Numerical methods of the preliminary evaluation of the role of admixed configurations in atomic calculations

Computer Physics Communications 134 (2001) 321–334 www.elsevier.nl/locate/cpc

Numerical methods of the preliminary evaluation of the role of admixed configurations in atomic calculations P. Bogdanovich 1 , R. Karpuškien˙e State Institute of Theoretical Physics and Astronomy, Goštauto st. 12, Vilnius 2600, Lithuania Received 21 April 2000

Abstract Methods of the preliminary evaluations of the averaged energy correlation corrections and averaged configuration mixing weights arising from the separate admixed configurations are presented in this paper. These methods are based on the usage of the second order of the perturbation theory and analytical expressions of the sums of the squares of the interconfigurational matrix elements and the averaged energy differences between configurations. Two possible ways to average these energy differences are discussed. Large number of calculations for different atomic states shows a good agreement of the preliminary evaluations with averaged results from diagonalization of energy matrices.  2001 Elsevier Science B.V. All rights reserved. PACS: 31.15.Ar; 31.25.Eb; 31.25.Jf Keywords: Admixed configuration; Energy correlation correction; Mixing weights; Configuration interaction; Perturbation theory

1. Introduction Multiconfigurational approach (MCA) is currently the main and the most universal method for taking into account correlation effects in calculation of energy spectra and other characteristics of many-electron atoms. The essence of the method is account for interconfigurational matrix elements of the energy operator (ICME) when calculating its matrix within the basis set involving the configuration being adjusted and a great number of additional (admixed) configurations. Depending upon the basis set of radial orbitals used, there are different levels of MCA; from the simplest superposition-of-configurations, when all electronic states are described by the conventional Hartree–Fock radial orbitals (RO), up to the multiconfigurational Hartree–Fock approximation when the multiconfigurational character of the sought wave function is introduced already at the stage of solving equations for radial orbitals [1,2]. The latter method has significant advantages in comparison with superpositionof-configurations, because it provides the most rapid convergence of the applied approximation with respect to the number of admixed configurations. However, whatever the RO basis set used, if the high accuracy of the energy of the atomic state under study is required and the multiconfigurational wave function describing adequately this state is necessary, one needs superposition of a great number of admixed configurations. This requires formation E-mail address: [email protected] (P. Bogdanovich). 1 Corresponding author.

0010-4655/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 0 - 4 6 5 5 ( 0 0 ) 0 0 2 1 4 - 9

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of quite an extensive RO basis set with a wide interval of orbital quantum numbers. Extension of the RO basis set used leads in its turn to fast growth of possible virtual excitations; the number of possible admixed configurations is proportional to the square of the number of additional RO. The ranges of energy operator matrices are increasing more rapidly. Calculation and diagonalization of extensive energy matrices may pose quite a difficult task even for powerful modern computers. However, the problem simplifies significantly, if one realizes that the influence of different admixed configurations upon the energy of the system under study and their contributions to its wave function are far from being similar. Thus, there appears a problem of selecting the most important admixed configurations from their whole set. Of course, anyone who applied MCA in the calculations was familiar with this problem. There is a vast amount of studies published during quite a long period of time where the role of admixed configurations was discussed and certain regularities of their application were found (see, e.g., [2–6]). Unfortunately, general recommendations are almost helpless when the number of possible admixed configurations becomes great and the number of those taken into account amounts to hundreds or even thousands. On the other hand, all recommendations are by no means absolute, because the role of different admixed configurations may vary quite strongly both within isoelectronic series and for different population of separate shells. It should also be noted that the role of different admixed configurations depends significantly upon the RO basis set employed to describe them. All the above leads us to a conclusion that it is in fact impossible to formulate any general recommendations that would allow one to determine a priori a sufficiently large set of admixed configurations important for the atomic state under study. Therefore, we encounter the problem of evaluating the role and selecting the most significant admixed configurations in each particular case, i.e. for a certain individual configuration under study and a given RO basis set. For several years, when performing particular calculations in many works (see, e.g., [7–9]) we used the technique that allowed evaluation of the influence of separate admixed configurations and their selection for MCA calculations based on the second order of the perturbation theory (PT). The present work deals both with the mentioned technique itself and its significant improvement. The following section presents the main ideas of the method. Section 3 compares the results of approximate evaluation with the corresponding values obtained by straightforward diagonalization of the energy matrix. The last section summarizes the main conclusions of this work.

2. The technique for evaluation of the contribution of admixed configurations For calculating ICME during the multiconfigurational procedure, the principal terms of the energy operator are commonly considered, namely, the kinetic energy operator and the operators of electrostatic electronic-nuclear and
is employed throughout this work as well. In this case the energy interelectron interactions. Such a Hamiltonian H of terms of configuration K does not split into levels, i.e. it depends only on the intermediate quantum numbers T and total orbital and spin momenta LS of the term. ICME behave in a similar way. In the second order of PT, the correction to the term energy caused by account for the admixed configuration K
is expressed via a well-known formula δE(KT LS, K
) =

 T



|K
T
LS2 KT LS|H .
|K
T
LS − KT LS|H
|KT LS K
T
LS|H

(2.1)

Here the square of ICME stands in the numerator and the energy differences of the corresponding terms in the denominator. For the sake of further simplicity, (2.1) is written so that the energy correction has a plus sign in the case when the term energy decreases.

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323

In order to evaluate the averaged influence of a particular admixed configuration under study, one should average the corrections (2.1) over all the terms of the configuration under study:  (2L + 1)(2S + 1)δE(KT LS, K
)
. δE(K, K ) = T LS  T LS (2L + 1)(2S + 1) The sum in the denominator is a statistical weight of the configuration considered, below it is mentioned as g(K). In this case the averaged correction is δE(K, K
) = g −1 (K)

 T LST



|K
T
LS2 (2L + 1)(2S + 1)KT LS|H .
|K
T
LS − KT LS|H
|KT LS K
T
LS|H

(2.2)

To treat further the expression (2.2), one should perform usual simplification based on the idea that the term energy difference in the denominator depends relatively slightly upon their quantum numbers and may be replaced by some average value E(K
, K). This replacing may be done correctly only if energy spectra of the configurations do not overlaps, i.e. energy difference between configurations is not smaller rather splitting of the spectra. Such an approach has been widely used in the atomic theory since the works [10,11], where the nature of semi-empirical corrections was explained. In this approximation, the average correction may be written as follows:  −1 
|K
T
LS2 . δE(K, K
) = g(K) E(K
, K) (2L + 1)(2S + 1)KT LS|H (2.3) T LST


Now, we can sum up in (2.3) over all intermediate and total momenta, i.e. to obtain analytical expressions for the sums of squares of ICME. These sums are denoted below as Θ(K, K
). The analytical expressions of Θ(K, K
) for all cases of possible types of two-electron and one-electron virtual excitations may be found in [12,13]. Thus, the averaged energy correction may be expressed in the following symbolic form: δE(K, K
) =

Θ(K, K
) , g(K) E(K
, K)

(2.4)

which must be further individualized depending on the virtual excitation employed to form the admixed configuration. For calculating the averaged correction by (2.4), along with Θ(K, K
), one should also calculate the energy difference between configurations E(K
, K). The simplest way is to use the well-known formula for the configuration energy averaged over all its terms. In this case, E(K
, K) = E(K
) − E(K) 
|K
T
L
S
 (2L
+ 1)(2S
+ 1)K
T
L
S
|H = g −1 (K
) T
L
S


−g

−1

(K)




|KT LS. (2L + 1)(2S + 1)KT LS|H

(2.5)

T LS

Thus, for the correction (2.4) we get the following expression: δE(K, K
) =

Θ(K, K
) g(K) E(K
, K)

.

(2.6)

The results obtained by (2.6) with the help of the analytical expressions of Θ(K, K
) and E(K
, K) were successfully used in the above-mentioned [7–9] and other calculations for determining the sets of configurations to participate in the superposition. The comparison of the diagonalization results with the data obtained in the preliminary evaluation has shown that the selection of configurations using this technique is in general quite reliable. However, some deviations from the preliminary estimates also occur. They mainly arise in the case when

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their own widths of the spectra of the adjusted and admixed configurations are comparable in magnitude with the averaged energy difference between the configurations. To make the estimates more accurate, one should use alternative definition of the energy difference E(K
, K)between the configurations. This may be achieved if not only the statistical weights of the terms themselves (leading to (2.5)), but also the ICME squares are used as the weight coefficients to describe the energy difference of the two configurations. In this case this difference may be presented as follows [12,13]: 
|K
P
LS2 (K
P
LS|H
|K
P
LS − KP LS|H
|KP LS)
(2L + 1)(2S + 1)KP LS|H
 . E(K , K) = LSP P  2


P LSP
(2L + 1)(2S + 1)KP LS|H |K P LS
is diagonal with This expression is written within the intermediate coupling scheme, in which the operator H ˜ respect to the intermediate quantum numbers P of the each configuration. As follows from [12,13], E(K, K
) may be written and in the standard LS-coupling scheme  
|K
T
LS 
, K) = Θ −1 (K, K
) (2L + 1)(2S + 1)KT LS|H E(K LST T
T






|K
T

LSK
T

LS|H
|KT LS ×K T LS|H 
|KT
LS − (2L + 1)(2S + 1)K
T LS|H LST T
T










×KT LS|H |KT LSKT LS|H |K T LS .


(2.7)


, K) also involves matrix elements of the energy operator As seen from (2.7), in this case the definition of E(K describing interactions inside the configurations, off-diagonal with respect to the intermediate quantum numbers T . So, as follows from [12,13], (2.7) makes it possible to obtain the energy difference that takes into account the results of Hamiltonian matrix diagonalization within individual configurations. The expression (2.7) allows more accurate description of the energy difference between the configurations for the purposes of the present work, because it takes into account the energy differences only between those terms that are coupled via ICME and for the desired correction (2.4) we get  δ E(K, K
) =

Θ(K, K
) . 
, K) g(K) E(K

(2.8)

Similarly, as in the case of sums of ICME squares, it is impossible to write down a unique analytical expression 
, K) for different types of virtual excitations. Such formulae have already been derived for all the types of E(K of admixed configurations and are currently verified. The expressions obtained contain the radial integrals, 6j- and 9j-coefficients and may be quite readily encoded and used in computer simulations, because they do not involve sums of products of parentage coefficients typical of the common matrix elements. In the second order of PT, one may evaluate not only energy corrections, but also expansion coefficients of the state function under study over the wave functions of admixed configurations: c(KT LS, K
T
LS) =

KT LS|H |K
T
LS K
T
LS|H |K
T
LS − KT LS|H |KT LS

.

Therefore, this leads to evaluation of the admixed configurations not only by their influence upon the level energies of the configuration under study, but also by their average contribution to its wave function. Since the expansion coefficients may have different signs, their squares should be averaged. The total averaged weight of the admixed configuration may be defined as W (K, K
) = g −1 (K)

 T LST



|K
T
LS2 (2L + 1)(2S + 1)KT LS|H .
|K
T
LS − KT LS|H
|KT LS)2 (K
T
LS|H

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325

The obtained expression differs from (2.2) only by the power of the energy difference in the denominator. By arguing as in the case of energy corrections, one may get the expression for the averaged weight of the admixed configuration as follows: W (K, K
) =

Θ(K, K
) 2

g(K) E (K
, K)

.

(2.9)

Eq. (2.9) was written for the case when the energy difference averaged over all terms was used (2.5). If we pass to the energy difference defined according to (2.7), then we shall get the following symbolic expression:  (K, K
) = W

Θ(K, K
) . 2 (K
, K) g(K) E

(2.10)


, K) for which rather simple analytical expressions Consequently, having found Θ(K, K
) , E(K
, K) and E(K exist, one may evaluate in two different approximations the average influence of the admixed configurations both upon the energy of the adjusted configuration and its wave function. It should be noted that this estimate is indeed averaged over all terms of the configuration under study. If the admixed configuration affects only the single of many terms of the adjusted configuration, then its obtained averaged contribution is underestimated. At the same time, the admixed configurations that do not influence upon the mutual arrangement of the term energies may nevertheless appear very important, though the account for them may be not very significant for some physical characteristics of the atom. However, such configurations are easily separated using simple physical considerations.

3. Comparison with the results of particular calculations In order to clarify to what extent the preliminary averaged estimates were realistic, we performed a number of calculations, where the values of the evaluated quantities were compared with the results of the diagonalization
matrix. Typical examples are discussed in the present section. To be able to compare, we averaged the of H diagonalization results as follows: 


−1
(2L + 1)(2S + 1)KT LS|H |KT LS − (2L + 1)(2S + 1)E(KP LS) . δE2 (K, K ) = g (K) T LS

P LS

(3.1) Here the first sum is the average energy of the configuration under study obtained before diagonalization and the second one is the energy of the same configuration obtained by diagonalizing the matrix written in the twoconfigurational (K and K
) approximation. Unfortunately, when diagonalizing the multiconfigurational matrix, it is impossible to separate energy corrections related to individual admixed configurations taken into account. The total weight of an admixed configuration in the wave function of the adjusted one was determined by expansion coefficients as follows:  W2 (K, K
) = g −1 (K) (2L + 1)(2S + 1)c2 (KT LS, K
T
LS). (3.2) T T
LS

In the calculations presented below, when describing virtual excitation to the shells absent in the adjusted configuration, we mainly used transformed radial orbitals (TRO) with a variable parameter. As follows from [14], they are readily generated and are very close by their properties to the solutions of the corresponding multiconfigurational Hartree–Fock equations. Since the comparison is performed within the same basis set, the choice of radial orbitals is of minor significance here. In all the calculations, when determining the energy difference, we took into account relativistic corrections within the Breit–Hartree–Fock approximation. In the calculations, along with our own computer codes, we employed the ones of [15,16].

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The averaged energy corrections obtained using three different methods for some ions with filling 2pN -shell (N = 2, 4, 6) are compared in Table 1. To make the comparison more obvious, the values in a.u. are given in this table only for diagonalization results, i.e. those calculated by (3.1). Instead of the values of δE(K, K
) and  δ E(K, K
) corrections, calculated using the preliminary evaluation technique of Eqs. (2.6) and (2.8), their relative errors δ(K, K
) and  δ(K, K
) are presented, expressed in per cent. Here and below the positive sign of the relative error corresponds to overestimation. In the table, all admixed configurations obtained by one- and two-electron excitations of 2s- and 2p-electrons to the state with n
3 and 4f are presented. When N = 6, part of these excitations are impossible or yield zero ICME due to the Brillouin’s theorem [17,18]. As seen from the table, in most cases even if the simplest difference of averaged energies is used (2.5), it allows the evaluation of the energy correction up to within several per cent. It is natural that the accuracy of the evaluation increases with the increase in the energy distance between the adjusted and admixed configurations. This is clearly seen when comparing the results obtained for different ionization degrees.  Transition to energy differences E(K, K
) taking into account only interacting terms (2.7) allows a significant increase in the accuracy of the preliminary evaluation results. This is particularly notable for two-electron excitations, when the admixed configuration contains not all the terms of the adjusted configuration, for instance, 2s 2 → 2p2 virtual excitation at N = 4, 2p2 → 3s 2 excitation at N = 2. Since in these cases the admixed  K
) leads to the accurate value of configurations have only one term 1 S, then the use of the difference E(K, the energy difference between interacting terms and the accuracy of the energy correction estimation is limited only by the accuracy of the second order of PT itself. At the same time, δE(K, K
) gives strongly underestimated corrections, because the single term of the adjusted configuration 1 S interacting with the admixed configuration is a high-lying one. A significantly overestimated correction δE(K, K
) is observed at rapid growth in the number of terms in the admixed configuration, e.g., 2p2 − 3l 2 excitations at N = 6, or the single-electron excitation 2s − 3d, which is an evidence that the terms of these configurations coupled with the terms of the adjusted one are in the upper part of the energy spectrum. The averaged weights obtained in the above calculations are presented in Table 2. Like in the previous table, absolute values of the weights are given only for the case of matrix diagonalization (3.2), and for estimates, the  (K, K
) in percent are presented. Since in the evaluation ω(K, K
) of W relative errors ω(K, K
) of W (K, K
) and  of the weights, the square averaged energy difference is used, then the relative errors obtained in this case are two to three times greater than in the previous table. Somewhat overestimated weights are also due to the fact that in the second order of the perturbation theory they are written for an unnormalized wave function, whereas the eigenfunctions obtained by diagonalization are exactly normalized. As seen from Table 2, even employment of the difference of average configuration energies allows in most cases the errors not exceeding 10%. Application of the energy difference obtained with account for ICME (2.7) leads usually to lower errors of estimation. Naturally, the same regularities are observed here, as in the previous table. The excited configuration 3d 6 4p for Fe II and Kr XII ions is another, more complicated example. Due to the known degeneracy of 3d- and 4s-electrons, in our calculations we additionally included in the basis set of Hartree– Fock radial orbitals of the configuration under study a solution of the usual Hartree–Fock equation for the function P (4s|r), calculated in the frozen-core field for configuration 3d 5 4p4s. Then, TRO with n = 5 was also added in this basis set. Obtained values and relative errors of the preliminary evaluations of the energy corrections and weight coefficients are presented in Table 3. To decrease the table dimensions, only virtual excitations from open shells were considered in this example. For the same reason, the table does not include admixed configurations that yield relatively smaller corrections. As seen from the table, the main trends indicated in the previous example remain here as well. In this case, on the average, the values determined using the energy difference E(K
, K) describe somewhat better the exact values than in the previous case. This is most probably associated with a great number of terms both in the adjusted and in the admixed configurations. Quite a unique situation occurs for admixed 3d 5 4p4s configuration in the iron ion. In this case, the errors of the estimates using the energy differences involving ICME (2.7) strongly increase. This is explained by the fact that in this case the energy spectra of the considered and admixed configurations strongly overlap, the energy differences between interacting terms may

P. Bogdanovich, R. Karpuškien˙e / Computer Physics Communications 134 (2001) 321–334

327

Table 1 Energy corrections δE2 (in a.u.) and relative errors of their preliminary evaluations δ (in %) for 1s 2 2s 2 2p N configurations K
1s 2 2p N+2

1s 2 2s 2 2p N−1 3p

1s 2 2s 2 2p N−1 4f

1s 2 2s2p N 3s

1s 2 2s2p N 3d

1s 2 2s 2 2p N−2 3s 2

1s 2 2s 2 2p N−2 3p 2

1s 2 2s 2 2p N−2 3d 2

1s 2 2s 2 2p N−2 4f 2

1s 2 2p N 3s 2

C I (N = 2)

Ca XV (N = 2)

O I (N = 4)

Ca XIII (N = 4)

Ne I (N = 6)

δE2

2.003E–02

7.148E–02

δ

3.60

1.98

−7.22

−8.42

–

 δ

3.60

1.98

4.14

2.56

–

δE2

2.947E–04

1.939E–04

1.166E–03

6.318E–04

–

3.614E–03

9.693E–03

–

δ

15.92

1.85

1.70

0.53

–

 δ

0.14

0.00

−1.75

−0.06

–

δE2

1.201E–03

1.556E–03

3.531E–03

4.560E–03

–

δ

2.49

0.48

0.68

0.19

–

 δ

0.14

0.01

0.06

0.01

–

δE2

2.017E–03

9.906E–04

1.977E–03

–

δ

10.18

1.47

6.97

1.58

–

 δ

−1.03

−0.01

−0.44

−0.01

–

δE2

8.586E–04

3.381E–02

4.366E–02

2.328E–02

4.259E–02

–

δ

18.55

3.63

15.51

4.81

–

 δ

1.70

0.10

0.78

0.09

–

δE2

1.811E–04

1.201E–04

1.023E–03

7.423E–04

2.480E–03

δ

−8.59

−1.06

0.05

0.01

4.78

 δ

0.23

0.00

0.05

0.01

0.05

δE2

6.211E–03

δ

0.17

−0.03

22.08

3.40

36.74

 δ

0.48

0.01

0.80

0.03

0.96

δE2

6.013E–03

7.614E–03

3.302E–02

4.372E–02

7.864E–02

3.898E–03

3.575E–02

2.424E–02

8.745E–02

δ

1.36

0.21

10.17

2.90

15.10

 δ

0.36

0.02

0.55

0.06

0.77

δE2

7.118E–04

9.019E–04

4.089E–03

5.256E–03

9.968E–03

δ

0.91

0.16

2.01

0.58

2.62

 δ

0.03

0.00

0.03

0.00

0.05

δE2

3.089E–04

2.508E–03

3.222E–03

2.581E–03

3.239E–03

δ

0.15

0.00

0.08

0.00

0.05

 δ

0.15

0.00

0.08

0.00

0.05

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P. Bogdanovich, R. Karpuškien˙e / Computer Physics Communications 134 (2001) 321–334

Table 1 Continued K
1s 2 2p N 3p 2

1s 2 2p N 3d 2

1s 2 2p N 4f 2

1s 2 2s 2 2p N−2 3s3d

1s 2 2s 2 2p N−2 3p4f

1s 2 2s2p N−1 3s3p

1s 2 2s2p N−1 3p3d

1s 2 2s2p N−1 3d4f

1s 2 2p N+1 3p

C I (N = 2)

Ca XV (N = 2)

O I (N = 4)

Ca XIII (N = 4)

Ne I (N = 6)

δE2

1.342E–03

6.524E–04

1.316E–03

6.440E–04

1.263E–03

δ

3.80

0.80

2.85

0.84

2.36

 δ

−0.17

0.00

−0.10

0.00

0.02

δE2

4.265E–03

5.256E–03

4.310E–03

5.198E–03

4.317E–03

δ

4.65

1.12

3.24

1.13

2.53

 δ

0.02

0.00

0.01

0.00

0.04

δE2

1.055E–03

1.240E–03

1.060E–03

1.236E–03

1.060E–03

δ

2.77

0.58

1.92

0.59

1.48

 δ

0.00

0.00

0.00

0.00

0.00

δE2

7.086E–05

5.440E–04

6.345E–04

3.023E–03

1.920E–03

1.99

0.50

3.37

−0.06

0.00

0.03

δ

−0.29

 δ

0.01

−0.4 0.00

δE2

2.782E–05

1.794E–05

1.918E–04

1.135E–04

5.177E–04

δ

0.23

0.06

0.11

0.07

0.05

 δ

0.00

0.00

−0.04

0.00

0.00

1.290E–02

9.477E–03

δE2

2.645E–02

1.959E–02

3.987E–03

δ

15.02

1.81

20.37

3.72

23.93

 δ

0.59

0.01

0.56

0.03

0.50

δE2

1.692E–03

5.695E–04

2.981E–03

6.756E–04

4.158E–03

δ

5.59

1.12

8.27

2.20

9.80

 δ

0.16

−0.01

−0.13

−0.02

0.01

δE2

5.252E–03

6.346E–03

1.034E–02

1.253E–02

1.534E–02

δ

5.11

1.00

4.59

1.30

4.33

 δ

0.06

0.00

0.07

0.00

0.11

δE2

2.605E–03

6.048E–04

1.185E–03

1.650E–04

–

δ

9.76

2.22

2.35

0.75

–

 δ

−0.44

−0.02

−0.20

−0.02

–

have opposite signs and, strictly speaking, the perturbation theory becomes inapplicable to describe this mixing. Quite a reasonable result of δE obtained in this case has an accidental nature. At the same time, the increase in the ionization degree rapidly removes the degeneracy of the 3d- and 4s-electrons, and for Kr XII the preliminary evaluation produces highly accurate results for this configuration. The similar results are obtaining for a lot of different ground and exited configurations including those with open shells of f -electrons.

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329

Table 2 Weights W2 of the admixed configurations K
and relative errors of their preliminary evaluations w (in %) for 1s 2 2s 2 2p N configurations K
1s 2 2p N+2

1s 2 2s 2 2p N−1 3p

1s 2 2s 2 2p N−1 4f

1s 2 2s2p N 3s

1s 2 2s2p N 3d

1s 2 2s 2 2p N−2 3s 2

1s 2 2s 2 2p N−2 3p 2

1s 2 2s 2 2p N−2 3d 2

1s 2 2s 2 2p N−2 4f 2

1s 2 2p N 3s 2

C I (N = 2) W2

2.51E–02

Ca XV (N = 2) 1.40E–02

O I (N = 4) 2.55E–03

Ca XIII (N = 4) 1.62E–03

Ne I (N = 6) –

ω

10.83

5.94

−10.50

−14.04

–

 ω

10.83

5.94

12.77

7.81

–

W2

3.48E–04

5.78E–06

8.44E–04

2.17E–05

–

ω

34.55

3.73

1.89

1.01

–

 ω

0.41

0.01

−4.90

−0.18

–

W2

4.73E–04

1.97E–05

6.98E–04

6.33E–05

–

ω

5.20

0.96

1.45

0.40

–

 ω

0.43

0.02

0.19

0.02

–

8.34E–04

5.55E–05

W2

6.05E–05

–

ω

20.17

2.95

13.94

3.18

–

 ω

−3.03

−0.03

−1.31

−0.03

–

W2

1.95E–02

1.12E–03

4.71E–04

9.71E–03

1.15E–03

–

ω

42.90

7.49

34.45

9.96

–

 ω

5.16

0.30

2.35

0.27

–

W2

1.52E–04

1.79E–06

3.78E–04

1.26E–05

5.44E–04

ω

−16.25

−2.11

0.16

0.00

9.84

 ω

0.69

0.01

0.16

0.00

0.16

W2

4.09E–03

ω

0.84

−0.06

50.13

6.96

88.43

 ω

1.45

0.02

2.36

0.09

2.73

W2

2.51E–03

1.07E–04

6.31E–03

6.51E–04

8.59E–03

ω

3.10

0.44

22.04

5.94

33.46

 ω

1.08

0.05

1.68

0.17

2.30

W2

1.42E–04

5.73E–06

3.98E–04

3.63E–05

5.75E–04

ω

1.86

0.33

4.09

1.16

5.36

 ω

0.08

0.00

0.08

0.01

0.16

W2

1.53E–03

3.47E–05

7.80E–04

3.98E–05

4.82E–04

ω

0.46

0.01

0.23

0.01

0.14

 ω

0.46

0.01

0.23

0.01

0.14

5.90E–05

1.03E–02

4.03E–04

1.42E–02

330

P. Bogdanovich, R. Karpuškien˙e / Computer Physics Communications 134 (2001) 321–334

Table 2 Continued K
1s 2 2p N 3p 2

1s 2 2p N 3d 2

1s 2 2p N 4f 2

1s 2 2s 2 2p N−2 3s3d

1s 2 2s 2 2p N−2 3p4f

1s 2 2s2p N−1 3s3p

1s 2 2s2p N−1 3p3d

1s 2 2s2p N−1 3d4f

1s 2 2p N+1 3p

C I (N = 2)

Ca XV (N = 2)

O I (N = 4)

Ca XIII (N = 4)

Ne I (N = 6)

W2

6.17E–04

9.23E–06

3.15E–04

1.00E–05

1.86E–04

ω

7.57

1.61

5.69

1.67

4.80

 ω

−0.49

−0.01

−0.30

−0.02

0.06

W2

1.39E–03

6.92E–05

7.01E–04

7.28E–05

4.20E–04

ω

9.55

2.25

6.60

2.27

5.16

 ω

0.06

0.00

0.03

0.00

0.13

W2

1.85E–04

7.66E–06

9.19E–05

8.23E–06

5.49E–05

ω

5.61

1.17

3.88

1.19

2.98

 ω

0.01

0.00

0.00

0.00

0.02

W2

3.93E–05

7.85E–06

1.69E–04

4.86E–05

3.05E–04

ω

−0.57

−0.07

3.97

1.00

6.89

 ω

0.03

0.01

−0.18

0.00

0.09

W2

8.65E–06

1.61E–07

2.99E–05

1.12E–06

4.87E–05

ω

0.46

0.13

0.18

0.13

0.11

 ω

0.01

0.00

−0.11

0.00

0.01

W2

6.77E–03

1.36E–04

3.10E–04

5.94E–03

6.59E–03

ω

33.07

3.67

45.68

7.61

54.64

 ω

1.79

0.04

1.69

0.08

1.50

W2

7.46E–04

8.02E–06

6.42E–04

1.03E–05

5.32E–04

ω

11.32

2.25

17.08

4.43

20.57

 ω

−0.48

−0.02

−0.38

−0.05

0.03

W2

5.43E–05

1.24E–03

1.16E–04

1.10E–03

ω

10.55

1.28E–03

2.01

9.48

2.63

8.96

 ω

0.17

0.01

0.22

0.02

0.33

W2

1.70E–03

1.58E–05

4.24E–04

4.70E–06

–

ω

19.98

4.46

4.59

1.49

–

 ω

−1.29

−0.06

−0.58

−0.05

–

In general, the above results demonstrate that the estimated role of an admixed configuration derived with the simplest energy difference E(K
, K) is somewhat larger, when we deal with the virtual excitations that leave the 
, K) makes the results essentially more accurate. This is due to the same orbital quantum number, and using E(K fact that, as may be seen from [19], the terms of configurations obtained this way and interacting with the terms in the initial configuration are in the upper part of the energy spectrum because of particular behavior of the coefficient

P. Bogdanovich, R. Karpuškien˙e / Computer Physics Communications 134 (2001) 321–334

331

Table 3 Energy corrections δE2 (in a.u.) and relative errors of their preliminary evaluations δ (in %) together with weights W2 of the admixed configurations K
and relative errors of their preliminary evaluations w (in %) for 3d 6 4p configuration K
3d 5 4p4s

3d 5 4p5d

3d 5 4p5g

3d 4 4p5p 2

3d 4 4p5d 2

3d 4 4p5f 2

3d 4 4p5g 2

3d 5 5p5d

3d 5 5d5f




δE2

Fe II

Kr XII

2.099E–03

4.947E–04

W2

Fe II

Kr XII

2.58E–02

8.92E–05

δ

−8.74

0.85

ω

−71.77

1.50

 δ

−25.60

−0.21

ω 

−81.24

−0.64

δE2

2.069E–03

9.007E–04

W2

1.02E–03

6.71E–05

δ

2.54

1.21

ω

3.62

2.26

 δ

−1.80

−0.17

ω 

−4.97

−0.51

δE2

5.810E–03

7.939E–03

W2

6.74E–04

2.19E–04

δ

1.55

0.85

ω

3.18

1.73

 δ

0.05

0.02

ω 

0.15

0.05

δE2

2.854E–03

1.162E–03

W2

6.99E–04

3.01E–05

δ

2.21

0.40

ω

4.49

0.80

 δ

0.03

0.00

ω 

0.09

0.00

δE2

6.731E–02

4.006E–02

W2

1.24E–02

1.384E–03

δ

32.73

8.34

ω

77.79

17.51

 δ

1.01

0.11

ω 

2.95

0.32

δE2

6.868E–02

W2

δ

10.41

5.67

ω

22.48

11.87

 δ

0.49

0.17

ω 

1.46

0.50

δE2

6.226E–03

8.203E–03

W2

3.59E–04

1.13E–04

5.196E–02

5.18E–03

1.78E–03

δ

1.81

9.83

ω

3.68

19.86

 δ

0.03

0.01

ω 

0.10

0.03

δE2

2.059E–03

3.561E–03

W2

4.64E–04

1.27E–04

δ

19.75

5.38

ω

42.89

11.05

 δ

−0.30

0.01

ω 

−0.94

0.02

3.782E–03

W2

δE2

2.302E–03

3.38E–04

1.41E–04

δ

7.99

4.51

ω

16.43

9.19

 δ

−0.17

−0.04

ω 

−0.50

−0.12

g0 (l N , l N ) describing the exchange interaction between shells, so that the usual average energy difference in (2.5) is underestimated. The above results indicate that preliminary evaluation of the role of admixed configurations allows rather accurate prediction of averaged diagonalization results in the two-configurational approximation. However, the

332

P. Bogdanovich, R. Karpuškien˙e / Computer Physics Communications 134 (2001) 321–334

Table 4 Weights W of the admixed configurations K
for Ne III 2s 2 2p 4 configuration K


WM

W

 W

W2

2s2p 4 3d

5.317E–03

7.071E–03

5.756E–03

5.679E–03

2s 2 2p 2 3p 2

3.158E–03

4.445E–03

3.602E–03

3.572E–03

2s 2 2p 2 3d 2

3.132E–03

3.979E–03

3.485E–03

3.454E–03

2p 6

2.282E–03

1.998E–03

2.531E–03

2.275E–03

2s2p 3 3s3p

2.235E–03

3.164E–03

2.551E–03

2.534E–03

2s2p 3 3d4f

6.660E–04

7.073E–04

6.643E–04

6.634E–04

2p 4 3d 2

4.137E–04

4.104E–04

3.909E–04

3.909E–04

2s 2 2p 3 4f

4.096E–04

3.664E–04

3.631E–04

3.627E–04

2s2p 4 3s

2.816E–03

3.439E–04

3.138E–04

3.152E–04

2p 4 3s 2

2.567E–04

3.156E–04

3.156E–04

3.153E–04

2s 2 2p 3 3p

2.360E–04

2.330E–04

2.244E–04

2.282E–04

2s 2 2p 2 4f 2

1.713E–04

2.129E–04

2.070E–04

2.069E–04

2s 2 2p 2 3s3d

1.586E–04

1.871E–04

1.822E–04

1.822E–04

2s 2 2p 2 3s 2

1.067E–04

1.225E–04

1.225E–04

1.224E–04

2p 4 3p 2

8.816E–05

9.031E–05

8.669E–05

8.680E–05

2s2p 3 3p3d

5.743E–05

6.256E–05

5.702E–05

5.717E–05

2p 4 4f 2

4.340E–05

4.991E–05

4.855E–05

4.855E–05

2p 5 3p

2.525E–05

3.176E–05

3.064E–05

3.073E–05

2s 2 2p 2 3p4f

9.191E–06

9.844E–06

9.817E–06

9.820E–06

real calculations require superposition of a large number of configurations. As already noted, in this case it is impossible to separate out the energy corrections associated with the account for individual configurations, because these quantities are not additive. At the same time, (3.2) presents a chance to separate the weight of an individual admixed configuration in the case of a large number of interacting configurations. For comparing the preliminary evaluation results with those of consistent superposition, Table 4 presents the weights of admixed configurations obtained in different approximations for the configuration 1s 2 2s 2 2p4 of Ne III. This table includes weights WM  obtained obtained by diagonalization of the multiconfigurational matrix and averaged by (3.2), weights W and W using the above-discussed methods of preliminary evaluation (2.9) and (2.10), and W2 obtained by diagonalization of the corresponding two-configurational matrix and averaged by (3.2). The data are arranged by descending WM . As seen from the table, the preliminary evaluations yield somewhat overestimated weights for all configurations. However, it is highly important that the preliminary evaluation yields almost the same order of configurations as diagonalization does. There is only one exception – the position of configuration 1s 2 2p6 . As already mentioned, the simple energy difference in this case produces a strongly underestimated value. Even the energy difference involving ICME does not restore the correct order of configurations, which, on the other hand, is consistent with the diagonalization results for the two-configurational matrix. However, these discrepancies are not so significant,  ) of this configuration and the following one occurs because the difference between the weights (for WM and W only in the third meaningful digit. Although the correlation corrections to the energy are not additive, the sum of their estimates over all admixed configurations gives an estimate of the magnitude of correlation effects taken into account. This quantity may also

P. Bogdanovich, R. Karpuškien˙e / Computer Physics Communications 134 (2001) 321–334

333

be defined as a shift of the average energy of the configuration under study obtained as a result of diagonalization of the multiconfigurational matrix. In the example of the Ne III ion considered above the diagonalization shifts the average energy by the value of δEM = 0.1628 a.u. The preliminary evaluation of the total energy correction yields the following values:    δE(K, K
) = 0.1824 a.u. and δ E(K, K
) = 0.1696 a.u. K


K


As seen from these figures, the application of the energy difference (2.7) taking into account the presence of ICME gives in this case the total energy correction within the accuracy of 4%. This error is comparatively low, but it is by an order of magnitude larger than the relative errors for the main admixed configurations obtained in the two-configurational approximation. Consequently, the error of the sum estimate of the correlation correction to the average energy is indeed to a greater extent an indication of nonadditivity of corrections than the result of errors in the estimates themselves. The calculations results presented in this section demonstrate the possibilities of the preliminary evaluation of the role of the most important and energetically close admixed configurations. In the real calculations, the number of the possible admixed configurations usually is considerably larger, whereas the accuracy of the preliminary evaluations grows together with the energy difference between employed and admixed configurations. Our code during the same calculation is generating all admixed configurations possible in the employed RO basis set and it is performing their preliminary evaluation. The number of such configurations can reach many thousands but the computational time on our PC does not exceed several tens of seconds even when the configurations with opened nd- and nf -electrons shells are considered. It is incomparably less than the computational time necessary for the calculation and diagonalization of all two-configurational energy matrices. When real evaluations are performed  the sum of the all averaged energy corrections K
δE(K, K
) are obtained together with the analogical sum for the configurations selected to take into account in the energy matrix calculations. It allows us to comprehend about the possible correlation corrections not included into the calculations. It is necessary to recognize that the preliminary evaluations are described in the second order of the PT so that the evaluations do not take into account interactions between admixed configurations. This leads to the nonadditivity of the real energy corrections and gives some small deviations from the evaluated weights.

4. Conclusion The results of the previous section and long and successful application of the method demonstrate that even using the simplest difference of average energies (2.5) we may get in most cases rather reliable averaged characteristics of the admixed configurations. Here, the evaluation accuracy grows for admixed configurations that have lower influence upon the adjusted one, which is important, because allows us to select only sufficiently important ones out of the great number of possible admixed configurations. In actual calculations, many thousands of admixed configurations were checked without notable consumption of CPU time. The time of computations is in fact determined by the time necessary to calculate radial integrals entering the expressions employed. Transition to the energy difference taking into account ICME (2.7) increases essentially the accuracy of evaluation. Here, the errors for practically important cases become considerably less than 1%. Such an accuracy is not necessary for the purposes of the present work (i.e. preliminary evaluation of the contribution of admixed configurations), but it may be quite useful when taking into account the correlation effects in the second order of the perturbation theory, as done, e.g., [20]. Naturally, both approximations considered work well only in the case if the perturbation theory is applicable at all, i.e. ICME are less than the energy difference between configurations. It should also be noted that the evaluation performed here does not answer the question whether the admixed configuration influences the mutual arrangement

334

P. Bogdanovich, R. Karpuškien˙e / Computer Physics Communications 134 (2001) 321–334

of terms or it mainly changes the total energy of the configuration. However, the answer may be readily found by analyzing the virtual excitation used to obtain this admixed configuration. For this purpose, the formulae of [20] may be useful. The technique described here not only evaluates the contribution of admixed configurations, but also makes it possible to compare the efficiency of different basis sets of radial orbitals. This may be performed both by comparing the influence of different configurations and via sums of energy corrections and weights of admixed configurations.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

A. Jucys, J. Exp. Theor. Fiz. (JETP) 23 (1952) 371 (in Russian). C.F. Fischer, The Hartree–Fock Method for Atoms, John Wiley and Sons, New York, 1977, p. 308. A. Jucys et al., Lietuvos TSR MA Darbai 2 (1958) 3 (in Russian). O. Sinanoglu, Adv. in Chem. Phys. VI (1964) 315. R.D. Cowan, The Theory of Atomic Structure and Spectra, University of California Press, Berkeley, 1981, p. 722. C.F. Fischer, T. Brage, P. Jönsson, Computational Atomic Structure. An MCHF Approach, Institute of Physics Publishing, Bristol/ Philadelphia, 1997, p. 279. P. Bogdanovich et al., Monthly Notices Roy. Astronom. Soc. 280 (1996) 95. P. Bogdanovich, R. Karpuškien˙e, Z. Rudzikas, Phys. Scripta 80 (1999) 474. P. Bogdanovich, I. Martinson, Phys. Scripta 60 (1999) 217. K. Rajnak, B.G. Wybourne, Phys. Rev. 132 (1963) 280. B.G. Wybourne, Phys. Rev. 137 (1965) 364. S. Kuˇcas, V. Jonauskas, R. Karazija, Phys. Scripta 55 (1997) 667. R. Karazija, Sums of Atomic Quantities and Mean Characteristics of Spectra, Mokslas, Vilnius, 1991, p. 272 (in Russian). P. Bogdanovich, R. Karpuškien˙e, Lith. J. Phys. 39 (1999) 193. C.F. Fischer, Comput. Phys. Commun. 43 (1987) 355. A. Hibbert, R. Glass, C.F. Fischer, Comput. Phys. Commun. 64 (1991) 455. C.F. Fischer, J. Phys. B 6 (1973) 1933. R. Karazija, L. Rudzikait˙e, Soviet Phys. Collection 27 (1987) 144. P. Bogdanovich, R. Karazija, J. Boruta, Soviet Phys. Collection 20 (1980) 11. P. Bogdanovich et al., Phys. Scripta 56 (1997) 230.