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Correlation Effects in Complex Spectra. II. Transition Probabilities for the Magnesium Isoelectronic Sequence
1.B. M. Glennon and W. L. Wiese, “Bibliography on Atomic Transition Probabilities,” Natl. Bur Std. (U.S.), Misc. Publ. 278, 1 (1966);
1.W. L. Wiese, M. W. Smith and B. M. Glennon, “Atomic Transition Probabilities,” NSRDS‐NBS 4 (U.S. Government Printing Office, Washington, D.C., 1966), Vol. 1.
2.D. R. Hartree, The Calculation of Atomic Structures (John Wiley & Sons, Inc., New York, 1957).
3.R. H. Garstang, Intern. Astron. Union 26, 57 (1966).
4.R. N. Zare, J. Chem. Phys. 45, 1966 (1966).
4.Errata: In the second form of Eq. (17) replace by in Eq. (18) replace by ; in Table I, entry B3, replace by and in entries B4 and B5 replace by Table I entries B2 and B3 are written for non‐equivalent electrons; for equivalent electrons an additional factor of √ must multiply I and A. A am indebted to Mr. Donald R. Beck, Department of Physics, LeHigh University, Bethlehem, Pa., for pointing out these corrections to me.
5.J. C. Slater, Phys. Rev. 81, 385 (1951).
6.F. Herman and S. Skillman, Atomic Structure Calculations (Prentice‐Hall, Inc., Englewood Cliffs, N.J., 1963).
7.A. W. Weiss, J. Chem. Phys. 47, 3573 (1967), following article.
8.E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge University Press, New York, 1935).
9.H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One‐and Two‐Electron Atoms (Academic Press Inc., New York, 1957);
9.S. Chandrasekhar, AstroPhys. J. 102, 223 (1945).
10.L. Goldberg, AstroPhys. J. 82, 1 (1935).
11.D. H. Menzel and L. Goldberg, AstroPhys. J. 84, 1 (1936).
12.C. W. Allen, Astrophysical Quantities (Athlone Press, London, 1955).
13.F. Rohrlich, AstroPhys. J. 129, 441, 449 (1959).
14.M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Wooten, Jr., The 3‐j and 6‐j Symbols (Technology Press, Cambridge, Mass., 1959).
15.J. C. Slater, “Accurate Methods of Atomic Calculation,” in Quantum Theory of Atomic Structure (McGraw‐Hill Book Co., New York, 1960), Vol. II, Chap. 18.
16.Equation (15) may also be shown to be equivalent to variationally determining the constants in (14) so that the energy of the atomic system is made an extremum. See J. C. Slater, Quantum Theory of Atomic Structure (McGraw‐Hill Book Co., New York, 1960), Vol. I, Appendix 9.
17.An expression for the calculation of absolute multiplet strengths from configuration interaction wavefunctions was first given by (a) C. Froese, AstroPhys. J. 140, 361 (1964).
17.A misprint in Froese’s equation was corrected by (b) R. J. S. Crossley and A. Dalgarno, Proc. Roy. Soc. (London) 286A, 510 (1965). Our Eq. (16) differs from Crossley and Dalgarno’s in that it contains the phase factor which is required when wavefunctions are used which do not belong to configurations built up from spin orbitals of the same principal quantum number.
18.Reference 8, pp. 168–169. See Table I of I for explicit values of for atoms having two electrons outside a core of closed shells.
19.E. U. Condon, Phys. Rev. 36, 1121 (1930).
20.W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
21.R. D. Cowan, A. C. Larson, D. Liberman, J. B. Mann, and J. Waber, Phys. Rev. 144, 5 (1966).
22.B. Y. Tong and L. J. Sham, Phys. Rev. 144, 1 (1966).
23.I. Lindgren, Arkiv Fysik 31, 59 (1965).
24.This parameter set was suggested by I. Lindgren and A. Rosén (private communication).
25.Our sources of spectroscopic data are the following: (a) for the Mg I spectrum G. Risberg, Arkiv Fysik 28, 381 (1964);
25.(b) for the AlII, PIV and Ca IX spectra, C. E. Moore, Natl. Bur. Std. (U.S.) Circ. 467, 1 (1949), and the references contained therein;
25.and (c) for the Si III spectrum, Y. G. Toresson, Arkiv Fysik 18, 389 (1960).
25.For the Si III spectrum see also C. E. Moore, “Selected Tables of Atomic Spectra” NSRDS‐NBS 3 Sec. 1 (U.S. Government Printing Office, Washington, 1965).
26.See J. E. Lennard‐Jones and J. A. Pople, Phil. Mag. 43, 581 (1952).
26.J. E. Lennard‐Jones, Proc. Natl. Acad. Sci. (U.S.) 38, 496 (1952) for a general discussion of spatial correlation in two‐electron atoms.
27.Table III indicates that the expansion coefficients change slowly in general along the isoelectronic sequence, so that the results shown in Fig. 1 for the ground state of Si III are quite similar to what is found for Mg I or Al II for example. Note in Fig. 1 for the curve the formation of a so‐called “Coulomb hole” due to the avoidance of the electrons.
28.For example, the lowest‐lying multiplet in the Ca IX spectra should be assigned to the configuration and not to the configuration as shown in Ref. 25(b). In Table II we have made the best assignments we could based on the configuration interaction wavefunctions. For further discussion see I.
29.Indeed this is confirmed by this work and the work of Weiss (Ref. 7). In particular Weiss has given the Hartree‐Fock values for many of the absolute multiplet strengths. In the case of the transition in Al II, the Hartree‐Fock and configuration interaction values differ by over two orders of magnitude.
30.A good portion of these programs have been documented in R. N. Zare, JILA Rept. No. 80 (Joint Institute for Laboratory Astrophysics, Boulder, Colorado, 1966). As an indication of the practicability of these calculations, it is worthwhile to mention some timing considerations. For example, the time required per element to calculate all energy levels, to find all wavefunctions and to compute dipole length and dipole velocity values of the absolute multiplet strengths for all possible transitions between the multiplets is about 10 min on a CDC‐3600 computer.
31.For example, see J. W. Swenson and G. Risberg, “Mg I lines in the Solar Spectrum,” Arkiv Fysik 31, 237 (1965).
32.Further information on correlated wavefunctions and oscillator strengths for the magnesium isoelectronic sequence may be obtained from the author on request.
33.S. Chandrasekhar, AstroPhys. J. 102, 223 (1945);
33.S. S. Huang, AstroPhys. J. 108, 354 (1948); , Astrophys. J.
33.D. R. Bates, J. Chem. Phys. 19, 1122 (1951);
33.H. Shull, J. Chem. Phys. 20, 18 (1952); , J. Chem. Phys.
33.M. Wolfsberg, J. Chem. Phys. 23, 793 (1955); , J. Chem. Phys.
33.G. Berthier, J. Chim. Phys. 51, 137 (1954);
33.S. Ehrenson and P. E. Phillipson, J. Chem. Phys. 34, 1224 (1961);
33.S. R. La Paglia, and O. Sinanoğlu, J. Chem. Phys. 44, 1888 (1966). I would like to thank Professor Paul Phillipson for suggesting to me the computation of the dipole length and dipole velocity forms of the absolute multiplet strength., J. Chem. Phys.
34.E. Trefftz, Z. AstroPhys. 28, 67 (1950). Trefftz’s work represents one of the earliest studies of correlation effects on the calculation of oscillator strengths, and she found significant deviations from Hartree‐Fock values when configuration interaction wavefunctions were used. The calculated values of S by Trefftz and this paper are in reasonable agreement by and large, at least if one includes only the configurations Trefftz used as her basis set. Besides the transition, another exception is the transition for which Trefftz found as compared to our value of in Table V.
35.A classic example of this effect is provided by the and states belonging to the configuration of the Be atom. If the self‐consistent field problem is solved separately for each state, it is found that the peak for the orbital for the state occurs at twice the distance that it occurs for the state. See D. R. Hartree and W. Hartree, Proc. Roy. Soc. (London) A154, 588 (1936).
36.A. Dalgarno, Proc. Phys. Soc. (London) 75, 439 (1960);
36.J. Linderberg and H. Shull, J. Mol. Spectry. 5, 1 (1960);
36.J. Linderberg, Phys. Rev. 121, 816 (1961);
36.M. Cohen and A. Dalgarno, Proc. Roy. Soc. (London) A261, 565 (1961).
37.D. Layzer, Ann. Phys. (N.Y.) 8, 271 (1959).
38.M. Cohen and A. Dalgarno, Proc. Roy. Soc. (London) A280, 258 (1964).
41.For as an example of the misleading nature of the Z‐expansion method for low Z, see E. Godfredson [AstroPhys. J. 145, 308 (1966)] who calculated atomic term energies for atoms and ions with 11 to 28 electrons by the Z‐expansion technique. In particular, Godfredson concludes that the lowest‐lying of the Mg I spectra should be reassigned to the configuration This is at variance with the findings of I.
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