20.
    W. Kohn, L. J. Sham, “Self-Consistent Equations Including Exchange and Correlation Effects”, Phys. Rev. 140 1133 (1965)CrossRefGoogle Scholar
  21. 21.
    R. Gaspar, “Uber eine Approximation des Hartree-Fockschen Potentials durch eine Universelle Potentialfunktion”, Acta. Phys. Acad. Sci. Hung. 3, 263 (1954)CrossRefGoogle Scholar
  22. 22.
    D. A. Liberman, “Exchange Potential for Electrons in Atoms and Solids”, Phys. Rev. 171, 1 (1968) L. J. Sham, W. Kohn, “One-Particle Properties of an Inhomogenous Interacting Electron Gas”, Phys. Rev. 145 561 (1966)CrossRefGoogle Scholar
  23. 23.
    D. J. Stukel, R. N. Euwema, T. C. Collins, V. H. Smith, “Exchange Study of Atomic Kiypton and Tetrahedral Semiconductors”, Phys. Rev. B1, 779 (1970)CrossRefGoogle Scholar
  24. 24.
    J. C. Slater, “Present Status of the Xα Statistical Exchange”, M.I.T. Semi-Annual Progress Report No. 71, p 1 (1969)Google Scholar
  25. 25.
    P. M. Raccah, R. N. Euwema, D. J. Stukel, T. C. Collins, “Comparison of Theoretical and Experimental Charge Densities for C. Si, Ge, and ZnSe”, Phys. Rev. B1, 756 (1970)CrossRefGoogle Scholar
  26. 26.
    J. C. Slater, Quantum Theory of Molecules and Solids, Vol. II, (McGraw Hill Book Co., 1965 )Google Scholar
  27. 27.
    G. Wepfer, T. C. Collins, R. N. Euvema, D. J. Stukel, “Symmetrization Techniques in Relativistic OPW Energy Band Calculations”, this volumeGoogle Scholar
  28. 28.
    We are grateful to G. Wepfer and J. Van Dyke for helpful discussions on this pointGoogle Scholar
  29. 29.
    G. Liebfried, Encyclopedia of Physics, Vol. 7, Part 1, p 132 ( Springer-Verlag, Berlin, 1955 )Google Scholar
  30. 30.
    R. I. Euvema, D. J. Stukelf T. C. Collins, J. S. DeWitt, D. G. Shankland, “Crystalline Interpolation with Applications to Brillouin-Zone Averages and Energy-Band Interpolation”, Phys. Rev, 178, 1419 (1969)CrossRefGoogle Scholar
  31. 31.
    D. G. Shankland, “Interpolation and Fourier Trans format ion with Functions Having Specified Smoothness Properties”, Boeing Document D6-2973G, The Boeing Company, Renton, Wash, D. G. Shankland, “k-Space Interpolation of Functions of Arbitrary Smoothness”, this volumeGoogle Scholar
  32. 32.
    F. Herman, S, Skillman, Atomic Structure Calculations, (Prentice-Hall, Inc. 1963)Google Scholar

Copyright information

© Plenum Press, New York 1971

Authors and Affiliations

  • R. N. Euwema
    • 1
  • D. J. Stukel
    • 1
  • T. C. Collins
    • 1
  1. 1.Aerospace Research LaboratoriesWright-Patterson Air Force BaseOhioUSA

Personalised recommendations