JOURNAL OF RESEARCH of the National Bureau of Standards

Vol. 86, No. 1, January-February 1981

The Configurations 3d "4p+3d4s4p+3d-24s24p in the First Spectra of the Iron Group

Charles Roth

Department of Mathematics, McGill University, Montreal, Quebec

September 1, 1980

Energy levels and Lande g-factors for the configurations 3d4p+3d-14s4p+3d-24s24p in the first spectra of the iron group were calculated and compared with experimental values, in both general and individual treatments. The calculations were done in intermediate coupling taking into account explicity the interactions between configurations, as well as complete effective interactions of the core, and effective interactions d - p. Due to a successful starting point based on Hartree-Fock calculations for the Slater parameters, as well as the insertion of the effective interactions, considerable improvement was obtained compared to previous results.

On fitting 1537 levels using 67 free interaction parameters a mean error of 182 cm ́1 was obtained. Altogether 3652 energy levels were calculated including all the levels for the configurations 3d-24s24p across the sequence. It was shown that all interaction parameters could be expressed either as linear functions, or linear functions with small quadratic corrections, of the atomic number. There was general qualitative agreement between the values of the parameters calculated using the semi-empirical method and those calculated using Hartree-Fock methods. There remained some isolated terms with large deviations. These are attributed to be due to the interactions with the configurations (3d+4s) 5p, that were not considered explicitly in this analysis.

Tables comparing the experimental and calculated energy levels and Lande g-factors, as well as detailed analyses for each spectrum are given in another paper.

Key words: Iron group elements; least squares optimization; theoretical spectroscopy.

1. Introduction.

Traditionally, theoretical spectroscopists consider the radial Slater integrals as unknown parameters, obtaining their values empirically by fitting the experimental data to the calculated energy levels, and then performing least-squares optimization calculations.

For the even configurations 3d+3d-14s, and the odd configurations 3d4p in the second and third spectra of the iron group, the results were excellent, [1-4]. Furthermore, it was shown that the radial parameters are either linear functions, or linear functions with small quadratic corrections, of the atomic number. For the odd configurations in neutral atoms, the interactions between configurations are very strong. Thus the algebraic matrices of the configurations (d+s)'p were calculated and checked by the author, [5-9]. Theoretical investigations were then performed for the configurations (3d+4s)'4p in neutral atoms of calcium, scandium, titanium, vanadium, chromium, manganese, iron, cobalt and nickel, [10-16]. Although the results were good (average r.m.s. error of 210cm 1); a very disturbing feature of the results was the fact that the behavior of the final values of the radial parameters was generally far from linear.

It would be highly anomalous to have the radial parameters behave so irregularly in the first spectra. Thus, in order to overcome this discrepancy it was essential to have improved initial values of the radial parameters. Hence the radial parameters were first calculated using the Hartree-Fock method. Leastsquares optimization calculations were then performed on these parameters forcing them to behave linearly, with possibly at most small quadratic corrections, as functions of the atomic number. The values thus computed were then compared with those obtained previously in individual treatments by the author, [10-16]. Whenever the Hartree-Fock values were uniformly higher or lower than those of the previous results, [10-16], appropriate scaling factors were utilized on the linearized Hartree-Fock values, and the latter were than used as initial parameters for this investigation.

'Figures in brackets indicate literature references at the end of this paper.

In this project were included the electrostatic and spin-orbit interactions of the individual configurations 3d4p, 3d-14s4p and 3d-24s24p; the explicit electrostatic interactions between configurations 3d4p-3d-14s4 p, 3d-14s4p-3d-24s24p and 3d4p-3d-24s24p; and the complete two- and three-body effective interactions of the core d electrons, as well as two-body mixed effective interactions between the 3d and 4p electrons.

The initial values of the radial parameters were then used to multiply the algebraic matrices on tape and the resulting matrices were diagonalized. Besides the eigenvalues, the diagonalization routine also yields the derivatives of the eigenvalues with respect to the parameters, the squares of the eigenvectors (percentage compositions) and the calculated Lande g values. The appropriate experimental levels were then fitted to the eigenvalues, and using the derivatives obtained in the diagonalization, least squares optimization calculations were performed. In these calculations, the improved values of the theoretical energy levels, the corrected values of the parameters including their statistical deviations and the sum of the squares of the differences between the observed and the calculated levels, were obtained. The rms error is then defined as

where the ▲, are the differences between the observed and calculated levels, n is the number of known levels and m is the number of free parameters. The mean error is quite different from the mean deviation

as the former takes into account the statistical effect of the number of free parameters. Hence in order for a new parameter to have physical significance, it should cause an essential decrease in the rms error, and not simply a decrease in the mean deviation.

The value of ▲ is also given by the least-squares routine. The same derivatives can be used for several variations in the least squares, either imposing different conditions on the parameters, inserting the experimental levels with different assignments, or even rejecting some levels from consideration. The parameters of that variation which yielded the best results were used to perform new diagonalizations. This iterative process was continued until mathematical convergence was attained. In the present project four complete iterations were required.

The use of the same assumptions and the same approximations in all the spectra made it possible to obtain a consistent set of interaction parameters and compare the results obtained from the spectra of different elements. Due to a successful choice of the initial values of the radial parameters, it was shown that the final values can indeed be expressed as simple functions of the atomic number. A consistent use of such interpolation formulas for all parameters, combined eleven problems, formerly independent, into one problem. This result, which is significant by itself, very much improved the reliability of the results for those spectra where the experimental data is still scarce, and which are thus most in need of reliable predictions of the unknown levels.

This is particularly true for the configurations 3d-24s24p. For each individual element there in not a sufficient number of experimental levels in order to predict even approximately the remaining levels. However, by considering a general treatment ALL the levels of the configurations 3d-24s24p for the entire sequence were calculated.

For completeness and comparison, individual least squares (ILS) were also performed for each element.

The procedures followed, a description of the various interactions considered, and an analysis of the results and significance of the different parameters are contained in this work. The tables comparing the experimental and calculated energy levels, values for all the theoretical levels specifying their percentage compositions, as well as detailed analyses for each spectrum are given in another paper, [17].

2. Effective electrostatic interactions

For the odd configurations in the first spectra of the iron group, both strong and weak configuration interactions are significant. The former arise when the perturbing and perturbed configurations are energetically close to each other and there is strong coupling of the configurations by the Coulomb field. These were taken into account by explicitly considering the configurations 3d4p, 3d-14s4p, 3d-24s24p and the electrostatic interactions between them. Weak interactions occur when the perturbing configurations are well separated from the perturbed configuration, and the coupling of the Coulomb field is weak.

The individual weak interactions may not be significant, but their cumulative influence may be quite large, due to the increasing density of states as the continuum is approached. As it would be completely futile to consider each of these effects individually, the aim should be to modify the energy matrices of the principal configurations so that the major part of all the weakly perturbing configurations be included.

By first order perturbation theory, different configurations do not interact. In second order only those configurations interact that differ in the quantum numbers of at most two electrons. Bacher and Goudsmit, [18], have shown that the terms of the configuration may be expressed as linear combinations of the terms Ւ of P, so that the perturbation of by all the configurations differing from it by the state of two electrons, and being distant from it, can be accounted for by suitably modifying the terms of P. Hence these perturbations can be described by two-body effective interactions.

The first correction of this kind for the configurations d was the aL(L+1) correction introduced by Trees, [19-20], in the configurations 3d4s of MnII and FeIII. Trees introduced his correction empirically, but Racah, [21], showed that the above effects can be described by a model or effective interaction of the form

where q12 is the seniority operator, [22]. For the configuration d" this becomes 912

where

a[L(L+1)-6n] + BQ

Q(n,v) = 14(n-v)(4l+4―n—v)
= 14(n-v)(12-n-v)

(1)

is the total seniority operator. Here n is the number of d electrons in the configurations dp, v is the seniority of the d core term, and I is 2 as we are dealing with d electrons. The constant −6na is usually incorporated into the height of the configuration. Racah, [21], showed that the aL(L+ 1) and the BQ corrections form a complete set of two-body effective interactions for the d configurations. This is due to the fact that together with the Slater integrals Fod2), F(d2) and F(d), they form a set of five independent parameters that can represent the five terms of ď2.

Bacher and Goudsmit, [18], also showed that if the far-lying perturbing configuration differs from by the state of only one electron, its effect can be described by expressing the terms of as linear combinations of P3, and modifying the values of these terms. Hence in the linear theory, the Hamiltonian in this case must be augmented by additional three-body interactions.

Rajnak and Wybourne, [23], obtained explicit formulas for the effective interactions representing the perturbation of an configuration by far-lying configurations differing from it by one or two electrons or holes. Racah and Stein, [24], subsequently, developed an elegant method that considerably simplified the calculations of Rajnak and Wybourne.

If A and B represent the perturbed and perturbing configurations, respectively, and if G is the operator e2 representing the Coulomb energy of repulsion between the electrons, then the matrix elements of Tij the second-order perturbation produced by B on A are given approximately by

1

(A¥ | W2│A¥') =

(A|G|By") (B¥" | G|4¥'),

(2) where AE is the distance between the centers of gravity of the two configurations, which are assumed to be well separated. According to Racah and Stein, [24], the operator G in the first factor is replaced by a "curtailed" operator g, whose matrix elements (A' | g| B'4") are equal to those of G if A' = A and B' = B, and vanish otherwise. Similarly, the operator G in the second factor is replaced by g, defined analogously to g. Then

and hence the electrostatic interaction between the configuration A and all the other distant configurations may be simply expressed as an effective interaction within the configuration A given by

Then using either the above method or that of Rajnak and Wybourne, [23], we obtain that the correction term W2, that must be added to the Hamiltonian of " caused by the perturbation of " by "-1 l'configurations is given by

The matrix elements of l, can be calculated by Racah algebra. The variables k and k' are even and nonzero integers that must satisfy the usual triangular conditions of the 6-j symbols. The variables k" can assume all integral values consistent with the triangular conditions for the 6-j symbols.

The parameter Trepresents the perturbation of the configuration 3d" by the configuration 3s3d"1. It was first considered by Trees, [25], when he investigated the configuration 3s23p°3ď, and took into account its

interaction with the configuration 3s3p 3d. Shadmi, [26] extended the work of Trees to all configurations 3d" +3d"-14s +3d"-24s2 in the sequence of the second spectra of the iron group by introducing a three-body effective interaction between 3d electrons which represented the perturbation of a configuration of the type 3s23d" by the configuration 3s3d"*1. Roth [3–4] included the parameters a, ẞ and T in the configurations 3d4p in the second and third spectra of the iron group, as well as for the configurations 3d34p+3d24s4p in V II, [27].

Further important investigations were carried out by Shadmi, Stein, Oreg, Caspi, Goldschmidt and Starkand [28-30].

Now from (10) with kk' = 2 here, we have W1⁄2= P(22;3d3d, 3d3s)' (22; dd,ds).

used in a previous work by the author, [7], we get for the coefficient operator of T the expression

The parameters T,, Ty, T, represent the perturbation of the configuration 3d" by a configuration of the type 3d"-1n'd, where n' ≥ 4. Similarly, to the above result for t their coefficient operators t、, t,, t, are

Only the parameters T and T, were used as T, and T, depend upon the other parameters.