Abstract

When Coulomb interactions between f3 and other electronic configurations are taken to third and higher orders of perturbation theory, it becomes necessary to augment the standard set of thirteen electrostatic parameters with eight more. The operators associated with these eight parameters are orthogonalized to the standard set of operators, and a complete set of the matrix elements of all 21 orthogonalized operators is given for f3. The group-theoretical properties of these operators are examined. Each operator corresponds to a unique pair of irreducible representations of R7 (the rotation group in the seven-dimensional orbital space of an f electron) and of G2 (Cartan’s first exceptional group), but the associated quasispin ranks and irreducible representations of Sp14 (the symplectic group in the 14-dimensional spin–orbital space of an f electron) are sometimes mixed. The combinations of quasispin ranks are odd for the three-electron operators t and even for the two-electron operators e.

© 1984 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Racah, "Theory of complex spectra. IV," Phys. Rev. 76, 1352–1365 (1949).
    [CrossRef]
  2. K. Rajnak and B. G. Wybourne, "Configuration interaction effects in lN configurations," Phys. Rev. 132, 280–290 (1963).
    [CrossRef]
  3. B. R. Judd, "Three-particle operators for equivalent electrons," Phys. Rev. 141, 4–14 (1966).
    [CrossRef]
  4. B. R. Judd, "Developments in atomic shell theory," in The Structure of Matter, B. G. Wybourne, ed. (U. Canterbury Press, Christchurch, New Zealand, 1972). The entry of -2272 given for the 2K term in the table on p. 409 should be -2772.
  5. 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).
    [CrossRef]
  6. B. R. Judd and H. Crosswhite, "Orthogonalized operators for the ƒ shell," J. Opt. Soc. Am. B 1, 255–260 (1984).
    [CrossRef]
  7. B. R. Judd, "Operator averages and orthogonalities," in the Proceedings of the XII Symposium on Group Theoretical Methods in Physics, Trieste (1983), to be published in Lecture Notes in Physics (Springer-Verlag, New York, 1984).
    [CrossRef]
  8. C. W. Nielson and G. F. Koster, Spectroscopic Coefficients for the pn,dn, and ƒn Configurations (MIT Press, Cambridge, Mass., 1963).
  9. B. G. Wybourne, Symmetry Principles and Atomic Spectroscopy (Wiley-Interscience, New York, 1970).
  10. B. R. Judd, Second Quantization and Atomic Spectroscopy (Johns Hopkins, Baltimore, Md., 1967).

1984 (1)

1983 (1)

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

1982 (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).
[CrossRef]

1972 (1)

B. R. Judd, "Developments in atomic shell theory," in The Structure of Matter, B. G. Wybourne, ed. (U. Canterbury Press, Christchurch, New Zealand, 1972). The entry of -2272 given for the 2K term in the table on p. 409 should be -2772.

1966 (1)

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

1963 (1)

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

1949 (1)

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

Crosswhite, H.

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).
[CrossRef]

Judd, B. R.

B. R. Judd and H. Crosswhite, "Orthogonalized operators for the ƒ shell," J. Opt. Soc. Am. B 1, 255–260 (1984).
[CrossRef]

B. R. Judd, "Operator averages and orthogonalities," in the Proceedings of the XII Symposium on Group Theoretical Methods in Physics, Trieste (1983), to be published 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).
[CrossRef]

B. R. Judd, "Developments in atomic shell theory," in The Structure of Matter, B. G. Wybourne, ed. (U. Canterbury Press, Christchurch, New Zealand, 1972). The entry of -2272 given for the 2K term in the table on p. 409 should be -2772.

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

B. R. Judd, Second Quantization and Atomic Spectroscopy (Johns Hopkins, Baltimore, Md., 1967).

Koster, G. F.

C. W. Nielson and G. F. Koster, Spectroscopic Coefficients for the pn,dn, and ƒn Configurations (MIT Press, Cambridge, Mass., 1963).

Nielson, C. W.

C. W. Nielson and G. F. Koster, Spectroscopic Coefficients for the pn,dn, and ƒn Configurations (MIT Press, Cambridge, Mass., 1963).

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).
[CrossRef]

Racah, G.

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]

Wybourne, B. G.

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

B. G. Wybourne, Symmetry Principles and Atomic Spectroscopy (Wiley-Interscience, New York, 1970).

J. Opt. Soc. Am. B (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).
[CrossRef]

Phys. Rev. (3)

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

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

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

Other (5)

B. R. Judd, "Developments in atomic shell theory," in The Structure of Matter, B. G. Wybourne, ed. (U. Canterbury Press, Christchurch, New Zealand, 1972). The entry of -2272 given for the 2K term in the table on p. 409 should be -2772.

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

C. W. Nielson and G. F. Koster, Spectroscopic Coefficients for the pn,dn, and ƒn Configurations (MIT Press, Cambridge, Mass., 1963).

B. G. Wybourne, Symmetry Principles and Atomic Spectroscopy (Wiley-Interscience, New York, 1970).

B. R. Judd, Second Quantization and Atomic Spectroscopy (Johns Hopkins, Baltimore, Md., 1967).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Tables (4)

Tables Icon

Table 1 Matrix Elements (ψ|ei|ψ′) and (ψ|ti|ψ′) of the Complete Orthogonalized Set of Two-Electron Operators ei and Three-Electron Operators ti for States ψ and ψ of f3

Tables Icon

Table 2 Group Labels and Multiplicative Factors for the Orthogonal Operators

Tables Icon

Table 3 Matrix Elements of the Components of t12 (with Specified 2K+1σ〉)

Tables Icon

Table 4 Matrix Elements of the Components of t12 for f2

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

( 2 S ψ + 1 ) ( 2 L ψ + 1 ) ψ H i ψ ψ H j ψ = 0             ( i j ) ,
t 11 = t 11 + ( 3 / 385 ) 1 / 2 e β .
ψ H ψ = [ ( N + 1 ) / ( N + 1 - n ) ] × ( ψ { ϕ ) ϕ H ϕ ( ϕ } ψ ) ,
f 3 S t 11 f 3 S = 18 ( 7 / 660 ) 1 / 2 ,
f 11 S t 11 f 11 S = - 18 ( 7 / 660 ) 1 / 2 , f 5 S t 11 f 5 S = - f 9 S t 11 f 9 S = 9 ( 7 / 660 ) 1 / 2 ,
f 7 S t 11 f 7 S = 0.
20 ( 105 ) 1 / 2 t 14 = 3 t 3 + 2 t 3 , 140 ( 15015 ) 1 / 2 t 15 = 3 t 4 + 2 t 4 .
49 ( 110 / 3 ) 1 / 2 t 12 = - ( 9 / 5 ) t 12 + ( 9 / 5 ) t 12 + ( 1 / 4 ) t 12 ,
[ ( a a a ) 1110000 ( aaa ) 1110000 ] χ σ ,
( ½ ½ , ½ - ½ 0 0 ) = - ( ½ - ½ , ½ ½ 0 0 ) = ( ½ ) 1 / 2 ,
( b b ) ( 0 ) = ( ½ ) 1 / 2 ( b 1 / 2 b ˜ - 1 / 2 - b - 1 / 2 b ˜ 1 / 2 )
= - ( ½ ) 1 / 2 ( b 1 / 2 b 1 / 2 + b - 1 / 2 b - 1 / 2 )
0000000 + 1100000 + 1111000 + 1111110 + 2200000 + 2211000 .
S 4 t i S 4 , D 4 t i D 4 , , L 2 t i L 2 ,
D 2 1 t i D 2 2 = + D 2 2 t i D 2 1 , G 2 1 t i G 2 2 = + G 2 2 t i G 2 1 , H 2 1 t i H 2 2 = + H 2 2 t i H 2 1 .
2000000 + 2110000 + 2111100 + 2220000 ,
2111100 ( 321 ) ( 22 ) , 2111100 ( 321 ) ( 40 ) , 2220000 ( 411 ) ( 40 ) .
F 12 f 4 D ¹ t 12 f 4 D ¹ = - 2772 / 5.
f 2 D ¹ t 12 f 2 D ¹ = 0.
- ( 9 / 5 ) ( 154 ) ( - 1 ) ( 5 / 2 3 5 / 2 - 3 / 2 0 3 / 2 ) / ( 5 / 2 3 5 / 2 - 5 / 2 0 5 / 2 ) + ( 9 / 5 ) ( 154 ) ( - 1 ) ( 5 / 2 1 5 / 2 - 3 / 2 0 3 / 2 ) / ( 5 / 2 1 5 / 2 - 5 / 2 0 5 / 2 ) = - 2772 / 5 ,

Metrics