Multiconfiguration Dirac – Fock calculations of Zn K ‐ shell radiative and nonradiative transitions

Zinc K ‐ shell radiative and radiationless transition rates are calculated using the multiconfiguration Dirac – Fock method. Correlation up to the 4p orbital is included in almost all transition rate calculations. Calculated radiative transition rates and transition probabilities are compared with Scofield's Dirac – Hartree – Slater and Dirac – Hartree – Fock calculations, presenting good agreement with the later. Radiative transition intensity ratios

when Scofield presented radiative rates calculated for the majority of transitions and elements, first using the Dirac-Hartree-Slater theory [1] and later using the Dirac-Hartree-Fock theory. [2] In the same decade, Chen et al. [3,4] presented rates of nonradiative transition calculated for the majority of possible transitions and elements, using the Dirac-Hartree-Fock theory. Scofield's and Chen et al.'s benchmark comprehensive calculations allowed for libraries, such as the Evaluated Atomic Data Library, [5] to present comprehensive sets of transition probabilities, which are useful regarding simulations of atomic relaxation and other applications. In the Hartree-Slater and Dirac-Hartree-Slater methods, it is considered that each electron is affected by an average field created by all other electrons, whereas in the Hartree-Fock and Dirac-Hartree-Fock methods, the average field is represented by the Coulomb and exchange operators, J C and K j , such that the average potential is given as where V C is the Coulomb potential and V ex is the exchange potential. With this inclusion, the Hartree-Fock method accounts some of the Coulomb repulsion between electrons and as generally been shown to be more accurate than those of the Hartree-Slater method. However, the Hartree-Fock approach does not fully describe the Coulomb repulsion between electrons. In fact, the major error source in this method arises from the lack of proper correlation between electrons. Thus, methods to fully account the electronic correlation have emerged, such as the multiconfiguration Hartree-Fock, configuration-interaction, many-body perturbation theory, and their relativistic versions, RCI (relativistic configuration-interaction), MCDF (multiconfiguration Dirac-Fock), RMBPT (relativistic many-body perturbation theory), respectively. In RCI (or configuration-interaction) and MCDF (or the multiconfiguration Hartree-Fock), electronic correlation is included by writing the antisymmetric wave function of the atomic system ψ as a linear combination of configuration wave functions φ, which are wave functions for different possible configuration states: ψð1; 2; …; NÞ ¼ ∑ i a i φ i , where a i are mixing coefficients. For example, the wave function for He in the ground state can be written as ψðHeÞ ¼ a 1 φð1s 2 Þ þ a 2 φð1s 1 2p 1 Þ þ a 3 φð2p 2 Þ; where a 1 , a 2 , and a 3 are the mixing coefficients, φ(1s 2 ) is the minimum configuration state function, and φ(1s 1 2p 1 ) and φ(2p 2 ) are extra correlations state functions. The configuration wave functions are written as a combination of one-electron orbitals.
In this work, we present Zn K-shell radiative and nonradiative transition MCDF calculations.

| MCDF CALCULATIONS
For this, the MCDFGME code, [6] developed by J. P. Desclaux and P. Indelicato, was used for the calculations of Zn K-shell transition rates. It implements the multiconfiguration Dirac-Fock method including various contributions self-consistently, such as Coulomb interaction, Breit corrections, and vacuum polarizations. It also includes quantum electrodynamics contributions. One of the many capacities of the code is the calculations of radiative and radiationless transition rates and the energies of the emitted X-rays and Auger electrons. The energy and wave function for the initial configuration, the Zn with a hole in the K-shell (1s 1 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10 ) and the energies and wave functions of all possible final configurations attained through radiative or radiationless transitions are calculated independently, including in most cases all possible extra correlation states up to the 4p orbital. From these, radiative and radiationless transition rates for all possible radiative and radiationless transitions are calculated. In the calculation of radiative transition, correlation up to the 4p orbital is included (for both the initial and final states) in all transitions, with the exception of the K-N 1 transition where in the final state (1s 2 2s 2 2p 6 3s 2 3p 6 4s 1 3d 10 ), only the selected extra correlation configurations up to the 4p orbital are included to be able to achieve convergence. Furthermore, also for the K-N 1 transition rate calculation, to achieve convergence, some orbitals are frozen during the iterative process. Due to convergence problems, several nonradiative transition rates are calculated including only   Note. Calculations were performed with relaxed orbitals unless if "calculation notes" state that orbitals were frozen. Extra correlation state functions from the 1s orbital up to the 4p orbital were included in the calculations unless "calculation notes" present a different orbital. In that case, the calculation was performed including extra correlation wavefunctions from the 1s orbital up to the orbital presented in "calculation notes." In the cases where no extra correlation state functions were included, "calculation notes" state that no correlation was considered. Transition rates were presented in milliatomic units: 1 milliatomic unit =4.134×10 −13 s −1 .
1s 1 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10 and 1s 2 2s 2 2p 6 3p 6 4s 2 3d 10 , no extra correlation states were included. Additionally, in many nonradiative transition rate calculations, the orbitals were frozen during the iterative process. In all calculations, Breit interaction (including its magnetic and retardation components) and vacuum polarization contributions are included in the self-consistent field method. The calculated transition rates are presented in Tables 1 and 2. From the calculated transition rates, K-shell fluorescence yield values and transition rate ratios are obtained.

| CONCLUSIONS
The multiconfiguration Dirac-Fock method calculations of Zn K-shell transition rates are performed. From these values, fluorescence yield, radiative transition probabilities, and transition intensity ratios are derived and compared with theoretical and experimental values. As presented in Table 3, radiative transition rates are in good agreement with Scofield's relativistic Dirac-Fock values. [2] As a consequence, as presented in Table 2, radiative transition probabilities are also in good agreement with Scofield's values. This work's MCDF transition probabilities for the K-L 2 , K-L 3 , K-M 2 , and K-M 3 transitions are in close agreement with the empirical values from NIST's Fundamental Parameters Database, [7] whereas for the K-L 1 and K-M 4,5 transitions, the same cannot be said. As for radiative transition ratios, as presented in Table 5, this work's values are in good agreement with Scofield's Hartree-Fock values. [2] When comparing against NIST's database ratios, [7] relatively good agreement is presented for the Kα 2 /Kα 1 , Kβ 3 /Kβ 1 , and Kβ 1 ′/ Kα 1 ratios, but strong disagreement is found for the Kα 3 /Kα 1 and Kβ 5 /Kβ 1 ratios.
From the comparison of Kβ/Kα ratios, presented in Table 7, it is interesting that Kup et al. [9] 's value is equal to the present work's MCDF value, even though in their calculations, no multiconfiguration wavefunctions were included except those of intermediate coupling. When comparing the present work's MCDF value with the theoretical values from Scofield, closer agreement is found with Scofield's Dirac-Hartree-Fock value. [2] Even though NIST's Fundamental Parameters Database [7] uses Salem et al.'s ratios, [8] their value differ from one another. This discrepancy can be due to NIST's database treatment to Salem et al.'s data. The present work value is in good agreement with Salem et al.'s empirical fit value [8] and with Kahoul et al.'s empirical fit value. [20] As presented in Table 8. this work's values of radiationless transition rates are often lower than Chen et al.'s [3] values and often higher than Safronova et al.'s values. [21] It is presented in Table 9 that the K-LL total value is lower than Chen et al.'s value and higher than Safronova. As for the K-LX and K-XY total rates, these presented values are higher than the other compared. Thus, the K-LX/K-LL and K-XY/K-LL ratios from this work are higher than the other values compared. The nonradiative transition intensities relative to the K-L 2 ( 1 D 2 ) transition, as presented in Table 10, are in most cases in good agreement with the experimental results presented.
The fluorescence yield value from this work is higher than all other theoretical and experimental values compared in Table 11. It is likely that the total radiationless transition rate A (TA) calculated in the present work is lower than it should be, and as a result, the K-shell fluorescence yield is calculated higher than it should be (as can be seen from Equation 4). Such is supported by the comparisons in Table 9, where it is shown that the calculated value for the total K-LL transitions rate is lower than the value from Chen et al.'s calculations. [3] Interestingly, the most recent values obtained through empirical fittings, which are those  [24] 0.466 HF [4] 0.488 Kup [9] 0.485