Abstract

The mass-velocity and Darwin terms of the one-electron-atom Pauli equation have been added to the Hartree-Fock differential equations by using the HX formula to calculate a local central field potential for use in these terms. Introduction of the quantum number j is avoided by omitting the spin-orbit term of the Pauli equation. The major relativistic effects, both direct and indirect, are thereby incorporated into the wave functions, while allowing retention of the commonly used nonrelativistic formulation of energy level calculations. The improvement afforded in calculated total binding energies, excitation energies, spin-orbit parameters, and expectation values of rm is comparable with that provided by fully relativistic Dirac-Hartree-Fock calculations.

© 1976 Optical Society of America

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References

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  1. J. C. Slater, Quantum Theory of Atomic Structure (McGraw-Hill, New York, 1960); E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge U. P., Cambridge, England, 1935).
  2. I. P. Grant, “Relativistic calculation of atomic structures,” Adv. Phys. 19, 747–811 (1970).
    [CrossRef]
  3. J. B. Mann and J. T. Waber, “SCF relativistic Hartree-Fock calculations on the superheavy elements 118–131,” J. Chem. Phys. 53, 2397–2406 (1970); “Self-consistent relativistic Dirac-Hartree-Fock calculations of lanthanide atoms,” Atomic Data 5, 201–229 (1973).
    [CrossRef]
  4. J. P. Desclaux, D. F. Mayers, and F. O’Brien, “Relativistic atomic wave functions,” J. Phys. B 4, 631–642 (1971); J. P. Desclaux, “A multiconfiguration relativistic Dirac-Fock program,” Computer Phys. Commun. 9, 31–45 (1975).
    [CrossRef]
  5. D. A. Liberman, J. T. Waber, and D. T. Cromer, “Self-consistent-field Dirac-Slater wave functions for atoms and ions. I. Comparison with previous calculations,” Phys. Rev. 137, A27–A34 (1965); D. A. Liberman and D. T. Cromer, “Relativistic self-consistent field program for atoms and ions,” Computer Phys. Commun. 2, 107–113 (1971).
    [CrossRef]
  6. H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One-and Two-Electron Atoms (Springer-Verlag, Berlin, 1957), Sec. 13, especially Eq. (13.6) and the paragraph following Eq. (13.11).
    [CrossRef]
  7. R. D. Cowan, “Atomic self-consistent-field calculations using statistical approximations for exchange and correlation,” Phys. Rev. 163, 54–61 (1967). The HX potential consists of the Hartree potential plus a modification of Slater’s ρ1/3exchange potential. The originally suggested value k1 = 0.7 in the exchange portion has been decreased to 0.65 on the basis of subsequent experience.
    [CrossRef]
  8. We used an extensively modified early version of the program developed by C. Froese Fischer, “A multi-configuration Hartree-Fock program with improved stability,” Computer Phys. Commun. 4, 107–116 (1972); Computer Phys. Commun. 7, 236 (1974).
    [CrossRef]
  9. D. F. Mayers, “Relativistic self-consistent field calculation for mercury,” Proc. R. Soc. Lond. A 241, 93–109 (1957); R. G. Boyd, A. C. Larson, and J. T. Waber, “Indirect relativistic effect on the 5f electrons in uranium,” Phys. Rev. 129, 1629–1630 (1963).
    [CrossRef]
  10. D. C. Griffin, K. L. Andrew, and R. D. Cowan, “Theoretical calculations of the d-, f-, and g-electron transition series,” Phys. Rev. 177, 62–71 (1969); “Instabilities in the iterative solution of the Hartree-Fock equations for excited electrons,” Phys. Rev. A 3, 1233–1242 (1971).
    [CrossRef]
  11. R. D. Cowan and J. B. Mann, “The atomic structure of superheavy elements,” in Atomic Physics, 2, Proceedings of the Second International Conference on Atomic Physics (Plenum, London, 1971), pp. 215–226.
  12. R. Zalubas, “Present state of analysis of the first spectrum of thorium (Th i),” J. Opt. Soc. Am. 58, 1195–1199 (1968); W. C. Martin, L. Hagan, J. Reader, and J. Sugar, “Ground levels and ionization potentials for lanthanide and actinide atoms and ions,” J. Phys. Chem. Ref. Data 3, 771–779 (1974); R. Zalubas and C. H. Corliss, “Energy levels and classified lines in the second spectrum of thorium (Th ii),” J. Res. Natl. Bur. Stand. (U.S.) 78A, 163–246 (1974).
    [CrossRef]
  13. C. E. Moore, Atomic Energy Levels, Natl. Bur. Stds. Circ. No. 467 (U.S. GPO, Washington, D. C., 1958), Vol. III.
  14. V. Kaufman and J. Sugar, “Spectrum and energy levels of five-times ionized tantalum (Ta vi),” J. Opt. Soc. Am. 65, 302–309 (1975).
    [CrossRef]
  15. Yong-Ki Kim and J. P. Desclaux, “Relativistic f values for the resonance transitions of Li- and Be-like ions,” Phys. Rev. Lett. 36, 139–141 (1976).
    [CrossRef]
  16. L. Armstrong, W. R. Fielder, and Dong L. Lin, “Relativistic effects on transition probabilities in the Li and Be isoelectronic sequences,” Phys. Rev. A 14, 1114–1128 (1976).
    [CrossRef]
  17. On the basis of a variety of trial calculations, we have found it suitable to evaluate d at r equal to one-quarter of the largest radius for which the series expansion is to be employed.

1976 (2)

Yong-Ki Kim and J. P. Desclaux, “Relativistic f values for the resonance transitions of Li- and Be-like ions,” Phys. Rev. Lett. 36, 139–141 (1976).
[CrossRef]

L. Armstrong, W. R. Fielder, and Dong L. Lin, “Relativistic effects on transition probabilities in the Li and Be isoelectronic sequences,” Phys. Rev. A 14, 1114–1128 (1976).
[CrossRef]

1975 (1)

1972 (1)

We used an extensively modified early version of the program developed by C. Froese Fischer, “A multi-configuration Hartree-Fock program with improved stability,” Computer Phys. Commun. 4, 107–116 (1972); Computer Phys. Commun. 7, 236 (1974).
[CrossRef]

1971 (1)

J. P. Desclaux, D. F. Mayers, and F. O’Brien, “Relativistic atomic wave functions,” J. Phys. B 4, 631–642 (1971); J. P. Desclaux, “A multiconfiguration relativistic Dirac-Fock program,” Computer Phys. Commun. 9, 31–45 (1975).
[CrossRef]

1970 (2)

I. P. Grant, “Relativistic calculation of atomic structures,” Adv. Phys. 19, 747–811 (1970).
[CrossRef]

J. B. Mann and J. T. Waber, “SCF relativistic Hartree-Fock calculations on the superheavy elements 118–131,” J. Chem. Phys. 53, 2397–2406 (1970); “Self-consistent relativistic Dirac-Hartree-Fock calculations of lanthanide atoms,” Atomic Data 5, 201–229 (1973).
[CrossRef]

1969 (1)

D. C. Griffin, K. L. Andrew, and R. D. Cowan, “Theoretical calculations of the d-, f-, and g-electron transition series,” Phys. Rev. 177, 62–71 (1969); “Instabilities in the iterative solution of the Hartree-Fock equations for excited electrons,” Phys. Rev. A 3, 1233–1242 (1971).
[CrossRef]

1968 (1)

1967 (1)

R. D. Cowan, “Atomic self-consistent-field calculations using statistical approximations for exchange and correlation,” Phys. Rev. 163, 54–61 (1967). The HX potential consists of the Hartree potential plus a modification of Slater’s ρ1/3exchange potential. The originally suggested value k1 = 0.7 in the exchange portion has been decreased to 0.65 on the basis of subsequent experience.
[CrossRef]

1965 (1)

D. A. Liberman, J. T. Waber, and D. T. Cromer, “Self-consistent-field Dirac-Slater wave functions for atoms and ions. I. Comparison with previous calculations,” Phys. Rev. 137, A27–A34 (1965); D. A. Liberman and D. T. Cromer, “Relativistic self-consistent field program for atoms and ions,” Computer Phys. Commun. 2, 107–113 (1971).
[CrossRef]

1957 (1)

D. F. Mayers, “Relativistic self-consistent field calculation for mercury,” Proc. R. Soc. Lond. A 241, 93–109 (1957); R. G. Boyd, A. C. Larson, and J. T. Waber, “Indirect relativistic effect on the 5f electrons in uranium,” Phys. Rev. 129, 1629–1630 (1963).
[CrossRef]

Andrew, K. L.

D. C. Griffin, K. L. Andrew, and R. D. Cowan, “Theoretical calculations of the d-, f-, and g-electron transition series,” Phys. Rev. 177, 62–71 (1969); “Instabilities in the iterative solution of the Hartree-Fock equations for excited electrons,” Phys. Rev. A 3, 1233–1242 (1971).
[CrossRef]

Armstrong, L.

L. Armstrong, W. R. Fielder, and Dong L. Lin, “Relativistic effects on transition probabilities in the Li and Be isoelectronic sequences,” Phys. Rev. A 14, 1114–1128 (1976).
[CrossRef]

Bethe, H. A.

H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One-and Two-Electron Atoms (Springer-Verlag, Berlin, 1957), Sec. 13, especially Eq. (13.6) and the paragraph following Eq. (13.11).
[CrossRef]

Cowan, R. D.

D. C. Griffin, K. L. Andrew, and R. D. Cowan, “Theoretical calculations of the d-, f-, and g-electron transition series,” Phys. Rev. 177, 62–71 (1969); “Instabilities in the iterative solution of the Hartree-Fock equations for excited electrons,” Phys. Rev. A 3, 1233–1242 (1971).
[CrossRef]

R. D. Cowan, “Atomic self-consistent-field calculations using statistical approximations for exchange and correlation,” Phys. Rev. 163, 54–61 (1967). The HX potential consists of the Hartree potential plus a modification of Slater’s ρ1/3exchange potential. The originally suggested value k1 = 0.7 in the exchange portion has been decreased to 0.65 on the basis of subsequent experience.
[CrossRef]

R. D. Cowan and J. B. Mann, “The atomic structure of superheavy elements,” in Atomic Physics, 2, Proceedings of the Second International Conference on Atomic Physics (Plenum, London, 1971), pp. 215–226.

Cromer, D. T.

D. A. Liberman, J. T. Waber, and D. T. Cromer, “Self-consistent-field Dirac-Slater wave functions for atoms and ions. I. Comparison with previous calculations,” Phys. Rev. 137, A27–A34 (1965); D. A. Liberman and D. T. Cromer, “Relativistic self-consistent field program for atoms and ions,” Computer Phys. Commun. 2, 107–113 (1971).
[CrossRef]

Desclaux, J. P.

Yong-Ki Kim and J. P. Desclaux, “Relativistic f values for the resonance transitions of Li- and Be-like ions,” Phys. Rev. Lett. 36, 139–141 (1976).
[CrossRef]

J. P. Desclaux, D. F. Mayers, and F. O’Brien, “Relativistic atomic wave functions,” J. Phys. B 4, 631–642 (1971); J. P. Desclaux, “A multiconfiguration relativistic Dirac-Fock program,” Computer Phys. Commun. 9, 31–45 (1975).
[CrossRef]

Fielder, W. R.

L. Armstrong, W. R. Fielder, and Dong L. Lin, “Relativistic effects on transition probabilities in the Li and Be isoelectronic sequences,” Phys. Rev. A 14, 1114–1128 (1976).
[CrossRef]

Froese Fischer, C.

We used an extensively modified early version of the program developed by C. Froese Fischer, “A multi-configuration Hartree-Fock program with improved stability,” Computer Phys. Commun. 4, 107–116 (1972); Computer Phys. Commun. 7, 236 (1974).
[CrossRef]

Grant, I. P.

I. P. Grant, “Relativistic calculation of atomic structures,” Adv. Phys. 19, 747–811 (1970).
[CrossRef]

Griffin, D. C.

D. C. Griffin, K. L. Andrew, and R. D. Cowan, “Theoretical calculations of the d-, f-, and g-electron transition series,” Phys. Rev. 177, 62–71 (1969); “Instabilities in the iterative solution of the Hartree-Fock equations for excited electrons,” Phys. Rev. A 3, 1233–1242 (1971).
[CrossRef]

Kaufman, V.

Kim, Yong-Ki

Yong-Ki Kim and J. P. Desclaux, “Relativistic f values for the resonance transitions of Li- and Be-like ions,” Phys. Rev. Lett. 36, 139–141 (1976).
[CrossRef]

Liberman, D. A.

D. A. Liberman, J. T. Waber, and D. T. Cromer, “Self-consistent-field Dirac-Slater wave functions for atoms and ions. I. Comparison with previous calculations,” Phys. Rev. 137, A27–A34 (1965); D. A. Liberman and D. T. Cromer, “Relativistic self-consistent field program for atoms and ions,” Computer Phys. Commun. 2, 107–113 (1971).
[CrossRef]

Lin, Dong L.

L. Armstrong, W. R. Fielder, and Dong L. Lin, “Relativistic effects on transition probabilities in the Li and Be isoelectronic sequences,” Phys. Rev. A 14, 1114–1128 (1976).
[CrossRef]

Mann, J. B.

J. B. Mann and J. T. Waber, “SCF relativistic Hartree-Fock calculations on the superheavy elements 118–131,” J. Chem. Phys. 53, 2397–2406 (1970); “Self-consistent relativistic Dirac-Hartree-Fock calculations of lanthanide atoms,” Atomic Data 5, 201–229 (1973).
[CrossRef]

R. D. Cowan and J. B. Mann, “The atomic structure of superheavy elements,” in Atomic Physics, 2, Proceedings of the Second International Conference on Atomic Physics (Plenum, London, 1971), pp. 215–226.

Mayers, D. F.

J. P. Desclaux, D. F. Mayers, and F. O’Brien, “Relativistic atomic wave functions,” J. Phys. B 4, 631–642 (1971); J. P. Desclaux, “A multiconfiguration relativistic Dirac-Fock program,” Computer Phys. Commun. 9, 31–45 (1975).
[CrossRef]

D. F. Mayers, “Relativistic self-consistent field calculation for mercury,” Proc. R. Soc. Lond. A 241, 93–109 (1957); R. G. Boyd, A. C. Larson, and J. T. Waber, “Indirect relativistic effect on the 5f electrons in uranium,” Phys. Rev. 129, 1629–1630 (1963).
[CrossRef]

Moore, C. E.

C. E. Moore, Atomic Energy Levels, Natl. Bur. Stds. Circ. No. 467 (U.S. GPO, Washington, D. C., 1958), Vol. III.

O’Brien, F.

J. P. Desclaux, D. F. Mayers, and F. O’Brien, “Relativistic atomic wave functions,” J. Phys. B 4, 631–642 (1971); J. P. Desclaux, “A multiconfiguration relativistic Dirac-Fock program,” Computer Phys. Commun. 9, 31–45 (1975).
[CrossRef]

Salpeter, E. E.

H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One-and Two-Electron Atoms (Springer-Verlag, Berlin, 1957), Sec. 13, especially Eq. (13.6) and the paragraph following Eq. (13.11).
[CrossRef]

Slater, J. C.

J. C. Slater, Quantum Theory of Atomic Structure (McGraw-Hill, New York, 1960); E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge U. P., Cambridge, England, 1935).

Sugar, J.

Waber, J. T.

J. B. Mann and J. T. Waber, “SCF relativistic Hartree-Fock calculations on the superheavy elements 118–131,” J. Chem. Phys. 53, 2397–2406 (1970); “Self-consistent relativistic Dirac-Hartree-Fock calculations of lanthanide atoms,” Atomic Data 5, 201–229 (1973).
[CrossRef]

D. A. Liberman, J. T. Waber, and D. T. Cromer, “Self-consistent-field Dirac-Slater wave functions for atoms and ions. I. Comparison with previous calculations,” Phys. Rev. 137, A27–A34 (1965); D. A. Liberman and D. T. Cromer, “Relativistic self-consistent field program for atoms and ions,” Computer Phys. Commun. 2, 107–113 (1971).
[CrossRef]

Zalubas, R.

Adv. Phys. (1)

I. P. Grant, “Relativistic calculation of atomic structures,” Adv. Phys. 19, 747–811 (1970).
[CrossRef]

Computer Phys. Commun. (1)

We used an extensively modified early version of the program developed by C. Froese Fischer, “A multi-configuration Hartree-Fock program with improved stability,” Computer Phys. Commun. 4, 107–116 (1972); Computer Phys. Commun. 7, 236 (1974).
[CrossRef]

J. Chem. Phys. (1)

J. B. Mann and J. T. Waber, “SCF relativistic Hartree-Fock calculations on the superheavy elements 118–131,” J. Chem. Phys. 53, 2397–2406 (1970); “Self-consistent relativistic Dirac-Hartree-Fock calculations of lanthanide atoms,” Atomic Data 5, 201–229 (1973).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Phys. B (1)

J. P. Desclaux, D. F. Mayers, and F. O’Brien, “Relativistic atomic wave functions,” J. Phys. B 4, 631–642 (1971); J. P. Desclaux, “A multiconfiguration relativistic Dirac-Fock program,” Computer Phys. Commun. 9, 31–45 (1975).
[CrossRef]

Phys. Rev. (3)

D. A. Liberman, J. T. Waber, and D. T. Cromer, “Self-consistent-field Dirac-Slater wave functions for atoms and ions. I. Comparison with previous calculations,” Phys. Rev. 137, A27–A34 (1965); D. A. Liberman and D. T. Cromer, “Relativistic self-consistent field program for atoms and ions,” Computer Phys. Commun. 2, 107–113 (1971).
[CrossRef]

R. D. Cowan, “Atomic self-consistent-field calculations using statistical approximations for exchange and correlation,” Phys. Rev. 163, 54–61 (1967). The HX potential consists of the Hartree potential plus a modification of Slater’s ρ1/3exchange potential. The originally suggested value k1 = 0.7 in the exchange portion has been decreased to 0.65 on the basis of subsequent experience.
[CrossRef]

D. C. Griffin, K. L. Andrew, and R. D. Cowan, “Theoretical calculations of the d-, f-, and g-electron transition series,” Phys. Rev. 177, 62–71 (1969); “Instabilities in the iterative solution of the Hartree-Fock equations for excited electrons,” Phys. Rev. A 3, 1233–1242 (1971).
[CrossRef]

Phys. Rev. A (1)

L. Armstrong, W. R. Fielder, and Dong L. Lin, “Relativistic effects on transition probabilities in the Li and Be isoelectronic sequences,” Phys. Rev. A 14, 1114–1128 (1976).
[CrossRef]

Phys. Rev. Lett. (1)

Yong-Ki Kim and J. P. Desclaux, “Relativistic f values for the resonance transitions of Li- and Be-like ions,” Phys. Rev. Lett. 36, 139–141 (1976).
[CrossRef]

Proc. R. Soc. Lond. A (1)

D. F. Mayers, “Relativistic self-consistent field calculation for mercury,” Proc. R. Soc. Lond. A 241, 93–109 (1957); R. G. Boyd, A. C. Larson, and J. T. Waber, “Indirect relativistic effect on the 5f electrons in uranium,” Phys. Rev. 129, 1629–1630 (1963).
[CrossRef]

Other (5)

J. C. Slater, Quantum Theory of Atomic Structure (McGraw-Hill, New York, 1960); E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge U. P., Cambridge, England, 1935).

H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One-and Two-Electron Atoms (Springer-Verlag, Berlin, 1957), Sec. 13, especially Eq. (13.6) and the paragraph following Eq. (13.11).
[CrossRef]

On the basis of a variety of trial calculations, we have found it suitable to evaluate d at r equal to one-quarter of the largest radius for which the series expansion is to be employed.

R. D. Cowan and J. B. Mann, “The atomic structure of superheavy elements,” in Atomic Physics, 2, Proceedings of the Second International Conference on Atomic Physics (Plenum, London, 1971), pp. 215–226.

C. E. Moore, Atomic Energy Levels, Natl. Bur. Stds. Circ. No. 467 (U.S. GPO, Washington, D. C., 1958), Vol. III.

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Figures (1)

FIG. 1
FIG. 1

Single-configuration Hartree-Fock multiplet oscillator strengths for 2s-2p and 2s-3p transitions in ions of the Li i isoelectronic sequence. Solid curves, nonrelativistic (HF); dashed curves, including relativistic energy-level shifts; dotted curves, including also relativistic (HFR) modifications in the radial wavefunctions. The horizontal line marked H indicates the nonrelativistic hydrogenic 2s-3p limit. The circles represent the fully relativistic (DHF, length-operator) values of Armstrong et al. (Ref. 16).

Tables (5)

Tables Icon

TABLE I Total binding energies (in rydbergs) of neutral closed-shell atoms by nonrelativistic (HF), semirelativistic (HFR), and relativistic (DHF) methods.

Tables Icon

TABLE II Expectation values of rm for U i 5f36d7s2 (atomic units).a

Tables Icon

TABLE III One-electron eigenvalues ∊nl or ∊nlj and experimental binding energiesa for U i 5f36d7s2 (Ry).

Tables Icon

TABLE IV Spin-orbit parameter values ζnl (Ry) for U i 5f36d7s2.

Tables Icon

TABLE V Miscellaneous spin-orbit parameter values ζnl (cm−1).

Equations (12)

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( n 1 l 1 ) w 1 ( n 2 l 2 ) w 2 ( n q l q ) w q
( n 1 l 1 j 1 - ) w 1 - ( n 1 l 1 j 1 + ) w 1 + ( n 2 l 2 j 2 - ) w 2 - ( n 2 l 2 j 2 + ) w 2 + .
E av ( p 2 s ) ,             F 2 ( p p ) ,             G 1 ( p s ) ,             ζ p .
p - 2 s :             E av ( p - 2 s ) ,             G 1 ( p - s ) , p - p + s :             E av ( p - p + s ) ,             G 2 ( p - p + ) ,             G 1 ( p - s ) ,             G 1 ( p + s ) , p + 2 s :             E av ( p + 2 s ) ,             F 2 ( p + p + ) ,             G 1 ( p + s ) ,
R 1 ( p - s , s p + ) ,             R 2 ( p - 2 , p + 2 ) ,             R 2 ( p - p + , p + 2 ) ,             R 1 ( p - s , s p + ) .
[ - d 2 d r 2 + l i ( l i + 1 ) r 2 + V i ( r ) - α 2 4 [ i - V i ( r ) ] 2 - δ l i 0 α 2 4 ( 1 + α 2 4 ( i - V i ( r ) ) ) - 1 × ( d V i d r ) ( d P i / d r P i - 1 r ) ] P i ( r ) = i P i ( r ) ,
0 r P i ( r ) P j ( r ) d r .
δ l 0 α 2 4 ( 1 + α 2 4 ( - V ) ) - 1 r 2 Z ( δ l 0 1 + ( + 4 / α 2 ) r / 2 Z ) .
F r λ + 1 + a 1 r λ + 2
λ + 1 = 1 2 ( 1 - d ) + [ l ( l + 1 ) + 1 4 ( 1 + d ) 2 - α 2 Z 2 ] 1 / 2
a 1 = - ( 2 + α 2 ) Z ( λ + 2 + d ) ( λ + 1 ) - l ( l + 1 ) + α 2 Z 2 .
λ + 1 = ( 1 - α 2 Z 2 ) 1 / 2 ,