Abstract

Orthogonalized operators are introduced in the atomic configurations fN in order to yield parameters that are more precisely defined and more stable than the conventional ones. Of the four Racah operators e0, e1, e2, and e3, only e1 needs adjusting. The set of two-electron scalars is made complete by the generalized Trees operators eα′, eβ′, and eγ′. Of the three-electron scalars ti, only t2 requires alteration. The theory is illustrated for f3 by adding the orthogonalized operators in successive steps and comparing the fits with those obtained if the conventional operators are used.

© 1984 Optical Society of America

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References

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  1. E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge U. Press, New York, 1935).
  2. R. E. Trees, "Configuration interaction in Mn II," Phys. Rev. 83, 756–760 (1951); "Term values in the 3d54s configuration of Fe III," Phys. Rev. 84, 1089–1091 (1951).
    [CrossRef]
  3. K. Rajnak and B. G. Wybourne, "Configuration interaction effects in lN configurations," Phys. Rev. 132, 280–290 (1963).
    [CrossRef]
  4. B. R. Judd, J. E. Hansen, and A. J. J. Raassen, "Parametric fits in the atomic d shell," J. Phys. B 15, 1457–1472 (1982). The entry −6√14 in the sixth column of Table A.6 of this paper should be interchanged with the zero standing to its immediate right.
    [CrossRef]
  5. B. R. Judd, "Operator averages and orthogonalities," in Proceedings of the XII Symposium on Group Theoretical Methods in Physics, Trieste (1983), in Lecture Notes in Physics (Springer-Verlag, New York, 1984).
    [CrossRef]
  6. G. Racah, "Theory of complex spectra. IV," Phys. Rev. 76, 1352–1365 (1949).
    [CrossRef]
  7. G. Racah, "Group theory and spectroscopy," in Springer Tracts in Modern Physics (Ergebnisse der Exakten Naturwissenschaf-ten) (Springer-Verlag, New York, 1965), Vol. 37.
    [CrossRef]
  8. P. Uylings (alias Uijlings), Zeeman Laboratorium, Amsterdam, the Netherlands (personal communication).
  9. D. J. Newman, "Operator orthogonality and parameter uncertainty," Phys. Lett. A 92, 167–169 (1982); Department of Physics, Hong Kong University, Hong Kong (personal communication).
    [CrossRef]
  10. B. R. Judd, "Three-particle operators for equivalent electrons," Phys. Rev. 141, 4–14 (1966).
    [CrossRef]
  11. J. Sugar, "Analysis of the third spectrum of praseodymium," J. Opt. Soc. Am. 53, 831–839 (1963); H. Crosswhite, H. M Cross-white, and B. R. Judd, "Magnetic parameters for the configuration ƒ3," Phys. Rev. 174, 89–94 (1968).
    [CrossRef]

1983 (1)

B. R. Judd, "Operator averages and orthogonalities," in Proceedings of the XII Symposium on Group Theoretical Methods in Physics, Trieste (1983), in Lecture Notes in Physics (Springer-Verlag, New York, 1984).
[CrossRef]

1982 (2)

B. R. Judd, J. E. Hansen, and A. J. J. Raassen, "Parametric fits in the atomic d shell," J. Phys. B 15, 1457–1472 (1982). The entry −6√14 in the sixth column of Table A.6 of this paper should be interchanged with the zero standing to its immediate right.
[CrossRef]

D. J. Newman, "Operator orthogonality and parameter uncertainty," Phys. Lett. A 92, 167–169 (1982); Department of Physics, Hong Kong University, Hong Kong (personal communication).
[CrossRef]

1966 (1)

B. R. Judd, "Three-particle operators for equivalent electrons," Phys. Rev. 141, 4–14 (1966).
[CrossRef]

1965 (1)

G. Racah, "Group theory and spectroscopy," in Springer Tracts in Modern Physics (Ergebnisse der Exakten Naturwissenschaf-ten) (Springer-Verlag, New York, 1965), Vol. 37.
[CrossRef]

1963 (2)

1951 (1)

R. E. Trees, "Configuration interaction in Mn II," Phys. Rev. 83, 756–760 (1951); "Term values in the 3d54s configuration of Fe III," Phys. Rev. 84, 1089–1091 (1951).
[CrossRef]

1949 (1)

G. Racah, "Theory of complex spectra. IV," Phys. Rev. 76, 1352–1365 (1949).
[CrossRef]

Condon, E. U.

E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge U. Press, New York, 1935).

Hansen, J. E.

B. R. Judd, J. E. Hansen, and A. J. J. Raassen, "Parametric fits in the atomic d shell," J. Phys. B 15, 1457–1472 (1982). The entry −6√14 in the sixth column of Table A.6 of this paper should be interchanged with the zero standing to its immediate right.
[CrossRef]

Judd, B. R.

B. R. Judd, "Operator averages and orthogonalities," in Proceedings of the XII Symposium on Group Theoretical Methods in Physics, Trieste (1983), in Lecture Notes in Physics (Springer-Verlag, New York, 1984).
[CrossRef]

B. R. Judd, J. E. Hansen, and A. J. J. Raassen, "Parametric fits in the atomic d shell," J. Phys. B 15, 1457–1472 (1982). The entry −6√14 in the sixth column of Table A.6 of this paper should be interchanged with the zero standing to its immediate right.
[CrossRef]

B. R. Judd, "Three-particle operators for equivalent electrons," Phys. Rev. 141, 4–14 (1966).
[CrossRef]

Newman, D. J.

D. J. Newman, "Operator orthogonality and parameter uncertainty," Phys. Lett. A 92, 167–169 (1982); Department of Physics, Hong Kong University, Hong Kong (personal communication).
[CrossRef]

Raassen, A. J. J.

B. R. Judd, J. E. Hansen, and A. J. J. Raassen, "Parametric fits in the atomic d shell," J. Phys. B 15, 1457–1472 (1982). The entry −6√14 in the sixth column of Table A.6 of this paper should be interchanged with the zero standing to its immediate right.
[CrossRef]

Racah, G.

G. Racah, "Group theory and spectroscopy," in Springer Tracts in Modern Physics (Ergebnisse der Exakten Naturwissenschaf-ten) (Springer-Verlag, New York, 1965), Vol. 37.
[CrossRef]

G. Racah, "Theory of complex spectra. IV," Phys. Rev. 76, 1352–1365 (1949).
[CrossRef]

Rajnak, K.

K. Rajnak and B. G. Wybourne, "Configuration interaction effects in lN configurations," Phys. Rev. 132, 280–290 (1963).
[CrossRef]

Shortley, G. H.

E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge U. Press, New York, 1935).

Sugar, J.

Trees, R. E.

R. E. Trees, "Configuration interaction in Mn II," Phys. Rev. 83, 756–760 (1951); "Term values in the 3d54s configuration of Fe III," Phys. Rev. 84, 1089–1091 (1951).
[CrossRef]

Uylings, P.

P. Uylings (alias Uijlings), Zeeman Laboratorium, Amsterdam, the Netherlands (personal communication).

Wybourne, B. G.

K. Rajnak and B. G. Wybourne, "Configuration interaction effects in lN configurations," Phys. Rev. 132, 280–290 (1963).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. B (1)

B. R. Judd, J. E. Hansen, and A. J. J. Raassen, "Parametric fits in the atomic d shell," J. Phys. B 15, 1457–1472 (1982). The entry −6√14 in the sixth column of Table A.6 of this paper should be interchanged with the zero standing to its immediate right.
[CrossRef]

Phys. Lett. A (1)

D. J. Newman, "Operator orthogonality and parameter uncertainty," Phys. Lett. A 92, 167–169 (1982); Department of Physics, Hong Kong University, Hong Kong (personal communication).
[CrossRef]

Phys. Rev. (4)

B. R. Judd, "Three-particle operators for equivalent electrons," Phys. Rev. 141, 4–14 (1966).
[CrossRef]

G. Racah, "Theory of complex spectra. IV," Phys. Rev. 76, 1352–1365 (1949).
[CrossRef]

R. E. Trees, "Configuration interaction in Mn II," Phys. Rev. 83, 756–760 (1951); "Term values in the 3d54s configuration of Fe III," Phys. Rev. 84, 1089–1091 (1951).
[CrossRef]

K. Rajnak and B. G. Wybourne, "Configuration interaction effects in lN configurations," Phys. Rev. 132, 280–290 (1963).
[CrossRef]

Other (4)

B. R. Judd, "Operator averages and orthogonalities," in Proceedings of the XII Symposium on Group Theoretical Methods in Physics, Trieste (1983), in Lecture Notes in Physics (Springer-Verlag, New York, 1984).
[CrossRef]

G. Racah, "Group theory and spectroscopy," in Springer Tracts in Modern Physics (Ergebnisse der Exakten Naturwissenschaf-ten) (Springer-Verlag, New York, 1965), Vol. 37.
[CrossRef]

P. Uylings (alias Uijlings), Zeeman Laboratorium, Amsterdam, the Netherlands (personal communication).

E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge U. Press, New York, 1935).

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Figures (2)

Fig. 1
Fig. 1

Values of the parameters E0 and Eo0 versus the total number of parameters N taken from Tables 4 and 5. The mean errors decrease as the new parameters γ, β, …, T8 are included in the fitting procedure. The convergence to the final values is much faster for the parameter Eo0 associated with the orthogonalized set of operators than it is for E0.

Fig. 2
Fig. 2

Values of E3 and Eo3 plotted in a similar way to the parameters of Fig. 1. The convergence and smaller mean errors for the orthogonal parameter Eo3 compared to E3 are not so striking as for Eo0 compared to E0 in Fig. 1. This is partly due to the fact that Eo3 is given [in Eqs. (9)] as a linear combination of only three conventional parameters, whereas Eo0 is given [in Eq. (10)] as a linear combination of as many as five conventional parameters.

Tables (5)

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Table 1 Orthogonal Scalar Operators with Well-Defined Irreducible Representations as Labels

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Table 2 Matrix Elements of the Operators oi in f2

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Table 3 Matrix Elements in f2 of Orthogonal Scalar Operators

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Table 4 Successively Included Parameters (in inverse centimeters) for Orthogonalized Operators for Pr iii in 4f3

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Table 5 Successively Included Parameters (in inverse centimeters) for Conventional Operators for Pr iii in 4f3

Equations (12)

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E ( S 1 ) = F 0 + 60 F 2 + 198 F 4 + 1716 F 6 , E ( P 3 ) = F 0 + 45 F 2 + 33 F 4 - 1287 F 6 , E ( D 1 ) = F 0 + 19 F 2 - 99 F 4 + 715 F 6 , E ( F 3 ) = F 0 - 10 F 2 - 33 F 4 - 286 F 6 , E ( G 1 ) = F 0 - 30 F 2 + 97 F 4 + 78 F 6 , E ( H 3 ) = F 0 - 25 F 2 - 51 F 4 - 13 F 6 , E ( I 1 ) = F 0 + 25 F 2 + 9 F 4 + F 6 .
( 2 S ψ + 1 ) ( 2 L ψ + 1 ) ψ H i ψ ψ H j ψ = 0             ( i j ) ,
E ( S 1 ) = E 0 + 9 E 1 , E ( P 3 ) = E 0 + 33 E 3 , E ( D 1 ) = E 0 + 2 E 1 + 286 E 2 - 11 E 3 , E ( F 3 ) = E 0 , E ( G 1 ) = E 0 + 2 E 1 - 260 E 2 - 4 E 3 , E ( H 3 ) = E 0 - 9 E 3 , E ( I 1 ) = E 0 + 2 E 1 + 70 E 2 + 7 E 3 .
U 14 S p 14 R 7 G 2 R 3 .
[ λ ] σ W U L [ 0 0 ] 0 0 ( 000 ) ( 00 ) 0.
e 1 = e 1 - 9 e 0 / 13 , e α = 1 2 e 3 + 5 L ( L + 1 ) / 4 - 30 G ( G 2 ) , e β = 5 G ( R 7 ) - 6 G ( G 2 ) , e γ = 25 G ( R 7 ) / 2 - 15 N / 2 + 3 2 e 0 - 1 2 e 1
t 2 t 2 - ( N - 2 ) e 3 / 70 2 .
H = e 0 E o 0 + e 1 E o 1 + e 2 E o 2 + e 3 E o 3 + α e α + β e β + γ e γ + t 2 T o 2 + t 3 T 3 + t 4 T 4 + t 6 T 6 + t 7 T 7 + t 8 T 8
f 12 ψ t 2 f 12 ψ = ( 2 / 14 ) f 12 ψ e 3 f 12 ψ ,
ζ = 663.298 ,             M 0 = 1.301 ,             P 2 = 195.428 ,
α = 4 α / 5 , β = - 4 α - β / 6 , γ = 8 α / 5 + β / 15 + 2 γ / 25 , T o 2 = T 2 , E o 3 = E 3 - 2 α / 5 + 2 T 2 / 140 , E o 2 = E 2 , E o 1 = E 1 + 4 α / 5 + β / 30 + γ / 25.
E o 0 = E 0 + 9 E 1 / 13 + [ 2 ( 14 - N ) / 13 ( N - 1 ) ] × ( 12 α + β / 2 + 3 γ / 5 ) .

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