Discrete-spectrum contributions to the Bauche-Arnoult

hyperfine structure parameters for the first row

transition metal atoms

W. Nowak, J. Karwowski, J. Dembczynski

To cite this version:

W. Nowak, J. Karwowski, J. Dembczynski. Discrete-spectrum contributions to the Bauche-Arnoult hyperfine structure parameters for the first row transition metal atoms. Journal dePhysique, 1984, 45 (4), pp.681-688. <10.1051/jphys:01984004504068100>. <jpa-00209798>

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Submitted on 1 Jan 1984

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681

Discrete-spectrum contributions to the Bauche-Arnoulthyperfine structure parameters for the first row transition metal atoms

W. Nowak, J. Karwowski

Instytut Fizyki, Uniwersytet Miko~aja Kopernika, Grudzi0105dzka 5, Toru0144, Poland

and J. Dembczy0144ski

Instytut Fizyki, Politechnika Pozna0144ska, Piotrowo 3, Pozna0144, Poland

(Reçu le 3 août 1983, révisé le 14 novembre, accepté le 2 décembre 1983)

Résumé. 2014 Les auteurs ont calculé par la technique Hartree-Fock les contributions du spectre discret aux para-mètres décrivant les effets croisés du second ordre des interactions électrostatiques et hyperfines (paramètres deBauche-Amoult) pour les configurations 3dN 4s et 3dN-1 4s2 des éléments du groupe du fer. Ils proposent uneméthode d’extrapolation pour estimer les contributions totales des états liés. Ils montrent que, dans le cas de laconfiguration 3dN 4s, les contributions les plus importantes proviennent des excitations 4s ~ ns, 3d ~ nd, ns ~ 3det ns ~ 4s; tandis que, dans le cas de la configuration 3dN-1 4s2, elles proviennent des excitations 3d ~ nd etns ~ 3d. Les contributions de 3d ~ ng et de 3d ~ ns sont toujours très petites.

Abstract. 2014 Hartree-Fock values of discrete spectrum contributions to the parameters describing crossed second-order effects of electrostatic and hyperfine interactions (Bauche-Amould parameters) have been calculated for3dN 4s and 3dN-1 4s2 configurations of the first row transition metal atoms. An extrapolation procedure is pro-posed to estimate the total bound state contributions. It is shown that in the case of 3dN 4s configuration the mostimportant are the 4s ~ ns, 3d ~ nd, ns ~ 3d and ns ~ 4s excitations, while, in the case of 3dN-1 4s2, they arethe 3d ~ nd and ns ~ 3d ones. The contributions from 3d ~ ng and 3d ~ ns excitations are always very small.

J. Physique 45 (1984) 681-688 AVRIL 1984,

Classification

Physics Abstracts31.20 - 35.1OF

1. Introduction

Recent developments in the laser spectroscopy [1, 2]allow to study with a great accuracy hyperfine struc-ture (hfs) of numerous levels in a given excited elec-tronic configuration. This provides new possibilities ofexploiting the effective-operator technique to studyconfiguration interaction (CI) effects on the hfs [3-6].The formal theory has been developed by Bauche-Amoult [7, 8] following the general method of Judd [9].It was shown that 13 parameters in the case of d’ sand 7 parameters in the case of dN - 1 s2 configurationdescribe all second-order effects of electrostatic and

hyperfine interactions. We shall refer to these para-meters as the Bauche-Amoult (BA) parameters. Theleast square fits of BA parameters for the groundconfigurations of some of the first row transition metalatoms have recently been determined by Dembczynski[3, 10]. Very little is however known on their Hartree-Fock (HF) values. The aim of this paper is to estimatethe discrete-spectrum HF values of BA parameters for

the case of 3dN 4s and 3dN - 1 4S2 configurations of theneutral iron group atoms. A comparison of the fittedvalues of the parameters to the HF ones contributes toan understanding of the absolute values of the indivi-dual parameters. On the other hand on the basis ofthe HF parameter values one may reduce the numberof independent parameters in the least squares proce-dure (e.g. by imposing a requirement for their ratios tobe equal to the HF ones).

2. Bauche-Arnoult parameters.

Four types of excitations contribute to the crossedsecond order hfs Hamiltonian for 3d’ 4s configura-tions : (a) excitation of a closed-shell electron to anempty shell, (b) excitation of an open-shell electron toan empty shell, (c) excitation of a closed-shell electronto an open shell, (d) an open to open shell excitation.Excitation types (a), (b) and (c) give contributions to3dN - 1 4s2 configurations Hamiltonian as well. Though

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01984004504068100

682

the closed to empty shell excitations (type (a)) givecontributions independent of the term and of thenumber N of 3d electrons, they are also taken intoaccount in our present study. For a discussion of thepossibility of an experimental estimation of their valuessee reference [3]. Excitation types (a), (b), (c) and (d)lead, respectively, to 4, 10, 2 and 2 radial hfs para-meters.

Detailed expressions defining the hfs parameterswere given by Judd [9] (excitation type (a)) and byBauche-Amoult [8] (excitations (b), (c), (d)). We shallmodify slightly the original notation in order to makeit more uniform and condensed Let no 10 and ni lidenote, respectively, the shells from and to which theexcitation takes place. Let us define the followingquantities :

where Rk is the Slater integral and I ð-E(no 10, n1 l1) is the absolute value of the difference between averageenergies of the two configurations involved. The second-order radial hfs parameters for the 3d’ 4s configurationsmay then be expressed in the following way :

(a) Closed to empty shell excitations :

where 10 = 1; li = 1, 3 ; k = 1, 3.(b) Open to empty shell excitations :

where

the values of A(k, 10, 11) coefficients are listed intable I. Moreover, in the case of 4s - nd excitations, an« exchange » parameter appears

where no 10 = 4s, h = d, nl = 3d; the prime in thesums means that both discrete and continuous spectracontributions should be included.

Table I. - Values of the A(k, 10, 11) coefficient.

(c) Closed to open shell excitations :

where 10 = s; n1 11 = 3d, 4s.(d) Open to open shell excitations :

here (no 10, ni 1,) pair is either (3d, 4s) or (4s, 3d).

683

A complete list of the radial parameters and thecorrespondence of the present notation to the one ofBauche-Arnoult [8] is given in table II. The parameterscorresponding to 3dN - 1 4S2 configurations are definedin the same way except that we have only one openshell in this case. In consequence, no open - open shellexcitations are possible. For open - empty shellexcitations no 10 = 3d and 11 = 0, 2, 4. In the case ofclosed -+ open shell excitations only one possibilityexists : 10 = s, n1 l1 = 3d.

3. Results of calculations.

3.1 HARTREE-FOCK VALUES. - Closed- and open-shell orbitals for the 3dN 4s and 3dN -1 4s2 configura-tions of Ti, V, Cr, Mn and Fe have been obtainedsolving the corresponding HF equations for the

average energy of the appropriate configuration. Thecalculations have been performed using the numericalHF program developed by Froese-Fischer [11]. Theempty shell orbitals are represented by a set of ortho-normal virtual orbitals, solutions of the HF equationsin which the Fock operator is built from the previouslyobtained, and then frozen, occupied orbitals. The

original Froese-Fischer program has been modified tocalculate Wk and Vk contributions to the radial hfsparameters. The HF values of the BA parameters givenby closed-form expressions (Eqs. (7) and (8)) . havebeen calculated exactly. The values of the other para-meters (Eqs. (4), (5) and (6)) have been estimated takingonly a few terms of the appropriate expansions. Allthese values are collected in table II, where the highestn 1 value taken into account is also specified in eachcase.

The convergence rate of the expansions (4), (5) and(6) may be estimated from the data displayed intable III, where the separate terms of the infinite sumsare listed. One can see that the convergence is rather

satisfactory. The nl = 9 term, in all the cases, consti-tutes 2-3 % of the parameter value. It is interesting tonote that the consecutive terms of the sum, for a givenexcitation, are equal to the same fraction of the totalparameter value. E.g., for the 3d -+ n, d excitation,n 1 = 4, 5, 6, 7, 8, 9 terms constitute respectively,51 %, 20 %, 10 %, 5 %, 3 %, 2 % of the four xknl(3d, d)parameters. Therefore it is reasonable to expect thatthe ratio of the hfs parameters, for each excitation type,is the same as that of the partial sums or even of W.k 1.3.2 ASYMPTOTIC EXTRAPOLATION VALUES. - Thoughthe convergence of the infinite series corresponding tothe excitations to the empty shells is fairly good, it isdesirable to estimate the contribution from the neglect-ed terms. One of the methods, rather commonly used,is based on the assumption that in equations (2) and (3)we may set

where DE is ni-independent. Then, using the expan-sion of the identity operator, we may replace equations(4), (5) and (6) by approximate closed-form expressions.

In the case under consideration this method suffersfrom an inconvenience - it leads to completely newtypes of integrals and, hence, its application is con-nected with an additional computational effort. On theother hand, due to an uncertainty in AE values, its

accuracy cannot exceed 10 %. Therefore we havechosen another, very simple method of estimation ofthe infinite sum contributions.For hydrogenlike orbitals, using results of [12], [13]

and [14], after some rather tedious algebra, we mayderive asymptotic expressions for Wknl and Vn in thelimit of n1 -+ oo, n, &#x3E;&#x3E; 11 :

where B, C and D are constants. Since virtual HF orbi-tals are rather well approximated by the properlyscaled hydrogenic ones, we may use (10) as an extra-polation formula for Wknl, with B, C and D beingadjustable parameters. The values of B, C and Dhave been determined by the least square method inwhich the expression (10) is adjusted to the HF valuesof Wknl(Vknl ). Since

is the two-parameters Riemann zeta function [15], wehave, according to (5),

Parameters xk3d(l0, l1) and y’,(no 10, l1) may be express-ed in a similar way using equations (4) and (5). Oneshould remember that the procedure is applicable tothe discrete spectrum contributions only.The accuracy of the extrapolation procedure has

been tested in a calculations of X23d(3d, d) parameterfor the 3d’ 4s1 configuration of Fel. First, the HFvalues of W;d(3d, nd) were calculated for n = 4,5,..., 12. Then a number of the least square calculationsof B, C and D were performed, taking into account onlya part of the available W23d parameters. The HF valuesof the remaining ones were compared with the extra-polated values calculated using equation (10). Theextrapolation error

where EP means that the quantity was extrapolatedand p is equal to the number of HF values used in theleast square fit of B, C and D, is a measure of the

accuracy of the method. The HF values of W23d, theextrapolation formula constants, the correspondingx23d values and the extrapolation errors are listed intable IV. As it is seen, the extrapolation procedure issurprisingly accurate. The extrapolation error does notexceed 4 % of a single W23d value and the final values ofthe BA parameters are practically independent of the

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Table II. - HF values of the second-order radial hfs parameters for the 3dN 4s and 3dN - 1 4s2 configurations ofTi, V, Cr, Mn and Fe. For parameters expressed by infinite sums, extrapolated values (EP) are also given.

a The highest excitation taken into account in the infinite expansion.b All the quantities have to be multipled by the power of 10 given in the corresponding entry.I The second-order perturbation theory is nonapplicable in this case. See text for a discussion.

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Table III. - Hartree-Fock values of A(k, lo, li) W:,(no 10, ni l1) and A(k, 10, l1) Vnl(n0 lo, ni 11), ni 9, forFe(3d7 4s1). In parentheses the percentage of the total parameter value is given.

a All the quantities in the same column have to be multiplied by the power of 1 o given in the corresponding entry.b Extrapolated values (Eq. (12)).

Table IV. - Hartree-Fock values of W23d(3d, n1 d) x 10-6 for Fe(3d’ 4s1), extrapolation formula parameters,values of X2 (3d, d) x 10-4 and the extrapolation formula errors.

a Eq. (5) partial sum with n1 = 4, 5, ..., 12.

b Eq. (12).

686

number of W2 parameters used in the least squareprocedure.The optimized values of B, C and D for the parame-

ters connected with no I --+ nl I excitations are col-lected in table V. The extrapolated (according toEq. (12)) hfs parameters are given in table II togetherwith the corresponding HF partial sums.

Since the differences between the partial sums andthe extrapolated values do not exceed 20 % of theparameter values and these differences are determinedwith an accuracy of about 4 %, we estimate, that thediscrete spectrum contributions to the hfs parametershave been determined with at least 1 % accuracy. Theextrapolation has not been performed for X3d(3d, g) andX’3d(PI 0. Their discrete spectrum HF values are by5-7 orders of magnitude smaller than the ones ofx!,(4s, s) or xt,(3d, d).

4. Discussion

The second-order hfs parameters enter the energyexpressions as contributions to the scaling factors ofthe standard (first-order) hfs parameters [16]. We have

where a is the first-order hfs parameter, identical forall levels in one configuration, ASL is a CI generatedcorrection and aSL is the SL-dependent parameterwhich includes the electron correlation effects. The CIcorrection L1SL’ within the second order perturbationtheory, is expressed as a linear combination of the BAsecond-order hfs parameters with coefficients rangingfrom 3 to 1/300 [3, 7, 8]. Hence, only the parameterswhich are not very small compared to 1 have to betaken into account in a parametric hfs theory. As it wasalready mentioned, the discrete spectrum contribu-tions to the second-order parameters connected with

no p - ni f and 3d - ng excitations are always negli-gible (I x:,(P, f) [ 3 x 10-6 and 1 x:,(3d, g) [ 5 x

10- 8). However an inspection of table II shows thatalso 3d -+ ns, 4d - nd and open - open shell (3d - 4s)excitations generate contributions to the pertinentparameters by 2-4 orders of magnitude smaller thanthe ones connected with no I -+ ni I and with closed

open shell excitations. In particular, ( x3d(3d, s) / 3 x 10-4, x 2 (4s, d) [ 4 x 10-5, y2 (4s, d) 2 x 10-4, and I x 2 3d (4s, 3d) [ 1 X2 3d (3d, 4s) 3d I 10-3.Then, the only second-order parameters containingappreciable discrete-spectrum contributions are thoseconnected with 3d -+ nd, 4s - ns, no p - n1 P andclosed -&#x3E; open shell excitations.The continuous spectrum states may rather substan-

tially influence the values of the hfs parametersdescribing excitations to the empty shells [17, 18]. Asit was shown by Wolter [17], the continuum contri-butes much stronger than the bound states to theStemheimer correction factor R [19], being closelyrelated to the second-order hfs parameters. AlsoBauche-Amoult and Labarthe [18] in their calculationof the second-order effects on the hfs parameters in Scand Ti, where continuum contributions were impli-citly taken into account, have shown that p - f and3d - g contributions are far from being negligible.Hence, a smallness of the discrete spectrum contribu-tions does not imply that the corresponding parametersmay be omitted in a parametric hfs calculation. On theother hand the continuum contributions do not

change values of the parameters connected withclosed -&#x3E; open and closed -&#x3E; closed shell excitations.

Hence, the present calculation gives a rather exactestimation of the values of these parameters.As it was already stated, the ratios of the parameter

values corresponding to the same excitation type arealmost independent of the number of terms taken intoaccount in the second-order perturbation series expan-sion. An inspection of table II shows that the ratios arealso almost independent of N. The ratios of the para-meters associated with the 3d - ni d excitations are

Table V. - Extrapolation formula constants for hfs parameters describing 3d - nd and 4s - ns excitations.

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given in table V I. A weak linear N-dependence appearsin the case of x4s(3d, d) IX2r 3,(3d, d) ratio only. It is

interesting to note that the parameter ratios for boththe considered configurations are quite similar, whiletheir absolute values are rather different. The N-depen-dence of the BA parameters is regular and may beapproximated by quadratic functions of N. The

corresponding expressions are collected in table VI.One can expect that these observations may be extend-ed to the exact (including continuum contributions)parameter values. Consequently, the expressions givenin table VI may be applied to reduce the number ofindependent parameters in semiempirical parametrichfs calculations.The experimental least square fits of the BA para-

meters for the 3d’ 4s configuration atoms have recentlybeen determined by Dembczynski [3, 10] and byJohann [20]. The set of fitted values of the parametersis still rather incomplete and not very accurate. Forexample, in the case of the 3dN 4s configuration ofiron, the fitted values of X23d(3d, d) and X23d(4s, s) vary,respectively, from 0.08 [3] to 0.12 and from 0.07 [3] to0.19 [20]. Therefore it is premature to perform a detail-ed comparison of the fitted and calculated parametervalues. In general, the fitted parameters are greater (inabsolute values) by a factor of 2-3 than the calculatedones. The general form of the N-dependence and therelative magnitude of the fitted parameters are howeverrather well reflected by the ones calculated using thediscrete spectrum states only. This result confirms ourexpectation concerning a practical usefulness of theexpressions collected in table VI.The most important factor responsible for the

differences between the fitted and the calculated para-meter values is certainly the neglect of the continuousspectrum contributions. Another reason of the discre-pancy is a limited applicability of the perturbationtheory. The CI effects may be treated by the secondorder perturbation theory under condition that the

Table VI. - Approximate N-dependence of the mostimportant (greater than 10- 3) Bauche-Arnoult para-meters and xk, (3d, d)/X2 (3d, d) ratios for the 3dN 4sand3dN - 1 4s2 configurations. All the parameter valueshave to be multiplied by 10- 3.

interacting configurations are separated well enough.This condition is usually fulfilled, except for the

open open shell excitations. The separation of theHF average energies of 3d’ 4s and 3d-’ 4S2 configu-rations is rather small but noticeable. However for Tiand for V they are practically degenerated. Therefore,for this kind of excitations, the perturbation theoryresults in the case of Fe, Mn and Cr are rather inaccu-rate. In the case of Ti and V the perturbation theory isnot applicable and consequently, the correspondingparameter values have not been included in table II.From this point of view, a uniform treatment of the(3d + 4S)N+1 configuration would be much moreadequate. An importance of this kind of effects hasalready been pointed out by Bauche and Bauche-Amoult [21].

Acknowledgments.

We wish to acknowledge a referee for useful comments.The work has been sponsored by the Polish Academyof Sciences, contract No. MR. 1. 5.

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