Abstract

The energies of the 37 observed levels of the 4f3 configuration of Pr iii have been calculated with an rms deviation of ±29 cm−1 by use of 13 adjustable parameters, including 5 parameters which account for nonlinear configuration-interaction effects. The physical significance of the nonlinear parameters is discussed in terms of the relative roles of various mechanisms of configuration interaction.

© 1965 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 University Press, New York, 1935).
  2. G. Racah, Phys. Rev. 76, 1352 (1949).
    [Crossref]
  3. R. F. Bacher and S. Goudsmit, Phys. Rev. 46, 948 (1934).
    [Crossref]
  4. D. R. Layzer, dissertation, Harvard University, Cambridge, Massachusetts (1950).
  5. R. E. Trees, Phys. Rev. 83, 756 (1951); Phys. Rev. 84, 1089 (1951); Phys. Rev. 85, 382 (1952).
    [Crossref]
  6. G. Racah, Phys. Rev. 85, 381 (1952).
    [Crossref]
  7. G. Racah, Lunds Univ. Arsskr. AVD. 2 50, 31 (1954).
  8. R. E. Trees and C. K. Jørgensen, Phys. Rev. 123, 1278 (1961).
    [Crossref]
  9. K. Rajnak and B. G. Wybourne, Phys. Rev. 132, 280 (1963).
    [Crossref]
  10. R. E. Trees, Phys. Rev. 129, 1220 (1963).
    [Crossref]
  11. J. Sugar, J. Opt. Soc. Am. 53, 831 (1963).
    [Crossref]
  12. C. W. Nielson and George F. Koster, Spectroscopic Coefficients for the pn, dn and fn Configurations (Technology Press, Cambridge, Massachusetts, 1963).
  13. B. G. Wybourne, J. Chem. Phys. 40, 1457 (1964).
    [Crossref]
  14. K. Rajnak, J. Chem. Phys. 37, 2440 (1962).
    [Crossref]
  15. G. F. Koster and C. W. Nielson, a magnetic tape entitled “Energy Matrices for all Configurations of Equivalent f Electrons,” Massachusetts Institute of Technology, Cambridge, Massachusetts.
  16. William C. Davidon, “Variable Metric Method for Minimization,” Argonne National Laboratory Report, (1959). This general procedure has been adapted for spectroscopic problems by T. Clements of our Math and Computing group.
  17. R. E. Trees, J. Opt. Soc. Am. 54, 651 (1964).
    [Crossref]
  18. R. E. Trees, Phys. Rev. 84, 1089 (1951).
    [Crossref]
  19. G. Racah, Phys. Rev. 89, 381 (1952).
    [Crossref]
  20. The present notation for the linear parameters is that of I. The β used by Trees17 is related to the present γ, although they are not identical. Trees’s γ is 1/12 of our β.
  21. B. R. Judd, Operator Techniques in Atomic Spectroscopy (McGraw–Hill Book Company, Inc., New York, 1963).
  22. B. R. Judd and I. Lindgren, Phys. Rev. 122, 1802 (1961).
    [Crossref]
  23. B. R. Judd, Proc. Roy. Soc. (London) A250, 562 (1959). These coefficients also appear in the tabulation of Nielson and Koster.12

1964 (2)

B. G. Wybourne, J. Chem. Phys. 40, 1457 (1964).
[Crossref]

R. E. Trees, J. Opt. Soc. Am. 54, 651 (1964).
[Crossref]

1963 (3)

K. Rajnak and B. G. Wybourne, Phys. Rev. 132, 280 (1963).
[Crossref]

R. E. Trees, Phys. Rev. 129, 1220 (1963).
[Crossref]

J. Sugar, J. Opt. Soc. Am. 53, 831 (1963).
[Crossref]

1962 (1)

K. Rajnak, J. Chem. Phys. 37, 2440 (1962).
[Crossref]

1961 (2)

B. R. Judd and I. Lindgren, Phys. Rev. 122, 1802 (1961).
[Crossref]

R. E. Trees and C. K. Jørgensen, Phys. Rev. 123, 1278 (1961).
[Crossref]

1959 (1)

B. R. Judd, Proc. Roy. Soc. (London) A250, 562 (1959). These coefficients also appear in the tabulation of Nielson and Koster.12

1954 (1)

G. Racah, Lunds Univ. Arsskr. AVD. 2 50, 31 (1954).

1952 (2)

G. Racah, Phys. Rev. 85, 381 (1952).
[Crossref]

G. Racah, Phys. Rev. 89, 381 (1952).
[Crossref]

1951 (2)

R. E. Trees, Phys. Rev. 84, 1089 (1951).
[Crossref]

R. E. Trees, Phys. Rev. 83, 756 (1951); Phys. Rev. 84, 1089 (1951); Phys. Rev. 85, 382 (1952).
[Crossref]

1949 (1)

G. Racah, Phys. Rev. 76, 1352 (1949).
[Crossref]

1934 (1)

R. F. Bacher and S. Goudsmit, Phys. Rev. 46, 948 (1934).
[Crossref]

Bacher, R. F.

R. F. Bacher and S. Goudsmit, Phys. Rev. 46, 948 (1934).
[Crossref]

Condon, E. U.

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

Davidon, William C.

William C. Davidon, “Variable Metric Method for Minimization,” Argonne National Laboratory Report, (1959). This general procedure has been adapted for spectroscopic problems by T. Clements of our Math and Computing group.

Goudsmit, S.

R. F. Bacher and S. Goudsmit, Phys. Rev. 46, 948 (1934).
[Crossref]

Jørgensen, C. K.

R. E. Trees and C. K. Jørgensen, Phys. Rev. 123, 1278 (1961).
[Crossref]

Judd, B. R.

B. R. Judd and I. Lindgren, Phys. Rev. 122, 1802 (1961).
[Crossref]

B. R. Judd, Proc. Roy. Soc. (London) A250, 562 (1959). These coefficients also appear in the tabulation of Nielson and Koster.12

B. R. Judd, Operator Techniques in Atomic Spectroscopy (McGraw–Hill Book Company, Inc., New York, 1963).

Koster, G. F.

G. F. Koster and C. W. Nielson, a magnetic tape entitled “Energy Matrices for all Configurations of Equivalent f Electrons,” Massachusetts Institute of Technology, Cambridge, Massachusetts.

Koster, George F.

C. W. Nielson and George F. Koster, Spectroscopic Coefficients for the pn, dn and fn Configurations (Technology Press, Cambridge, Massachusetts, 1963).

Layzer, D. R.

D. R. Layzer, dissertation, Harvard University, Cambridge, Massachusetts (1950).

Lindgren, I.

B. R. Judd and I. Lindgren, Phys. Rev. 122, 1802 (1961).
[Crossref]

Nielson, C. W.

G. F. Koster and C. W. Nielson, a magnetic tape entitled “Energy Matrices for all Configurations of Equivalent f Electrons,” Massachusetts Institute of Technology, Cambridge, Massachusetts.

C. W. Nielson and George F. Koster, Spectroscopic Coefficients for the pn, dn and fn Configurations (Technology Press, Cambridge, Massachusetts, 1963).

Racah, G.

G. Racah, Lunds Univ. Arsskr. AVD. 2 50, 31 (1954).

G. Racah, Phys. Rev. 85, 381 (1952).
[Crossref]

G. Racah, Phys. Rev. 89, 381 (1952).
[Crossref]

G. Racah, Phys. Rev. 76, 1352 (1949).
[Crossref]

Rajnak, K.

K. Rajnak and B. G. Wybourne, Phys. Rev. 132, 280 (1963).
[Crossref]

K. Rajnak, J. Chem. Phys. 37, 2440 (1962).
[Crossref]

Shortley, G. H.

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

Sugar, J.

Trees, R. E.

R. E. Trees, J. Opt. Soc. Am. 54, 651 (1964).
[Crossref]

R. E. Trees, Phys. Rev. 129, 1220 (1963).
[Crossref]

R. E. Trees and C. K. Jørgensen, Phys. Rev. 123, 1278 (1961).
[Crossref]

R. E. Trees, Phys. Rev. 83, 756 (1951); Phys. Rev. 84, 1089 (1951); Phys. Rev. 85, 382 (1952).
[Crossref]

R. E. Trees, Phys. Rev. 84, 1089 (1951).
[Crossref]

Wybourne, B. G.

B. G. Wybourne, J. Chem. Phys. 40, 1457 (1964).
[Crossref]

K. Rajnak and B. G. Wybourne, Phys. Rev. 132, 280 (1963).
[Crossref]

J. Chem. Phys. (2)

B. G. Wybourne, J. Chem. Phys. 40, 1457 (1964).
[Crossref]

K. Rajnak, J. Chem. Phys. 37, 2440 (1962).
[Crossref]

J. Opt. Soc. Am. (2)

Lunds Univ. Arsskr. AVD. 2 (1)

G. Racah, Lunds Univ. Arsskr. AVD. 2 50, 31 (1954).

Phys. Rev. (10)

R. E. Trees and C. K. Jørgensen, Phys. Rev. 123, 1278 (1961).
[Crossref]

K. Rajnak and B. G. Wybourne, Phys. Rev. 132, 280 (1963).
[Crossref]

R. E. Trees, Phys. Rev. 129, 1220 (1963).
[Crossref]

G. Racah, Phys. Rev. 76, 1352 (1949).
[Crossref]

R. F. Bacher and S. Goudsmit, Phys. Rev. 46, 948 (1934).
[Crossref]

R. E. Trees, Phys. Rev. 83, 756 (1951); Phys. Rev. 84, 1089 (1951); Phys. Rev. 85, 382 (1952).
[Crossref]

G. Racah, Phys. Rev. 85, 381 (1952).
[Crossref]

R. E. Trees, Phys. Rev. 84, 1089 (1951).
[Crossref]

G. Racah, Phys. Rev. 89, 381 (1952).
[Crossref]

B. R. Judd and I. Lindgren, Phys. Rev. 122, 1802 (1961).
[Crossref]

Proc. Roy. Soc. (London) (1)

B. R. Judd, Proc. Roy. Soc. (London) A250, 562 (1959). These coefficients also appear in the tabulation of Nielson and Koster.12

Other (7)

C. W. Nielson and George F. Koster, Spectroscopic Coefficients for the pn, dn and fn Configurations (Technology Press, Cambridge, Massachusetts, 1963).

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

The present notation for the linear parameters is that of I. The β used by Trees17 is related to the present γ, although they are not identical. Trees’s γ is 1/12 of our β.

B. R. Judd, Operator Techniques in Atomic Spectroscopy (McGraw–Hill Book Company, Inc., New York, 1963).

G. F. Koster and C. W. Nielson, a magnetic tape entitled “Energy Matrices for all Configurations of Equivalent f Electrons,” Massachusetts Institute of Technology, Cambridge, Massachusetts.

William C. Davidon, “Variable Metric Method for Minimization,” Argonne National Laboratory Report, (1959). This general procedure has been adapted for spectroscopic problems by T. Clements of our Math and Computing group.

D. R. Layzer, dissertation, Harvard University, Cambridge, Massachusetts (1950).

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Tables (3)

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Table I Angular dependence of nonlinear configuration interaction parameters (×102).

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Table II Parameter values for the 4f3 configuration of Pr iii (in cm).

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Table III Calculated energy levels, eigenvectors and g factors for the 4f3 configuration.

Equations (14)

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C ( ψ , ψ ) = k k k even ( k k , l ) ( 2 k + 1 ) { k k k l l l } × ( ψ h i j ( { u h ( k ) u i ( k ) } ( k ) u j ( k ) ) ( 0 ) ψ ) ,
( k k , l ) = P ( k k , l ) - P ( k k , l ) .
C ( ψ , ψ ) = k k X ( k k , l ) Y ( k k , l ) ,
X ( k k , l ) = k even > 0 ( 2 k + 1 ) { k k k l l l } × ( ψ h i j ( { u h ( k ) u i ( k ) } ( k ) u j ( k ) ) ( 0 ) ψ ) .
Z ( k k k ) = all permutations of k , k , and k × l Y ( k k , l ) ( 2 k + 1 ) { k k k l l l } .
{ k k k l l l } { k k k l l l } ,
σ = [ i Δ i 2 / ( N - K ) ] 1 2 ,
( ψ h , i , j ( { u h ( k ) u i ( k ) } ( k ) u j ( k ) ) ( 0 ) ψ )
X ( k k , l ) = k even > 0 ( 2 k + 1 ) { k k k l l l } × { ( ψ h i j ( { u h ( k ) u i ( k ) } ( k ) u j ( k ) ) ( 0 ) ψ ) - { k k k l l l } [ ( ψ ( u ( k ) ) 2 ψ ) + ( ψ ( u ( k ) ) 2 ψ ) + ( ψ ( u ( k ) ) 2 ψ ) - 2 N 2 l + 1 δ ( ψ , ψ ) ] } ,
( ψ ( { u h ( k ) u i ( k ) } ( k ) u j ( k ) ) ( 0 ) ψ ) = ψ ¯ ψ ¯ { k k k L ¯ L ¯ L } ( ψ u ( k ) ψ ¯ ) × ( ψ ¯ u ( k ) ψ ¯ ) ( ψ ¯ u ( k ) ψ ) ,
( ψ ( u ( k ) ) 2 ψ ) = 1 2 L + 1 ψ ¯ ( - 1 ) L - L ¯ × ( ψ u ( k ) ψ ¯ ) ( ψ ¯ u ( k ) ψ ) .
X ( k k , l ) = 6 k even > 0 ( 2 k + 1 ) { k k k l l l } × ψ ¯ ψ ˜ ( - 1 ) L + l + L ˜ [ ( 2 L ¯ + 1 ) ( 2 L ˜ + 1 ) ] 1 2 × ( ψ { ψ ¯ ) ( ψ ˜ } ψ ) { L ¯ L ˜ k l l L } { l l k l l k L ¯ L ˜ k } .
k > 0 ( 2 k + 1 ) { l l k l l k L ˜ L ¯ k } { l l k k k l } = { L ¯ l l k l l } { l k l L ¯ l L ˜ } - ( - 1 ) L ˜ + l + l ( 2 k + 1 ) ( 2 l + 1 ) { L ¯ k L ˜ l l l } δ ( k , k ) ,
X ( k k , l ) = 6 ψ ψ ˜ ( - 1 ) L + l + L [ ( 2 L ¯ + 1 ) ( 2 L ˜ + 1 ) ] 1 2 × ( ψ { ψ ¯ ) ( ψ ˜ } ψ ) { L ¯ L ˜ k l l L } × [ { L ¯ l l k l l } { L ¯ k L ˜ l l l } - ( - 1 ) L ˜ + l + l ( 2 k + 1 ) ( 2 l + 1 ) { L ¯ k L ˜ l l l } ] .