Skip to main content
SpringerLink
Log in
Book cover

Relativistic Quantum Theory of Atoms and Molecules pp 259–324Cite as

Analysis and approximation of Dirac Hamiltonians

  • Chapter

  • 1967 Accesses

Part of the book series: Springer Series on Atomic, Optical, and Plasma Physics ((SSAOPP,volume 40))

Abstract

Computational problems in atomic and molecular structure and processes require understanding of the mathematics of Dirac operators and of methods for constructing numerical solutions of the Dirac equation. Whilst simple problems such as the structure of hydrogenic atoms can be solved analytically, the majority of applications to many-electron systems can only be solved approximately. Some progress can be made using methods of functional analysis and symmetry properties, but the interpretation of experiments often needs highprecision numerical predictions. Meeting these demands requires cost-effective and reliable algorithms for constructing solutions of the Dirac equation.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   259.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   329.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Titchmarsh E C 1948 Introduction to the Theory of Fourier Integrals (2nd. ed.) (Oxford: Clarendon Press).

    Google Scholar 

  2. Kato T 1976 Perturbation Theory for Linear Operators (Berlin: Springer-Verlag).

    MATH  Google Scholar 

  3. Reed M and Simon B 1972–1979 Methods of Modern Mathematical Physics (4 vols.) (New York: Academic Press).

    MATH  Google Scholar 

  4. Kalf, H, Schmincke, U-W, Walter, J and Wüst, R 1975 On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials, in Spectral theory and differential equations, (Lecture Notes in Mathematics No. 448) ed. W N Everitt, pp. 182–226. (Berlin: Springer-Verlag).

    Chapter  Google Scholar 

  5. Greiner W, Müller B and Rafelski J 1985 Quantum Electrodynamics of Strong Fields (Berlin: Springer-Verlag).

    Google Scholar 

  6. Richtmyer R D 1978 Methods of Advanced Mathematical Physics (2 vols) (New York: Springer-Verlag).

    Google Scholar 

  7. Fischer C F, Brage T and Jönsson P 1997 Computational Atomic Structure. An MCHF approach (Bristol and Philadelphia: Institute of Physics).

    Google Scholar 

  8. Pauling L and Wilson Jr E B 1985 Introduction to Quantum Mechanics with applications in chemistry (Reprint of the 1935 edition) (New York: Dover).

    Google Scholar 

  9. Dirac P A M 1958 The Principles of Quantum Mechanics 4th ed. (Oxford: Clarendon Press).

    MATH  Google Scholar 

  10. Kellner G W 1927 Z. Phys. 44 91, 110.

    Article  ADS  Google Scholar 

  11. Hylleraas E A 1928 Z. Phys. 48, 469.

    Article  ADS  Google Scholar 

  12. Hylleraas E A 1929 Z. Phys. 54, 347.

    Article  ADS  Google Scholar 

  13. Hylleraas E A 1930 Z. Phys. 65, 209.

    Article  ADS  Google Scholar 

  14. Hartree D R 1957 The calculation of atomic structures (New York: John Wiley).

    MATH  Google Scholar 

  15. Fischer C F 1977 The Hartree-Fock Method for Atoms (New York: John Wiley).

    Google Scholar 

  16. Swirles B 1935 Proc Roy Soc (Lond) A 152 625; — 1936 Proc Roy Soc (Lond) A 157 680.

    Article  MATH  ADS  Google Scholar 

  17. Grant I P 1961 Proc Roy Soc (Lond) A 262 555; — 1965 Proc Phys Soc 86 523.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. Grant I P 1970 Adv. Phys. 19 747.

    Article  ADS  Google Scholar 

  19. Joachain C J (1983) Quantum collision theory (3rd edition) (Amsterdam: North-Holland Publishing Company).

    Google Scholar 

  20. Synek M 1964 Phys. Rev. A 136 1552.

    Article  ADS  Google Scholar 

  21. Kim Y K 1967 Phys. Rev. A 154 17.

    Article  ADS  Google Scholar 

  22. Kim Y K 1967 ibid. 155 190.

    Google Scholar 

  23. Kagawa T 1975 Phys. Rev. A 12, 2245.

    Article  ADS  Google Scholar 

  24. Kagawa T1980 Phys. Rev. A 22, 2340.

    Google Scholar 

  25. Schwarz W H E and Wallmeier H 1982 Mol. Phys. 46, 1045.

    Article  ADS  Google Scholar 

  26. Wallmeier H and Kutzelnigg W 1981 Chem. Phys. Lett. 78, 341.

    Article  ADS  Google Scholar 

  27. Schwarz W H E and Wechsel-Trakowski E 1982 Chem. Phys. Lett. 85, 94.

    Article  ADS  Google Scholar 

  28. Mark F and Rosicky F 1980 Chem. Phys. Lett. 74, 562.

    Article  ADS  MathSciNet  Google Scholar 

  29. Mark F and Schwarz W H E 1982 Phys. Rev. Lett. 48, 673.

    Article  ADS  Google Scholar 

  30. Kutzelnigg W 1984 Int. J. Quant. Chem. 25, 107.

    Article  Google Scholar 

  31. Klahn B and Bingel W A 1977 Theoret. Chim. Acta (Berl.) 44 9.

    Article  MathSciNet  Google Scholar 

  32. Klahn B and Bingel W A 1977 Theoret. Chim. Acta (Berl.) 44 27.

    Article  MathSciNet  Google Scholar 

  33. Klahn B and Morgan III J D 1984 J. Chem. Phys. 81 410.

    Article  ADS  Google Scholar 

  34. Mikhlin S G 1964 Variational Methods in Mathematical Physics (transl. T. Boddington) (Oxford: Pergamon Press).

    MATH  Google Scholar 

  35. Bonitz G 1971 Zum Ritzschen Verfahren. (Berlin: VEB Deutscher Verlag der Wissenschaften).

    Google Scholar 

  36. Kato T 1951 Trans. Am. Math. Soc. 70, 195.

    Article  MATH  Google Scholar 

  37. Eckart C 1930 Phys. Rev. 36 878.

    Article  ADS  Google Scholar 

  38. Löwdin P O 1959 Adv. Chem. Phys. 2 207.

    Article  Google Scholar 

  39. Grant I P 1982 Phys. Rev. A 25, 1230.

    Article  ADS  Google Scholar 

  40. Grant I P 1986 J. Phys. B: Atom. molec. phys. 19, 3187.

    Article  ADS  MathSciNet  Google Scholar 

  41. Grant I P and Quiney H M 2000 Phys. Rev. A 62, 022508.

    Article  ADS  Google Scholar 

  42. Stakgold 1979 Green’s functions and Boundary Value Problems (New York: John Wiley & Sons).

    MATH  Google Scholar 

  43. Dyall K G, Grant I P and Wilson S 1984 J. Phys. B 17, L45.

    Article  ADS  MathSciNet  Google Scholar 

  44. Dyall K G, Grant I P and Wilson S 1984 J. Phys. B 17, 493.

    Article  ADS  MathSciNet  Google Scholar 

  45. Dyall K G, Grant I P and Wilson S 1984 J. Phys. B 17, 1201.

    Article  ADS  MathSciNet  Google Scholar 

  46. Lee Y S and McLean A D 1982 J. Chem. Phys. 76, 735.

    Article  ADS  Google Scholar 

  47. Quiney H M, Grant I P and Wilson S 1989, in [48], p. 307.

    Google Scholar 

  48. Kaldor U (ed.) 1989 Many-Body Methods in Quantum Chemistry (Lecture Notes in Chemistry 52) (Berlin: Springer-Verlag).

    Google Scholar 

  49. Grant I P and Quiney H M 1988 Adv. At. Mol. Phys. 23, 37.

    Article  Google Scholar 

  50. Grant I P 1989 in Relativistic, Quantum Electrodynamic and Weak Interaction Effects in Atoms (ed. W R Johnson, P J Mohr and J Sucher), p. 235.

    Google Scholar 

  51. Drake G W F ed 2005 Springer Handbook of Atomic, Molecular and Optical Physics Second edn (New York: Springer-Verlag).

    Google Scholar 

  52. Szmytkowski R 1997 J. Phys. B: At. Mol. Opt. Phys. 30, 825.

    Article  ADS  MathSciNet  Google Scholar 

  53. Rotenberg M 1962 Ann. Phys. (N. Y.) 19, 262.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  54. Rotenberg M 1970 Adv. Atom. Molec. Phys. 6, 233.

    Article  Google Scholar 

  55. Abramowitz M and Stegun I A 1970 Handbook of Mathematical Functions (New York: Dover).

    Google Scholar 

  56. Klahn B and Bingel W A 1977 Theoret. Chim. Acta (Berl.) 44, 9, 27.

    Article  MathSciNet  Google Scholar 

  57. Smith B T et al. 1976 Matrix Eigensystem Routines-EISPACK Guide, (2e) (Lecture Notes in Computer Science, Vol. 6) (New York: Springer-Verlag).

    MATH  Google Scholar 

  58. Press W H, Teukolsky S A, Vetterling W T and Flannery B P 1992 Numerical Recipes in FORTRAN (2e) (Cambridge: University Press).

    MATH  Google Scholar 

  59. Quiney H M 1987 Finite basis set studies of the Dirac equation (Unpublished D Phil Dissertation, Oxford University).

    Google Scholar 

  60. Fischer C F 1972 Atomic Data 4 302; — 1973 At Data Nucl Data Tab 12 87.

    Article  ADS  Google Scholar 

  61. Mayers D F 1957 Proc. Roy. Soc. A 241 93.

    Article  ADS  MathSciNet  Google Scholar 

  62. Mayers D F and O’Brien F 1968 J. Phys. B 1, 145.

    Article  ADS  Google Scholar 

  63. Desclaux J P, Mayers D F and O’Brien F 1971 J. Phys. B: At. Mol. Phys. 4 631.

    Article  ADS  Google Scholar 

  64. Cowan R D 1981 The Theory of Atomic Structure and Spectra (Berkeley: University of California Press).

    Google Scholar 

  65. Wilson S 1984 Electron correlation in molecules (Oxford: Clarendon Press).

    Google Scholar 

  66. Gladwell I and Wait R 1979 A survey of numerical methods for partial dfferential equations. (Oxford: Clarendon Press).

    Google Scholar 

  67. Shore B 1974 J. Phys. B: At. Mol. Phys. 7, 2502.

    Article  ADS  Google Scholar 

  68. Altenberger-Siczek A and Gilbert T L 1976 J. Chem. Phys. 64, 432.

    Article  ADS  Google Scholar 

  69. Bottcher C and Strayer M R 1987 Ann. Phys. (N.Y.) 175, 64.

    Article  ADS  Google Scholar 

  70. Johnson W R and Sapirstein J 1986 Phys. Rev. Lett. 57, 1126.

    Article  ADS  Google Scholar 

  71. Johnson W R, Idrees M and Sapirstein J 1987 Phys. Rev. A 35, 3218.

    Article  ADS  Google Scholar 

  72. Johnson W R, Blundell S A and Sapirstein J 1988 Phys. Rev. A 37 307.

    Article  ADS  MathSciNet  Google Scholar 

  73. Froese Fischer C and Idrees M 1989 Comput. Phys. 3, 53.

    ADS  Google Scholar 

  74. Froese Fischer C and Idrees M 1990 J. Phys. B: At. Mol. Phys. 23, 679.

    Article  ADS  Google Scholar 

  75. Froese Fischer C and Guo W 1990 J. Comput. Phys. 90, 486.

    Article  MATH  ADS  Google Scholar 

  76. Froese Fischer C, Guo W and Shen Z1992 Int. J. Quant. Chem. 42, 849.

    Article  Google Scholar 

  77. Sapirstein J and Johnson W R 1996 J. Phys. B: At. Mol. Phys. 29, 5213.

    Article  ADS  Google Scholar 

  78. de Boor C 1978 A practical guide to splines (New York: Springer Verlag).

    MATH  Google Scholar 

  79. Powell M J D 1981 Approximation theory and methods (Cambridge: Cambridge University Press).

    MATH  Google Scholar 

  80. Davies A J 1980 The finite element method: a first approach (Oxford: Clarendon Press).

    MATH  Google Scholar 

  81. Chodos A, Jaffe R L, Johnson K, Thorn C B and Weisskopf V F 1974 Phys. Rev. D 9 3471.

    Article  ADS  MathSciNet  Google Scholar 

Download references

Editor information

Editors and Affiliations

  1. Mathematical Institute, Oxford University, Oxford, UK

    I. P. Grant

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

(2007). Analysis and approximation of Dirac Hamiltonians. In: Grant, I.P. (eds) Relativistic Quantum Theory of Atoms and Molecules. Springer Series on Atomic, Optical, and Plasma Physics, vol 40. Springer, New York, NY. https://doi.org/10.1007/978-0-387-35069-1_5

Download citation

Publish with us

Policies and ethics

Search

Navigation