Screening Constant by Unit Nuclear Charge Calculations of Resonance Energies of the 3(K,

We report accurate energies of the 3(2,0,+)n D, G, F°, 3(0, 2,+)n D, 3(1,1,+)n D, P, F, 3(−1,1,+)n P, and 3(1,1,+)n D°, F°, series of Helike ions using the Screening constant by unit nuclear charge method. It is shown that the angular correlation quantum number K is effectively related to the cosinus of the angle between the position vectors of the two electrons.


Introduction
Helium-like systems are very rich in structures attributed to doubly excited states with mixed configurations. Studies of autoionizing states in the helium isoelectronic sequence are very useful prototypes for the analysis of "manyparticles" investigations. As the independent particles model is unappropriated for interpreting doubly excited states (DES) of He-like systems, most atomic spectra are treated on the classification scheme with the set of correlation quantum numbers K, T and A. For a given state of He-like systems, the classification scheme is labelled as For the DES converging to the N = 2 hydrogenic threshold, the Screening constant by unit nuclear charge (SCUNC) has been used previously [2,3] to report accurate results for Helike systems. Using the complex-coordinate rotation (CCR) method, Chung and Lin [4] studied over 540 doubly excited states of Li + resonances below the N = 2 and N = 3 thresholds and tabulated relativistic resonances parameters by grouping the DES into Rydberg series labelled in the (K, T, A) scheme classification. Using the Feshbach formalism, Bachau et al., [5] reported a complete set of data belonging to the lowest resonances of 1,3 S e , 1,3 P°, 1,3 D e , 1,3 F° and 1,3 G e symmetries of He-like doubly excited states lying between the N = 2 and N = 3 thresholds with Z = 2-10. But, energy positions are missing for Z >10 and in the NIST database, no data is quoted for the Rydberg series of the helium-isoelectronic series converging to the N ≥ 3 hydrogenic thresholds. In addition, it is also challenging to succeed on interpreting quantitatively the physical meaning of the K correlation quantum number. In fact, if T is roughly speaking the projection of L onto the interelectronic axis and describes then the orientations between the orbitals of the two electrons, K is related to the cosinus of the interelectronic angle θ 12 as K ≈ -〈r < cos θ 12 〉 where r < denotes the radius of the inner electron. Physically, the larger the positive value of K, the value of − cosθ 12 is closer to unity. In addition for states with positive K, r 12  Energies of the 3 (K, T, A) n 1,3 L π Rydberg Series of He-Like Ions of new quantum models for interpreting atomic spectra must first be appropriated in the well description of electronelectron correlation and relativistic effects in two electron systems. So a complete description of all the Rydberg series belonging to the He-like systems is necessary. The goal of the present study is to extend recent calculations of Sakho [6] on the 1,3 P° series to the 3 (2,0,+) n 1 D e , 3 (0,2,+) n 1 D e , 3 (1,1,+) n 3 D e , 3 (1,1,+) n 3 P e , 3 (−1,1,+) n 3 P e , 3 (1,1,+) n 3 F e , 3 (2,0,+) n 1 G e , 3 (1,1,+) n 1 D°, 3 (1,1,+) n 1 F°, and 3 (2,0,+) n 3 F° Rydberg series of Helium -like ions (Z =3-40) applying the Screening constant by unit nuclear charge (SCUNC) method. Quantitative interpretation of the K-angular correlation quantum number is also aimed in this work. In section 2 presents the theoretical procedure adopted in this work. The results obtained are discussed in section 3,

Brief Description of the SCUNC Formalism
In the framework of Screening constant by unit Nuclear charge formalism, total energy of ( ) In this equation, the principal quantum numbers N and n, are respectively for the inner and the outer electron of Heisoelectronic series. In this equation, the β-parameters are screening constant by unit nuclear charge expanded in inverse powers of Z and given by are parameters to be evaluated. With respect to the new classification scheme, equation (1) takes the form [2,3] [ Using equation (2), we get from (3)

Expressions of the Resonance Energies
For all the Rydberg series investigated in the present work and lying to the N = 3 hydrogenic threshold, total energy is expressed as follows using Eq. (4) The -screening constants in Eq. (5) are evaluated from accurate relativistic data of Chung and Lin [4] on Li + . To take into account the effect of the nucleus volume with increasing Z, a tiny correction of type we have introduced in Eq. (5) -screening constants in Eq. (5) are evaluated from accurate relativistic data of Chung and Lin [4] on Li the energy positions measured with respect to the ground state E 0 of Li + , total energy (5) is given by For Li + , E 0 = − 198.0977 eV and the reduced Rydberg is equal to 13.604635 eV [4].
The present study is limited to the Rydberg series starting with the lowest n = 3 resonance.

Results and Discussion
The results obtained in this work are listed in Tables 1-6 Table 1. The order of the other series are of similar. To clarify accuracy in the present calculations, some data are compared with the Feshbach formalism (FF) computations of Bachau et al., [5].
For Table 1 where total energies of the ( ) +     [5]. Here again, the agreements between the calculations are very good up to Z = 10.   The very good agreements between the SCUNC and the FF [5] calculations are also obtained comparing the results quoted in Table 3 Table 5 presents calculations of excitation energies for doubly 1,3 L e (L = P, D, F, G) and 1,3 L° (L = D, F) excited states of He-like (Z = 4-10) systems. Energies are calculated with respect to the ground state of the corresponding system. Comparison is done with some literature data such as those from Density functional theory (DFT) of Roy et al., [11], Complex rotation method (CCR) of Ho and Bathia [12], time-dependent variation perturbation (TDVP) of Ray and Mukherjee [13] and those from Multiconfiguration (MC) of Lipsky et al., [14]. It should be underlined that, in the DFT and TDVP models, the doubly excited states are labeled in the Nlnl' 2S+1 L π notation disregarding the appropriated N (K, T, A) n 2S+1 L π classification scheme. Overall good agreement are obtained between the calculations.  [5] have been calculated here by using the same ground state from NIST [15]. The other results published in a. u are taken from Roy et al., [11].   [5]. DFT, Density functional theory, Roy et al., [11]. CCR, complex rotation method, Ho and Bathia [12]. TDVP, time-dependent variation perturbation, Ray and Mukherjee [13]. MC, Multiconfiguration, Lipsky et al. [14].
On the other hand, according to the physical meaning of the K correlation angular quantum number, for states with positive K, the two electrons tend to stay on the opposite sides of the nucleus while in states with negative K the two electrons tend to stay on the same side of the nucleus. In addition, the angular correlation quantum number K is related to the cosinus of the interelectronic angle θ12 as K ≈ -〈r< cos θ12〉 where r< denotes the radius of the inner electron. Physically, the larger the positive value of K, the value of − cosθ12 is closer to unity. These statements can be verified quantitatively in the framework of the SCUNC method by evaluating the radial expectation values n N r 1 12 − given by (in a.u) [6] ( ) For the N = 3 threshold, we get

Conclusion
In this paper, accurate resonance energies of the 3 (K, T, A) n 1,3 L π Rydberg series of the helium-like ions (Z = 3-12) are reported. Calculations are performing in the framework of the Screening constant by unit nuclear charge formalism. Good agreements are obtained with various literature data. In contrast with all the existing ab initio methods for which resonance energies cannot be calculated directed from analytical formula, it is demonstrated in this work the possibilities to report accurate resonance from a simple and single analytical formula. It should be underlined that, no resonance energies are listed in the NIST database for the N > 2 thresholds for many He-like systems. The present calculations may then be very useful for the NIST team as far as critical evaluation of atomic data relative to the doubly 3 (K, T, A) n 1,3 L π excited states in the Helium-like systems are concerned