Core Effects on Transition Energies for 3 d k Configurations in Tungsten Ions

All energy levels of the 3dk, k = 1,2,. . . , 8, 9, configurations for tungsten ions, computed using the GRASP2K fully relativistic code based on the variational multiconfiguration Dirac–Hartree–Fock method, are reported. Included in the calculations are valence correlation where all 3s, 3p, 3d orbitals are considered to be valence orbitals, as well as core–valence and core–core effects from the 2s, 2p subshells. Results are compared with other recent theory and with levels obtained from the wavelengths of lines observed in the experimental spectra. It is shown that the core correlation effects considerably reduce the disagreement with levels linked directly to observed wavelengths, but may differ significantly from the NIST levels, where an unknown shift of the levels could not be determined from experimental wavelengths. For low values of k, levels were in good agreement with relativistic many-body perturbation levels, but for 2 < k < 8, the present results were in better agreement with observation.


Introduction
Because of their importance for the ITER project [1], spectra of tungsten ions have recently received much attention over a wide range of wavelengths.Of special interest are the NIST EBIT experiments reported by Ralchenko et al. [2], who studied tungsten ions with the ground states 3d, 3d 2 , . .., 3d 8 , and 3d 9 .Detailed collisional-radiative modelling was undertaken to identify the measured spectral lines.For the modelling they relied on energy levels, radiative transition probabilities, and electron-impact collisional cross-sections obtained using the relativistic Flexible Atomic Code (FAC) [3].They found that many of the strong lines arose from magnetic dipole (M1) transitions.These lines were located in a narrow range of wavelengths, mostly well isolated with line ratios that could infer plasma properties, and were sensitive to electron densities.All these features make the M1 lines useful for plasma diagnostics.The measured observed wavelengths for M1 transitions and the FAC energy levels were analyzed by Kramida [4] for spectra for these ions, and form the basis for the energy levels included in the Atomic Spectra Database (ASD) [5].
At the same time, highly charged ions are of special interest for theory in that both correlation and relativistic effects are interrelated, and additional quantum electrodynamic (QED) corrections are needed for accurate results.Quinet [6] reports an extensive summary of a large variety of theoretical energy levels and forbidden transitions for all levels of 3d k ground configurations, and compared their energy levels with the NIST energies.Included among the various methods were results that he obtained using the GRASP code developed by Norrington [7].Most of the correlation included in the calculation was valence correlation restricted to the n = 3 complex.More recently, Guo et al. [8] computed energy levels, wavelengths, and transition probabilities for the same configurations for a number of ions, including tungsten.The theoretical basis for their work was the relativistic many-body perturbation theory (RMBPT) as described in [9], but small corrections for finite nuclear size, nuclear recoil, vacuum polarization, and self-energy correction were also included using standard procedures such as those in GRASP2K [10].All basis orbitals were determined from the same central field, and all three types of correlation-valence-valence (VV), core-valence (CV), and core-core (CC)-where the core consists of the the full 1s, 2s, 2p core were included .Statistically, their energy levels were in much better agreement with NIST values than those of Quinet [6].
The purpose of the present work was to evaluate the accuracy of energy levels obtained from variational multconfiguration Dirac-Hartree-Fock methods as implemented in the GRASP2K code [10].Included are all three correlation types as in the RMBPT calculation-except for the 1s 2 core, that will be assumed to be inactive.

Multiconfiguration Dirac-Hartree-Fock (MCDHF) and Configuration Interaction Methods
In the MCDHF method [11,12], as implemented in the GRASP2K program package [10], the wave function Ψ(γPJ M J ) for a state labeled γPJ M J , where J and M J are the angular quantum numbers and P is the parity, is expanded in antisymmetrized and coupled configuration state functions (CSFs) Ψ(γPJ M J ) = M ∑ j=1 c j Φ(γ j PJ M J ). ( The labels {γ j } denote other appropriate information about the CSFs, such as orbital occupancy and coupling of the subshells.The CSFs are built from products of one-electron orbitals, having the general form where χ ±κ,m (θ, ϕ) are two-component spin-orbit functions.The radial functions {P nκ (r), Q nκ (r)} are represented numerically on a grid.Wave functions for a number of targeted states are determined simultaneously in the extended optimal level (EOL) scheme.Given initial estimates of the radial functions, the energies E and expansion coefficients c = (c 1 , . . ., c M ) t for the targeted states are obtained as solutions to the configuration interaction (CI) problem Hc = Ec, where H is the CI matrix of dimension M × M with elements Radial functions are solutions of systems of differential equations that define a stationary state of an energy functional for a wave function expansion.
Two types of expansions may be used.In the past, both usually were the same, but for large calculations, there are advantages to relaxing this restraint.The first is the expansion that determines the radial functions using the RMCDHF program of the GRASP2K package.For occupied orbitals, optimized radial functions can be obtained by applying the variational principal of an energy expression.However, when correlation orbitals are to be determined, the most effective orbitals are those that are in the same region of space as the occupied orbitals for a given type of correlation, as has been shown in partitioned configuration interaction (PCFI) studies [13].In this work, we consider two regions: the 3s, 3p, 3d region for valence-valence (VV) correlation and the 2s, 2p region for core-valence (CV) and core-core (CC) correlations.
The second is an expansion for the relativistic configuration interaction (RCI) program that determines the wavefunction and its associated energy for a given Hamiltonian and based on a given orbital basis.In the present work, the Hamiltonian for RCI was the Dirac-Coulomb Hamiltonian (DC) plus the transverse photon interaction (DCB), the vacuum polarization effects as accounted for by the Uehling potential, and electron self-energies as calculated with the screened hydrogenic formula [12,14], namely the DCBQ Hamiltonian.The RCI program is relatively simple to parallelize efficiently [15,16] using message passing.As a result, much larger expansions are possible for RCI calculations than RMCDHF ones that build the orbital basis.Present calculations were done with forty-eight (48) processors for the larger cases.
The computational procedure was essentially the same for all ions.The first step was to perform Dirac-Hartree-Fock (DHF) calculations (in the EOL approximation) for all states associated with the 3s 2 3p 6 3d k configuration.This calculation determined the 1s, 2s, 2p orbitals for all subsequent calculations.Then, sequentially, orbital sets of increasing size, with maximum principal quantum numbers n = 3, 4, 5, were determined from expansions that defined valence-valence correlation expansions.The latter were obtained from single-and double-excitations from the valence shells to those of the orbital set.Since the 3d shell is unfilled, excitations such as 3s 2 → 3d 2 are allowed and increase the generalized occupation number for the 3d orbitals but decrease those of 3s.Variational methods determined the new orbitals introduced at each stage using the Dirac-Coulomb Hamiltonian.The n = 6 orbitals were targeted for core correlation effects.They were obtained from calculations that included CV correlation from the n = 2 shell where one orbital from the active core (either 2s or 2p) and one 3s, 3p, or 3d orbital were excited, as well as CC, where two n = 2 orbitals were excited.At the same time, excitations from 3s, 3p subshells were limited to single excitations for 3s or 3p, thereby contracting the n = 6 orbitals to overlap more strongly with the n = 2 orbitals and reducing the size of the expansions.For the configurations 3d k , k = 3, 4, 5, 6, 7, the expansions were still exceedingly large and additional restrictions on interactions were imposed that define the energy functional.First, what might be considered a zero-order approximation was obtained that consisted of the CSFs of the n = 5 VV expansion that accounted for 99.9 percent of the normalized expansion.All other terms of the n = 6 expansions were treated as first-order corrections.In deriving the energy expression that determines the radial factors of the n = 6 orbitals, it was assumed that the interaction between CSFs of the first-order corrections could be neglected.This procedure optimizes the interaction of the n = 6 orbitals with the zero-order wave function, and has the effect of contracting the core-valence orbitals.
Each of these four orbital sets were then used in relativistic configuration interaction (RCI) calculations that included VV, CV, and CC correlation effects (excluding the 1s shell) for the three Hamiltonians-DC, DCB, and DCBQ.Again, for the cases where k = 3, 4, 5, 6, 7, the RCI calculations were performed under the assumption that interactions between CSF of the first-order correction could be ignored.
Table 1 summarizes the size of various expansions for the different 3d k configurations, whereas Table 2 shows how the mean radii of the n = 6 orbitals are contracted relative to the valence correlation orbitals.Note that the size increases rapidly as the number of electrons (or holes) increases from one to five, as well as the number of J values and levels.The number of CSFs defining 99.9% of the wave function composition is relatively small.Increasing this percentage to 99.99% would include some higher order corrections.As for mean radii, it should be noted the the 3d orbitals (in non-relativistic notation) have a mean radius closer to the core than either 3s or 3p.Listed in Table 2 are typical values for the 3d 5 configuration.The mean radii are also depicted graphically in Figure 1.Correlation increases the generalized orbital occupation number of the 3d orbitals, but decreases those of all other occupied orbitals.The n = 4 and n = 5 orbitals have mean radii similar to those of the valence orbitals, whereas the n = 6 orbitals that are used to represent CC and CV correlation have mean radii either similar to n = 2 orbitals or between n = 2 and n = 3, as in CV correlation.

Results and Their Comparison
Table 3 reports some of the results for all levels of the 3d k configurations of tungsten ions from RCI calculations for the DCBQ Hamiltonian.The classification of energy levels are presented in the LSJand jj-couplings.A set of three quantum numbers L, S, and seniority ν allows a one-to-one classification of 3d k (k = 3, 4, 5, 6, 7) energy levels in LSJ-coupling.These quantum numbers are presented in Table 3 as (2S+1) L ν .The n = 5 results include only VV correlation, whereas n = 6 include all three correlation effects.The next column is the energy levels as reported by NIST [5].Included here are the different types of results.Energies with no square brackets are directly related to observed wavelengths-often these are in the lower portion of the spectrum.Then, there are levels that may be linked to an observed wavelength but the shift of the energy levels relative to the ground state is not known from experiment.These levels include a +x or +y in the table.Thus, the difference between two levels with the same +x is known accurately, but not the levels themselves.Taking these factors into account, it is clear that the inclusion of core effects has reduced the discrepancy with NIST values by about a factor of 1/2.In the next column, the values found by Quinet [6] are generally like the VV results.From a general theoretical point of view, the the RMBPT results of Guo et al. [8] should be the most accurate.In the case of 3d 2 , RMBPT results have also been reported by Safronova and Safronova [17], and are reported in the last column.These results are not as accurate as those of Guo et al.In these tables, all energies are reported in the units of 1000 cm −1 .
Table 3. Energy level results for 3d, 3d 2 , . .., 3d 8 , 3d 9 ground configuration of tungsten ions.Shown is a unique label in LSJand jj-notation, the J value, the present n = 5 result for valence-valence (VV) correlation, and n = 6 result for all three types of correlation, the Atomic Spectra Database (ASD) value [5], the Quinet value [6], the Guo et al.RMBPT g value [8], and the Safronova & Safronova RMBPT s value [17].All energy levels are reported in 1000 cm −1 .The uncertainties of NIST energy levels not based on observed wavelengths are estimated as being less than 5000 cm −1 , or 5.00 in our table.In order to better understand the importance of various effects in Table 4, we report the NIST energy levels that are based on observation and differences of various theories for only those levels where NIST values are accurate, although there may be an unknown shift.Table 4. Difference from NIST energy levels derived from observation.Shown is the LS label, the J value, the present n = 5 result for VV correlation, and n = 6 result for all three types of correlation, the ASD value [5], the Quinet value [6], the Guo et al.RMBPT g value [8], and the Safranova & Safronova RMBPT s value [17].All energy levels are reported in 1000 cm −1 .Table 4 shows clearly that the uncertainties of the present n = 6 results are smaller by about a factor of a half when no shifts are indicated in the NIST value.For these levels, the n = 6 results statistically differ less than the Quinet values that are similar to the less accurate n = 5 values.The most accurate results for 3d and 3d 9 are the RMCDHF g results, although for 3d 9 , the n = 6 are almost of the same accuracy.RMBPT g is the more accurate for 3d 2 , with n = 6 almost the same.For 3d 8 , the two lower levels, RMBPT g is the more accurate, whereas n = 6 is the more accurate for the two upper levels.A similar pattern seems to hold for other spectra.An interesting case is 3d 7 4 P J = 5/2 and 1/2, where both levels have an unknown shift.An exact theoretical value and an exact NIST value (except for the shift) would have the same difference for the two levels.In the present case, the n = 6 differences are more similar than the RMCDHF g differences.In fact, from this table, we can conclude that any NIST value for which the theoretical difference from NIST for both methods is more than 1.00 has a noticeable error.Thus, for example, the 3 P 1 level of 3d 8 with an energy level of 644.70 Kcm −1 suggests that the NIST values is not accurate to two decimal places.

Label
The errors in different theoretical results are shown in Figure 2. Note the similarity in accuracy of the present n = 6 results and values reported by Guo et al. [8].The accuracy of theoretical energy levels are best evaluated by comparing theoretical wavelengths with wavelengths of observed lines in the spectrum.In Table 5, all wavelengths for M1 transitions between the 3d k levels for the present n = 5, 6 results are compared with experimental results and other theory, when available.This table clearly shows the improvement in accuracy of n = 6 calculations over n = 5, as well as the GRASP results reported by Quinet [6], and in many cases the very close agreement with Guo et al. [8].Two exceptions are the 3d 7 4 F 3 − 3d 7 2 F 3 (J = 9/2 to J = 7/2) transition, for which the observed wavelength is 14.166(3) nm, the present n = 6 is 14.170 nm, and the Guo et al. value is 14.187 nm.Similarly, the 3d 8 3 F − 3d 8 1 G (J = 4 to J = 4) transition has an observed wavelength of 15.511(3) nm, whereas the present value is 15.518 nm and the Guo et al. value is 15.463 nm.

Conclusions
The present study has shown that the inclusion of core correlation effects improves the accuracy of theoretical transition wavelengths for M1 transitions in 3d k configurations of tungsten ions.Omitted in our work were correlation effects arising from the 1s 2 core.Further studies are needed to determine whether the discrepancy with observation arises from the limited orbital set for core correlation or from the inactive 1s 2 shell in our present work.

Figure 2 .
Figure 2. Plot comparing the accuracy of different theoretical methods.

Table 1 .
Tableshowingthe size (M) of the n = 6 relativistic configuration interaction (RCI) expansions and the size of the zero-order space (m) for the different tungsten ions.

Table 2 .
Mean radii in a.u. of orbitals for the 3d 5 configuration and their generalized occupation number w.
Figure1.Plot of the mean radii of orbitals of the 3d 5 configuration in the order listed in Table2.

Table 5 .
Wavelengths from theory for observed M1 transitions compared with observed wavelengths (in nm).Included are some long wavelengths for transitions between close-lying levels.