Abstract

Theoretical calculation of energy levels and spectra ab initio are described for transitions of the type pmpm−1l, especially 3pm–3pm−13d transitions in the Ar i and Cl i isoelectronic sequences. Strong changes of wavefunction composition occur in going from neutral atoms to ions as a result of collapse of the 3d wavefunction; for m = 2 and 6 these take the form of coupling changes, and for m = 3–5 they take the form of changes of parentage composition. Abnormally low Ti v 3p6–3p54d intensities are explained as an interference effect in evaluation of the dipole radial integral.

© 1968 Optical Society of America

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  1. R. D. Cowan, J. Opt. Soc. Am. 58, 808 (1968).
    [Crossref]
  2. R. D. Cowan, Phys. Rev. 163, 54 (1967).
    [Crossref]
  3. In this paper, most energies will be given in units of one kilo-Kayser = 1000 cm−1.
  4. In collaboration with M. Wilson, some (unpublished) calculations have been made for La ii and Ce iii, each of which has a large number of closely-spaced and overlapping two-electron configurations. It was found to be essential to include relativistic and correlation corrections to predict even the correct ordering of the centers of gravity of most of the configurations; in such complicated cases, the present computational methods give errors about fivefold greater than that quoted above.
  5. Cf. B. Edlén, “Atomic Spectra,” in Handbuch der Physik, edited by S. Flügge (Springer-Verlag, Berlin, 1964), Vol. XXVII, p. 186, (Ref. 1).
  6. The author is indebted to K. Hallén, A. Borgström, and L. Minnhagen for providing values of the K ii and Ca iii 3p5nd energy levels in advance of publication.
  7. B. G. Wybourne, Spectroscopic Properties of Rare Earths (Interscience Publishers, John Wiley & Sons, Inc., New York, 1965), pp. 69–78.
  8. The existence of LS coupling conditions in Ca iii has been pointed out by Y. Shadmi (quoted by C. E. Moore), J. Opt. Soc. Am. 53, 886 (1963).
    [Crossref]
  9. A. H. Gabriel, B. C. Fawcett, and C. Jordan, Nature 206, 390 (1965).
    [Crossref]
  10. A. H. Gabriel, B. C. Fawcett, and C. Jordan, Proc. Phys. Soc. (London) 87, 825 (1966).
  11. For Ca iii, computed values are R1(3d,4d) = 23.0, R1(3d,5d) = 13.6, R1(3d,6d) = 9.4, R1(4d,5d) = 4.6, R1(4d,6d) = 3.2, and R1(5d,6d) = 2.0 kK. Corresponding values for Fe ix are 17.1, 4.6, 1.4, 5.8, 3.9, and 2.6 kK, respectively. (r1=g1=43.)
  12. C. E. Moore, Atomic Energy Levels, Natl. Bur. Std. (U. S.) Circ. No. 467, 3 vols. (U. S. Government Printing Office, Washington, 1949, 1952, 1958).
  13. J. C. Slater, Quantum Theory of Atomic Structure (McGraw–Hill Book Co., New York, 1960), Vol. II, pp. 287, 290 f.
    [Crossref]
  14. The parentage changes from one limit to the other are entirely analogous to the coupling changes in two-electron spectra discussed by R. D. Cowan and K. L. Andrew, J. Opt. Soc. Am. 55, 502 (1965) [see especially Fig. 4].
  15. Even viewing parentage notation as a serial-numbering device, the highest 3p4 3d2D in K iii to Ni xii has been misidentified9 as (1D) 2D. This is the result of mis-extrapolation to the (1D) 2D of Ar ii at 172 kK as listed in AEL [where (1S) 2D is erroneously given at 186 kK] instead of to the correct (1S) 2D at 179 kK [L. Minnhagen, Arkiv Fysik 14, 123 (1958)].
  16. The low reliability of theoretical dipole integrals when interference effects are large is well known; see, for example, D. R. Bates and A. Damgaard, Phil. Trans. Roy. Soc. (London) A242, 117 (1949).
    [Crossref]
  17. Though at first glance it would appear to be a strong coincidence that destructive interference should occur at a physically detectable integral value of Z, observations show only that intensities in Ti are no more than 10−2 of those in neighboring elements [B. Edlén, private communication]; since intensities are proportional to P2, this implies only that P= 0 occurs within ±0.1 of an integral value of Z so that there is a 20% chance of intensities being down by a factor of 10−2 in some element. Much greater coincidences have been observed—for example, a line in the spectrum of Pb i has been found [D. R. Wood, K. L. Andrew, A. Giacchetti, and R. D. Cowan, J. Opt. Soc. Am. 58, 830 (1968)] to suffer a purely fortuitous intensity loss by a factor smaller than 10−6!
  18. L. Å. Svensson and J. O. Ekberg, Arkiv Fysik 37, 65 (1968).
    [Crossref]
  19. In an earlier published version of Fig. 12 [R. D. Cowan and N. J. Peacock, Astrophys. J. 142, 390 (1965); Astrophys. J. 143, 283 (1966)], the ordinates (values of S∝ gfλ) were computed by hand on the assumption of pure LS coupling and 100% pure parentage. Comparison of the two figures gives an indication of the importance of parentage and LS mixing of the wavefunctions. (Wavelengths are somewhat different in the two figures because hfs wavefunctions were employed for the old calculations.)
    [Crossref]
  20. G. Racah, Phys. Rev. 63, 367 (1943), Eq. (19); or C. W. Nielson and G. F. Koster, Spectroscopic Coefficients for the pn, dn, and fn Configurations (The MIT Press, Cambridge, Mass., 1963), pp. viii and 4.

1968 (3)

1967 (1)

R. D. Cowan, Phys. Rev. 163, 54 (1967).
[Crossref]

1966 (1)

A. H. Gabriel, B. C. Fawcett, and C. Jordan, Proc. Phys. Soc. (London) 87, 825 (1966).

1965 (3)

In an earlier published version of Fig. 12 [R. D. Cowan and N. J. Peacock, Astrophys. J. 142, 390 (1965); Astrophys. J. 143, 283 (1966)], the ordinates (values of S∝ gfλ) were computed by hand on the assumption of pure LS coupling and 100% pure parentage. Comparison of the two figures gives an indication of the importance of parentage and LS mixing of the wavefunctions. (Wavelengths are somewhat different in the two figures because hfs wavefunctions were employed for the old calculations.)
[Crossref]

A. H. Gabriel, B. C. Fawcett, and C. Jordan, Nature 206, 390 (1965).
[Crossref]

The parentage changes from one limit to the other are entirely analogous to the coupling changes in two-electron spectra discussed by R. D. Cowan and K. L. Andrew, J. Opt. Soc. Am. 55, 502 (1965) [see especially Fig. 4].

1963 (1)

The existence of LS coupling conditions in Ca iii has been pointed out by Y. Shadmi (quoted by C. E. Moore), J. Opt. Soc. Am. 53, 886 (1963).
[Crossref]

The existence of LS coupling conditions in Ca iii has been pointed out by Y. Shadmi (quoted by C. E. Moore), J. Opt. Soc. Am. 53, 886 (1963).
[Crossref]

1958 (1)

Even viewing parentage notation as a serial-numbering device, the highest 3p4 3d2D in K iii to Ni xii has been misidentified9 as (1D) 2D. This is the result of mis-extrapolation to the (1D) 2D of Ar ii at 172 kK as listed in AEL [where (1S) 2D is erroneously given at 186 kK] instead of to the correct (1S) 2D at 179 kK [L. Minnhagen, Arkiv Fysik 14, 123 (1958)].

1949 (1)

The low reliability of theoretical dipole integrals when interference effects are large is well known; see, for example, D. R. Bates and A. Damgaard, Phil. Trans. Roy. Soc. (London) A242, 117 (1949).
[Crossref]

1943 (1)

G. Racah, Phys. Rev. 63, 367 (1943), Eq. (19); or C. W. Nielson and G. F. Koster, Spectroscopic Coefficients for the pn, dn, and fn Configurations (The MIT Press, Cambridge, Mass., 1963), pp. viii and 4.

Andrew, K. L.

Bates, D. R.

The low reliability of theoretical dipole integrals when interference effects are large is well known; see, for example, D. R. Bates and A. Damgaard, Phil. Trans. Roy. Soc. (London) A242, 117 (1949).
[Crossref]

Cowan, R. D.

R. D. Cowan, J. Opt. Soc. Am. 58, 808 (1968).
[Crossref]

Though at first glance it would appear to be a strong coincidence that destructive interference should occur at a physically detectable integral value of Z, observations show only that intensities in Ti are no more than 10−2 of those in neighboring elements [B. Edlén, private communication]; since intensities are proportional to P2, this implies only that P= 0 occurs within ±0.1 of an integral value of Z so that there is a 20% chance of intensities being down by a factor of 10−2 in some element. Much greater coincidences have been observed—for example, a line in the spectrum of Pb i has been found [D. R. Wood, K. L. Andrew, A. Giacchetti, and R. D. Cowan, J. Opt. Soc. Am. 58, 830 (1968)] to suffer a purely fortuitous intensity loss by a factor smaller than 10−6!

R. D. Cowan, Phys. Rev. 163, 54 (1967).
[Crossref]

The parentage changes from one limit to the other are entirely analogous to the coupling changes in two-electron spectra discussed by R. D. Cowan and K. L. Andrew, J. Opt. Soc. Am. 55, 502 (1965) [see especially Fig. 4].

In an earlier published version of Fig. 12 [R. D. Cowan and N. J. Peacock, Astrophys. J. 142, 390 (1965); Astrophys. J. 143, 283 (1966)], the ordinates (values of S∝ gfλ) were computed by hand on the assumption of pure LS coupling and 100% pure parentage. Comparison of the two figures gives an indication of the importance of parentage and LS mixing of the wavefunctions. (Wavelengths are somewhat different in the two figures because hfs wavefunctions were employed for the old calculations.)
[Crossref]

Damgaard, A.

The low reliability of theoretical dipole integrals when interference effects are large is well known; see, for example, D. R. Bates and A. Damgaard, Phil. Trans. Roy. Soc. (London) A242, 117 (1949).
[Crossref]

Edlén, B.

Cf. B. Edlén, “Atomic Spectra,” in Handbuch der Physik, edited by S. Flügge (Springer-Verlag, Berlin, 1964), Vol. XXVII, p. 186, (Ref. 1).

Ekberg, J. O.

L. Å. Svensson and J. O. Ekberg, Arkiv Fysik 37, 65 (1968).
[Crossref]

Fawcett, B. C.

A. H. Gabriel, B. C. Fawcett, and C. Jordan, Proc. Phys. Soc. (London) 87, 825 (1966).

A. H. Gabriel, B. C. Fawcett, and C. Jordan, Nature 206, 390 (1965).
[Crossref]

Gabriel, A. H.

A. H. Gabriel, B. C. Fawcett, and C. Jordan, Proc. Phys. Soc. (London) 87, 825 (1966).

A. H. Gabriel, B. C. Fawcett, and C. Jordan, Nature 206, 390 (1965).
[Crossref]

Giacchetti, A.

Jordan, C.

A. H. Gabriel, B. C. Fawcett, and C. Jordan, Proc. Phys. Soc. (London) 87, 825 (1966).

A. H. Gabriel, B. C. Fawcett, and C. Jordan, Nature 206, 390 (1965).
[Crossref]

Minnhagen, L.

Even viewing parentage notation as a serial-numbering device, the highest 3p4 3d2D in K iii to Ni xii has been misidentified9 as (1D) 2D. This is the result of mis-extrapolation to the (1D) 2D of Ar ii at 172 kK as listed in AEL [where (1S) 2D is erroneously given at 186 kK] instead of to the correct (1S) 2D at 179 kK [L. Minnhagen, Arkiv Fysik 14, 123 (1958)].

Moore, C. E.

The existence of LS coupling conditions in Ca iii has been pointed out by Y. Shadmi (quoted by C. E. Moore), J. Opt. Soc. Am. 53, 886 (1963).
[Crossref]

C. E. Moore, Atomic Energy Levels, Natl. Bur. Std. (U. S.) Circ. No. 467, 3 vols. (U. S. Government Printing Office, Washington, 1949, 1952, 1958).

Peacock, N. J.

In an earlier published version of Fig. 12 [R. D. Cowan and N. J. Peacock, Astrophys. J. 142, 390 (1965); Astrophys. J. 143, 283 (1966)], the ordinates (values of S∝ gfλ) were computed by hand on the assumption of pure LS coupling and 100% pure parentage. Comparison of the two figures gives an indication of the importance of parentage and LS mixing of the wavefunctions. (Wavelengths are somewhat different in the two figures because hfs wavefunctions were employed for the old calculations.)
[Crossref]

Racah, G.

G. Racah, Phys. Rev. 63, 367 (1943), Eq. (19); or C. W. Nielson and G. F. Koster, Spectroscopic Coefficients for the pn, dn, and fn Configurations (The MIT Press, Cambridge, Mass., 1963), pp. viii and 4.

Shadmi, Y.

The existence of LS coupling conditions in Ca iii has been pointed out by Y. Shadmi (quoted by C. E. Moore), J. Opt. Soc. Am. 53, 886 (1963).
[Crossref]

Slater, J. C.

J. C. Slater, Quantum Theory of Atomic Structure (McGraw–Hill Book Co., New York, 1960), Vol. II, pp. 287, 290 f.
[Crossref]

Svensson, L. Å.

L. Å. Svensson and J. O. Ekberg, Arkiv Fysik 37, 65 (1968).
[Crossref]

Wood, D. R.

Wybourne, B. G.

B. G. Wybourne, Spectroscopic Properties of Rare Earths (Interscience Publishers, John Wiley & Sons, Inc., New York, 1965), pp. 69–78.

Arkiv Fysik (2)

Even viewing parentage notation as a serial-numbering device, the highest 3p4 3d2D in K iii to Ni xii has been misidentified9 as (1D) 2D. This is the result of mis-extrapolation to the (1D) 2D of Ar ii at 172 kK as listed in AEL [where (1S) 2D is erroneously given at 186 kK] instead of to the correct (1S) 2D at 179 kK [L. Minnhagen, Arkiv Fysik 14, 123 (1958)].

L. Å. Svensson and J. O. Ekberg, Arkiv Fysik 37, 65 (1968).
[Crossref]

Astrophys. J. (1)

In an earlier published version of Fig. 12 [R. D. Cowan and N. J. Peacock, Astrophys. J. 142, 390 (1965); Astrophys. J. 143, 283 (1966)], the ordinates (values of S∝ gfλ) were computed by hand on the assumption of pure LS coupling and 100% pure parentage. Comparison of the two figures gives an indication of the importance of parentage and LS mixing of the wavefunctions. (Wavelengths are somewhat different in the two figures because hfs wavefunctions were employed for the old calculations.)
[Crossref]

J. Opt. Soc. Am. (4)

Nature (1)

A. H. Gabriel, B. C. Fawcett, and C. Jordan, Nature 206, 390 (1965).
[Crossref]

Phil. Trans. Roy. Soc. (London) (1)

The low reliability of theoretical dipole integrals when interference effects are large is well known; see, for example, D. R. Bates and A. Damgaard, Phil. Trans. Roy. Soc. (London) A242, 117 (1949).
[Crossref]

Phys. Rev. (2)

G. Racah, Phys. Rev. 63, 367 (1943), Eq. (19); or C. W. Nielson and G. F. Koster, Spectroscopic Coefficients for the pn, dn, and fn Configurations (The MIT Press, Cambridge, Mass., 1963), pp. viii and 4.

R. D. Cowan, Phys. Rev. 163, 54 (1967).
[Crossref]

Proc. Phys. Soc. (London) (1)

A. H. Gabriel, B. C. Fawcett, and C. Jordan, Proc. Phys. Soc. (London) 87, 825 (1966).

Other (8)

For Ca iii, computed values are R1(3d,4d) = 23.0, R1(3d,5d) = 13.6, R1(3d,6d) = 9.4, R1(4d,5d) = 4.6, R1(4d,6d) = 3.2, and R1(5d,6d) = 2.0 kK. Corresponding values for Fe ix are 17.1, 4.6, 1.4, 5.8, 3.9, and 2.6 kK, respectively. (r1=g1=43.)

C. E. Moore, Atomic Energy Levels, Natl. Bur. Std. (U. S.) Circ. No. 467, 3 vols. (U. S. Government Printing Office, Washington, 1949, 1952, 1958).

J. C. Slater, Quantum Theory of Atomic Structure (McGraw–Hill Book Co., New York, 1960), Vol. II, pp. 287, 290 f.
[Crossref]

In this paper, most energies will be given in units of one kilo-Kayser = 1000 cm−1.

In collaboration with M. Wilson, some (unpublished) calculations have been made for La ii and Ce iii, each of which has a large number of closely-spaced and overlapping two-electron configurations. It was found to be essential to include relativistic and correlation corrections to predict even the correct ordering of the centers of gravity of most of the configurations; in such complicated cases, the present computational methods give errors about fivefold greater than that quoted above.

Cf. B. Edlén, “Atomic Spectra,” in Handbuch der Physik, edited by S. Flügge (Springer-Verlag, Berlin, 1964), Vol. XXVII, p. 186, (Ref. 1).

The author is indebted to K. Hallén, A. Borgström, and L. Minnhagen for providing values of the K ii and Ca iii 3p5nd energy levels in advance of publication.

B. G. Wybourne, Spectroscopic Properties of Rare Earths (Interscience Publishers, John Wiley & Sons, Inc., New York, 1965), pp. 69–78.

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Figures (12)

Fig. 1
Fig. 1

An outline of two computer programs for ab initio theoretical calculation of atomic structure and atomic spectra.

Fig. 2
Fig. 2

Theoretical (HX) excitation energies 3p n –3p m −1nd and 3p m –3p m −1ns in the isoelectronic sequences Ar i to Al i, for Ar ivi, Ti vx, and Fe ixxiv.

Fig. 3
Fig. 3

Theoretical center-of-gravity ionization energies from the excited configuration 3p5 nl (expressed as quantum defect) for the Ar i isoelectronic series. [The label Cr viii should read Cr vii.]

Fig. 4
Fig. 4

Expectation values of r for the 3p and 3d electrons in the Ar i isoelectronic sequence 3p5 3d, and for the 4s electron in 3p5 4s.

Fig. 5
Fig. 5

Theoretical (HX) values of energy-interaction parameters for the configuration 3p5 3d in the isoelectronic sequence Ar i–Ni xi.

Fig. 6
Fig. 6

The effect of d-wavefunction collapse on coupling conditions in the configurations 3p5 3d and 3p5 4d. The curves represent theoretical predictions; the plotted points are results deduced by least-squares fitting of experimental energy levels.

Fig. 7
Fig. 7

The effect of d- and f-wavefunction collapse on coupling conditions in p5l configurations. The plotted points are experimental results deduced by least-squares fitting of energy levels.

Fig. 8
Fig. 8

Coupling changes in the configuration 2p5 3d of the Ne i isoelectronic sequence, due to modest increases of the 2p–3d interaction strength, with small values of the 2p spin–orbit interaction.

Fig. 9
Fig. 9

The dependence of the energies of the three 2D terms of p4d on the strength of the pd Coulomb interaction, both with [solid curves, F2(pp) = 60] and without [dashed curves, F2(pp) = 0] the presence of pp interactions. The predominant parentage composition of each 2D term is shown at various points along the curves. Conditions in Cl i 3p4 3d, and in more highly excited nd configurations for all members of the Cl i isoelectronic sequence, lie to the left of the vertical dotted line, where the lowest 2D is mainly of (3P) parentage and the highest is mainly (1S). The configuration 3p4 3d in all ions lies to the right of the dotted line, where parentage compositions are quite different.

Fig. 10
Fig. 10

Theoretical (HX) values of the reduced dipole-matrix element P3 p , ns ≡ (3p||r||ns) for 3p6–3p5ns transitions.

Fig. 11
Fig. 11

Theoretical values of the reduced dipole-matrix element P3 p , nd ≡(3p||r||nd) for 3p6–3p5 nd transitions. The dashed curve shows results evaluated from Hartree–Fock radial wavefunctions.

Fig. 12
Fig. 12

Schematic theoretical spectra of resonance transitions 3p m 3d n −1–3p m −1 3d n in ionized iron. The value of ΔEav is about 450 kK in all cases (Fig. 2), so that the left-hand portion of each spectrum lies outside the range of the figure; however, there is only one line in the omitted range that has a computed value of gf>0.1 (the minimum value used for plotting), namely, the line Fe xiii 3p2 1D–3p3d 1D at 401.6 kK (gf = 0.34). Multiplet designations for the upper configuration p m −1 d n include the parent term of d2 (Fe viii) or p m −1 (Fe xxii) which contributes most strongly to the calculated wave function.

Tables (5)

Tables Icon

Table I Theoretical (HX) excitation and ionization energies (in kK), relative to the center of gravity of the ground configuration. a

Tables Icon

Table II Theoretical (HX) ionization energies a (in kK).

Tables Icon

Table III Comparison of theoretical and empirical parameter values (in kK) for the configuration 3p53d.

Tables Icon

Table IV Computed energies in 3p53d configurations (kK), relative to 3p6 1S0.

Tables Icon

Table V Computed wavelengths (Å), oscillator strengths, and transition probabilities (sec−1) of the resonance lines 3p6 1S0–3p53d in the Ar i isoelectronic sequence.

Equations (3)

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

H b b = E av δ b b + i j k [ f k F k ( n i l i , n j l j ) + g k G k ( n i l i , n j l j ) ] + i d i ζ ( n i l i ) .
E av = [ Σ b ( 2 J b + 1 ) · E b ] / [ Σ b ( 2 J b + 1 ) ] ,
P l , l ( l r l ) = δ l , l ± 1 l > 1 2 ( - 1 ) l + l > 0 r R l ( r ) R l ( r ) r 2 d r