Theoretical energy level spectra and transition data for 4p4d, 4p4f and 4p4d configurations of W ion
Abstract
The ab initio quasirelativistic HartreeFock method developed specifically for the calculation of spectral parameters of heavy atoms and highly charged ions was applied to determine atomic data for tungsten ions. The correlation effects were included by adopting configuration interaction method. The BreitPauli approximation for quasirelativistic HartreeFock radial orbitals was employed to take into account relativistic effects. The energy level spectra, radiative lifetimes, Lande factors were calculated for the , and configurations of W ion. The atomic data, namely, the transition wavelengths, spontaneous emission rates and oscillator strengths for the electric dipole, electric quadrupole and magnetic dipole transitions among and within the levels of these configurations are tabulated.
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[cor1]Corresponding author.
Contents:
1 Introduction
Due to its unique features, the metallic tungsten is widely employed in devices which operate at extremely high temperatures. These properties facilitate adoption of tungsten in fusion reactors including internationally developed ITER tokamak [[1], [2]]. High temperatures within fusion plasma cause tungsten to evaporate. Furthermore, the tungsten atoms in plasma are ionized to very high degrees. This process has a negative consequences on plasma since it makes plasma temperatures to drop significantly.
To perform plasma processes modeling and diagnostics one needs huge amount of spectroscopic data describing various ionization stages of tungsten and other atoms [[3], [4]]. Atomic spectroscopy can help to determine important properties of plasma [[5], [6], [7]]. Nevertheless, the spectra of the tungsten ions present in fusion plasma devices are still not investigated properly. This conclusion is evident from review [[8]] where all available experimental wavelengths and energy level spectra for ions from W III to W LXXIV are provided.
It is evident from the analysis of [[8]] and references within, that only a few data for the spectra of tungsten ions with open 4d shells are available. In present work we investigate a relatively simple system, namely W. This ion has only one valence 4d electron in its ground state. It follows from [[8]] that only 11 excited levels from the possible 48 ones for the configurations under consideration have been determined experimentally. On the other hand, a theoretical study of energy level spectra was presented in [[9]]. However, this work suffers from the fact that the data are provided just for some fifty percent of existing levels representing the configurations under consideration. Furthermore, the transition data were determined for the highly excited levels only.
In present work we employ the quasirelativistic method [[10], [11], [12]] developed specifically for the ab initio calculation of spectral parameters of heavy atoms and highly charged ions. A preliminary analysis of the suitability of this method for the tungsten ions with open 4d shell was presented in [[13]]. It was demonstrated that our quasirelativistic approach is appropriate for the inclusion of the relativistic effects, and the further improvement of accuracy of the calculated data can be achieved with inclusion of the correlation effects.
In Section 2 we provide a short description of our calculation method. The more detailed description of the applied approach and the analysis of the convergence within the configuration interaction (CI) approximation applied for the inclusion of correlation effects is presented in [[14]]. Obtained results are discussed in Section 3.
2 Calculation method
2.1 Quasirelativistic approximation
We use a quasirelativistic approximation for ab initio calculations of ion energy spectra. This approach significantly differs from widely used method described in [[15]]. In our method, the radial orbitals (RO) are obtained by solving the quasirelativistic equations of the following form:
(1) 
The first line of Eq.(1) represents the traditional HartreeFock equations, where denotes the exchange part of the potential and represents the direct part of the potential including the interaction of an electron with nucleus and with other electrons. We take into account the finite size of a nucleus within the nucleus potential [[16]]. This allow us to represent the radial orbitals in powers of a radial variable in the nucleus region. Next two terms with the multiplier describe the relativistic correction of the massvelocity dependence. The last term of Eq.(1) represents the potential of the electron contact interaction with the nucleus. In our approach we include the contact interaction with the nucleus not only for the s electrons but also some part of it for the p electrons:
(2) 
A detailed discussion of the particular features of Eq.(1) is given in [[10], [11]], whereas their solution techniques are described in [[17], [18]].
Concluding the description of applied approximation, we want to articulate the unique features of our quasirelativistic method which significantly differs from widely used approach described in [[15]]. The main differences arise from our adopted set of HartreeFock quasirelativistic equations (QRHF) featuring several distinctive properties, namely:

No statistical potentials are used. There are only conventional selfconsistent field direct and exchange potentials in QRHF.

The massvelocity term is divided into two parts  the direct potential and the exchange one.

The contact interaction term contains only the nucleus potential derivative in the numerator. There are no twoelectron potentials in the numerator.

Only the direct part of the potential is included into the denominator of the contact interaction with a nucleus term.

The contact interaction with a nucleus is taken into account not only for selectrons, but also for pelectrons with some additional corrections made [[11]].
One must look in [[10], [11]] for the most complete description of the way to determine QRHF equations employed in this work. The methods to calculate the energy level spectra were discussed extensively in [[12]].
For the energy level spectra calculation, we include all twoelectron interactions in the same way as it is done in conventional BreitPauli approximation. This similarity makes it possible to apply widely used code mchf breit [[19]] for the angular integration of the BreitPauli Hamiltonian matrix elements. We employ the computer program mchf mltpol [[20]] to determine the matrix elements of transition operators along with the code mchf lsjtr [[21]] which has been adopted for use with the quasirelativistic radial orbitals.
2.2 Correlation effects
We use configuration interaction (CI) approximation to include electron correlation effects. In present work, depending on the type of oneelectron state, two sorts of radial orbitals are exploited in the CI expansion of wavefunctions. For the configurations under consideration (adjusted configurations), we employ oneelectron radial orbitals determined from solutions of Eq.(1). This type of RO was applied for all electrons with and . We adopt the transformed radial orbitals (TRO) to describe virtually excited electrons included in CI expansion.
The method of transformed radial orbitals to account for the configuration interaction effects was developed in [[22]] and successfully applied for the nonrelativistic radial orbitals (see, e.g. [[23], [24], [25], [26]]). The comparison of TRO properties with those of the solutions of the multiconfigurational HartreeFockJucys equations [[27]] demonstrates their close similarity [[28], [29]]. Such a way to include correlation effects within our quasirelativistic approach was successfully applied for the studies of the tungsten spectral parameters in [[30]].
In current work the TRO were employed to describe the admixed configurations, having virtually excited electrons with the principal quantum number and with all allowed values of orbital quantum number . These orbitals are determined from the RO of the configurations under consideration by ensuring the effective inclusion of the correlation effects. For the present calculations, we employ the form of TRO with two free variational parameters and :
(3) 
Here the multiplier ensures the normalization of determined TRO, the first term in parenthesis performs the necessary transformation, and the second term provides the orthogonality of all employed radial orbitals. The parameters and are varied in order to ensure the maximum of averaged energy correction to the energy of the adjusted configuration in the second order of perturbation theory:
(4) 
We apply the same set of radial orbitals to describe both the even and odd configurations. This approach enables us to avoid any problems caused by the nonorthogonality of RO in calculation of the electron transition parameters.
As a consequence of adopted TRO, the basis of radial orbitals and the number of possible admixed configurations increases rapidly. Therefore one needs to perform a selection of the admixed configurations by including only those which have the greatest influence on the configurations being adjusted. As a selection criteria we apply the mean weight of the admixed configuration within the CI expansion of the adjusted configuration wavefunction determined in the second order of perturbation theory:
(5) 
We include only those configurations which have larger than the specified small parameter in our calculation of energy levels and transition parameters. The methods and computer codes to calculate the mean weight from Eq. (5) have been described in [[31], [32]]. Such a way to select the interacting configurations in the nonrelativistic approximation was sucessfully exploited (see, e.g., earlier mentioned [[23], [24], [25], [26]] and many others).
The configuration state functions (CSFs) of the selected configurations are used to form the energy operator matrices. By diagonalizing these Hamiltonian matrices, we can determine the energy eigenvalues and eigenfunctions for the investigated configurations and exploit them for calculation of transition parameters. More details of employed approach one can find in [[12]] and references therein, whereas some examples of application of our method are presented in [[30], [33]].
In present work we have selected only those admixed configurations which have the mean weights in the wave functions of the configurations under consideration larger than . This procedure has allowed us to reduce the number of admixed configurations in the CI expansion of the wavefunctions to 102 for even configurations and to 217 for odd configurations. So here we use an approximation denoted as F in [[14]]. It was demonstrated in [[14]] that any further expansion of the admixed configurations basis does not lead to significant improvement in the calculated energy level spectra accuracy.
We have determined energy eigenvalues and eigenfunctions by diagonalizing Hamiltonian matrices obtained in the coupling scheme in BreitPauli approximation. Using these results, we have calculated the transition parameters. We have computed data for the electric dipole (E1), electric quadrupole (E2) and magnetic dipole (M1) operators for the transitions both between the configurations under consideration and within these configurations. The emission transition rates of all above mentioned types were used to determine the radiative lifetimes of excited states.
3 Results and discussion
We present the energy level spectra with the percentage contribution of eigenfunctions together with Lande factors and the radiative lifetimes for the investigated configurations in Table 2. It was demonstrated in [[14]] that deviations of our theoretical level energy values from the experimental spectra data do not exceed in most cases. Consequently, we can conclude that these discrepancies in transition energies do not affect significantly the accuracy of calculated values of the transition rates or oscillator strengths. In Table 1 we compare our calculated energy levels with data obtained from the completely relativistic calculations RELAC [[9]] and with available experimental data [[7]] presented in compilation [[8]]. This evaluation favorably demonstrates a better agreement of our data with the experimental level energies, especially for highlyexcited states, comparing to the data from relativistic calculations RELAC. This is caused mainly by considerably improved inclusion of correlation effects.
Such a good agreement with experimental data leads to conclusion that our quasirelativistic approach adequately accounts for the relativistic effects even for highlyionized heavy atoms. It is worth to mention a very good agreement of theoretical finestructure level splitting for the ground term 4p4d D pointing to close spinorbit interaction values obtained in our work and in fully relativistic calculations [[9]].
As it was mentioned before, all our calculations were performed in coupling scheme. It is well established fact that the spinorbit interaction induces a very strong mixing of terms for highlycharged ions. Therefore in Table 1 we present only the level numbers (indices) taken from Table 2 where one can find a corresponding CI wavefunction expansion. It is easy to notice from the CSFs contributions presented in Table 2 that the strong mixing occurs both for the terms of the same configuration and for terms of different configurations.
A wellestablished case of a strong interaction between the and configurations (see [[34]]) can be confirmed from the analysis of the wave function percentage contributions. For example, a contribution from the configuration for the level in Table 2 is merely . Moreover, the largest contribution for the level comes from the configuration although the level itself clearly must be attributed to the configuration .
In Table 3 we present the transition wavelengths , the emission transition rates and the weighted oscillator strength values . We use only the level indices and the total angular momentum from Table 2 to describe both the initial and final levels along with the type of transition operator (E1, E2 or M1) applied to calculate the transition parameters. We present the transitions from the excited level to lowerlying levels in decreasing order of their rates. We limit the number of presented transitions by removing those having the transition rate values by 3 orders of magnitude lower comparing to the strongest transition from the particular energy level. This condition would remove completely the E2 transitions. Nonetheless, we have placed some most important E2 transitions in Table 3.
The magnetic dipole M1 transitions within configuration were computed using a completely relativistic approximation in [[35]]. As one can see from Table 3, their value of transition rate is very close to the value of from our calculation.
From analysis of data presented in Table 3, we can notice that the electric dipole transition rates are the most strong ones for all excited configuration levels when the total angular momentum . Furthermore, the strong E1 transitions define the values of radiative lifetimes for these levels presented in Table 2. The only exemption from this rule demonstrates the level with total angular momentum . Even if the E1 transitions from this level have the largest rates , the values of two M1 transition rates are of the same order of magnitude as for the E1 transitions. This causes the significant decrease of the radiative lifetime value for this energy level.
The electric dipole transitions from the excited levels with are forbidden because the largest value of total angular momentum of the ground state . Consequently, the radiative lifetimes for these levels are determined by the magnetic dipole transitions since the E2 transition rates are significantly smaller.
Term [[8]]  Level Name [[9]]  EXP [[8]]  QRHF  RELAC [[9]]  

1  4p4d D  (4p)(4d)  1.5  0  0  0 
2  4p4d D  (4p)(4d)  2.5  1546  1538  1537 
3  4p4d F (3/2,2)  0.5  12276  12311  
4  4p4d F (3/2,2)  1.5  12276  12312  
5  (4p)(4p)(4d)  2.5  12591  12569  
6  (4p)(4p)(4d)  3.5  12752  12720  
10  (4p)(4p)(4d)(4d)  4.5  13922  13861  
13  (4p)(4p)(4d)(4d)  5.5  14255  14201  
16  (4p)(4p)(4d)(4d)  3.5  14631  14646  
17  (4p)(4p)(4d)(4d)  2.5  14680  14691  
18  (4p)(4p)(4d)(4d)  1.5  14756  14794  
19  (4p)(4p)(4d)(4d)  4.5  14853  14861  
20  4p4d G (3/2,4)  (4p)(4p)(4d)(4d)  2.5  15085  15220  15220 
21  (4p)(4p)(4d)  5.5  15500  15443  
22  4p4d F (3/2,3)  (4p)(4p)(4d)(4d)  1.5  15426  15618  15680 
24  (4p)(4p)(4d)  3.5  15671  15649  
25  (4p)(4p)(4d)  4.5  15854  15851  
28  (4p)(4p)(4d)  3.5  16561  16567  
30  4p4d F (3/2,4)  (4p)(4p)(4d)  2.5  17315  17535  17630 
32  4p4f F  (4p)(4f)  2.5  17581  17752  17840 
33  4p4f F  (4p)(4f)  3.5  17696  17889  17951 
34  4p4d F (1/2,2)  (4p)(4p)(4d)  2.5  20145  20319  20396 
44  4p4d D (1/2,2)  1.5  21843  22069  
46  4p4d G (1/2,4)  (4p)(4p)(4d)(4d)  3.5  23255  23274  23488 
47  (4p)(4p)(4d)(4d)  1.5  23309  23525  
48  (4p)(4p)(4d)(4d)  2.5  23369  23347 
Table 1. Energy levels and radiative lifetimes of and configurations of W.
Throughout the table we present all the eigenfunction components which have their percentage contributions higher than
The energy level number (index)  
The level energy in cm  
The Lande factor . The notation a(b) means  
The radiative lifetime in s  
The total angular momentum  
Contribution  The percentage contribution of CSF in level eigenfunction 
Table 2. Electron transitions among and within the , and configurations of W.
The transitions from the upper level to the lower ones are presented in the descending order of the transition rate values . The number of presented transitions is limited to those having the transition rate values less than 3 orders of magnitude smaller comparing to the strongest transition from the particular energy level.
The total angular momentum of the initial level  
The initial energy level number  
The total angular momentum of the final level  
The final energy level number  
Type  The electron transition type 
(Å)  The transition wavelength in Å 
(s  The emission transition rate in s. The notation aEb means 
The weighted oscillator strength . The notation aEb means 
Contribution  

1  0  0.80  1.5  98 4d D  
2  1538  1.20  2.55(4)  2.5  98 4d D  
3  12311  1.00  8.44(1)  1.5  42 4p4d(F) D  +23 4p4d(F) F  +10 4p4d(D) D  
4  12312  0.12  2.44(1)  0.5  84 4p4d(F) D  +11 4p4d(D) P  
5  12591  1.03  3.83(1)  2.5  36 4p4d(F) F  +18 4p4d(D) D  +12 4p4d(F) G  +10 4p4d(F) D  
6  12752  1.03  1.75(1)  3.5  32 4p4d(F) G  +31 4p4d(F) G  +23 4p4d(D) F  +7 4p4d(F) F  
7  13284  1.36  5.37(0)  1.5  28 4p4d(F) P  +21 4p4d(S) P  +15 4p4d(P) D  
8  13636  1.30  1.95(1)  2.5  32 4p4d(F) D  +22 4p4d(P) P  +15 4p4d(P) D  
9  13915  1.29  2.88(1)  3.5  35 4p4d(F) F  +27 4p4d(P) D  +16 4p4d(F) D  +10 4p4d(D) F  
10  13922  1.18  3.04(4)  4.5  70 4p4d(F) G  +15 4p4d(F) G  +13 4p4d(F) F  
11  14052  1.09  1.29(1)  1.5  40 4p4d(F) D  +14 4p4d(F) D  +14 4p4d(D) D  +11 4p4d(D) P  
12  14104  1.23  4.29(1)  2.5  40 4p4d(P) P  +17 4p4d(F) F  +15 4p4d(G) F  
13  14255  1.17  2.68(6)  5.5  57 4p4d(G) H  +42 4p4d(F) G  
14  14290  1.06  1.15(4)  3.5  29 4p4d(F) G  +15 4p4d(F) F  +15 4p4d(F) F  +15 4p4d(G) G  +14 4p4d(G) F 
15  14408  1.66  4.37(2)  0.5  33 4p4d(P) P  +29 4p4d(P) S  +19 4p4d(D) P  
16  14631  1.17  3.20(1)  3.5  20 4p4d(P) D  +13 4p4d(D) F  +13 4p4d(G) G  +12 4p4d(G) F  +11 4p4d(F) D 
17  14680  1.22  1.60(1)  2.5  36 4p4d(P) D  +25 4p4d(P) D  +15 4p4d(D) D  
18  14756  1.39  2.77(0)  1.5  30 4p4d(P) S  +25 4p4d(P) D  +17 4p4d(P) P  
19  14853  1.13  1.47(5)  4.5  38 4p4d(G) G  +24 4p4d(F) F  +18 4p4d(F) G  +17 4p4d(G) H  
20  15220  0.88  6.50(3)  2.5  41 4p4d(G) F  +28 4f F  +13 4p4d(F) F  
21  15500  1.20  1.70(4)  5.5  57 4p4d(F) G  +42 4p4d(G) H  
22  15618  0.85  2.16(3)  1.5  32 4p4d(F) F  +22 4p4d(P) D  +12 4p4d(F) D  
23  15619  1.21  1.60(0)  2.5  24 4p4d(D) D  +18 4p4d(P) P  +15 4p4d(F) D  +13 4p4d(P) D  
24  15671  1.22  5.32(2)  3.5  34 4p4d(D) F  +29 4p4d(P) D  +13 4fF  
25  15854  1.17  1.20(4)  4.5  38 4p4d(F) F  +30 4p4d(G) G  +18 4p4d(F) G  +13 4p4d(G) H  
26  15972  1.08  6.58(3)  0.5  65 4p4d(D) P  +14 4p4d(P) P  +12 4p4d(P) S  
27  16350  0.76  1.49(3)  0.5  52 4p4d(P) P  +24 4p4d(P) D  +18 4p4d(P) S  
28  16551  1.18  4.48(2)  3.5  17 4f F  +16 4p4d(F) F  +15 4p4d(D) F  +14 4p4d(F) D  +12 4p4d(P) D 
29  16897  1.27  5.82(1)  1.5  50 4p4d(S) P  +14 4p4d(P) D  
30  17535  1.09  1.02(3)  2.5  29 4p4d(F) D  +15 4p4d(F) F  +13 4f F  
31  17553  1.36  1.58(3)  1.5  20 4p4d(S) P  +17 4p4d(P) S  +16 4p4d(P) P  +13 4p4d(D) D  +12 4p4d(P) D 
32  17752  0.93  1.21(3)  2.5  49 4f F  +18 4p4d(G) F  
33  17889  1.12  1.72(3)  3.5  63 4f F  +18 4p4d(G) G  
34  20319  0.77  8.80(4)  2.5  39 4p4d(F) G  +28 4p4d(F) F  +15 4p4d(D) F  
35  21314  1.02  7.13(1)  3.5  46 4p4d(F) G  +19 4p4d(F) G  +14 4p4d(F) F  
36  21387  1.07  1.70(2)  2.5  18 4p4d(F) F  +16 4p4d(D) F  +14 4p4d(F) G  +14 4p4d(P) D  +12 4p4d(F) D 
37  21478  1.07  9.10(3)  1.5  37 4p4d(P) D  +16 4p4d(F) D  
38  21547  1.96  1.39(3)  0.5  41 4p4d(P) P  +33 4p4d(P) S  +13 4p4d(P) P  
39  21555  0.43  9.08(4)  0.5  56 4p4d(P) D  +21 4p4d(P) P  +13 4p4d(S) P  
40  21754  1.04  1.39(2)  4.5  45 4p4d(G) H  +19 4p4d(G) G  +16 4p4d(F) G  
41  22069  0.88  3.03(4)  1.5  43 4p4d(F) D  +17 4p4d(G) D  +12 4p4d(F) F  
42  22794  1.29  2.73(3)  3.5  45 4p4d(F) D  +32 4p4d(F) F  +18 4p4d(F) F  
43  23116  1.15  3.88(4)  2.5  34 4p4d(F) D  +12 4p4d(P) D  +10 4p4d(F) D  
44  23159  1.20  4.30(3)  1.5  33 4p4d(D) P  +26 4p4d(D) D  +12 4p4d(P) D  +11 4p4d(P) P  
45  23221  1.11  1.30(2)  4.5  31 4p4d(F) G  +23 4p4d(G) H  +17 4p4d(F) G  +15 4p4d(F) F  +11 4p4d(G) G 
46  23274  1.06  5.17(4)  3.5  53 4p4d(G) F  +29 4p4d(G) G  
47  23309  1.48  4.69(4)  1.5  52 4p4d(P) P  +20 4p4d(P) S  +11 4p4d(P) P  
48  23369  1.11  4.68(1)  2.5  35 4p4d(D) F  +24 4p4d(P) D  +17 4p4d(P) D  
49  24058  0.67  3.13(2)  0.5  75 4p4d(S) P  +13 4p4d(P) D 
Type  ( Å)  (s)  

2.5  2  1.5  1  M1  650.2  3.93E04  1.49E05 
1.5  1  E2  650.2  1.89E01  7.19E09  
1.5  3  1.5  1  E1  81.23  1.15E09  4.56E03 
2.5  2  E1  92.83  3.38E07  1.75E04  
0.5  4  1.5  1  E1  81.22  4.09E09  8.09E03 
2.5  5  1.5  1  E1  79.42  2.37E09  1.34E02 
2.5  2  E1  90.47  2.47E08  1.82E03  
3.5  6  2.5  2  E1  89.18  5.72E07  5.45E04 
1.5  7  2.5  2  E1  85.14  1.78E08  7.74E04 
1.5  1  E1  75.28  8.25E06  2.80E05  
2.5  8  1.5  1  E1  73.34  5.13E07  2.48E04 
2.5  2  E1  82.66  3.11E04  1.91E07  
3.5  9  2.5  2  E1  80.80  3.47E09  2.72E02 
4.5  10  3.5  6  M1  854.6  3.29E04  3.60E05 
3.5  6  E2  854.6  2.13E00  2.33E09  
1.5  11  1.5  1  E1  71.17  5.76E09  1.75E02 
2.5  2  E1  79.91  1.97E09  7.56E03  
2.5  12  2.5  2  E1  79.58  2.31E09  1.31E02 
1.5  1  E1  70.90  2.44E07  1.11E04  
5.5  13  4.5  10  M1  2999  3.68E02  5.95E06 
3.5  6  E2  665.1  5.94E00  4.73E09  
3.5  14  2.5  2  E1  78.42  4.94E04  3.65E07 
2.5  5  M1  588.5  2.32E04  9.64E06  
3.5  6  M1  650.0  1.42E04  7.21E06  