Calculations Energy of the (nl2) 1Lπ Doubly Excited States of Two-Electron Systems via the Screening Constant by Unit Nuclear Charge Formalism

In this work, the total energies of doubly excited states (ns) S, (np) D, (nd) G, (nf) I, (ng) K, and (nh) M of the helium isoelectronic sequence with Z ≤ 10 are calculated in the framework of the variational method of the Screening Constant by Unit Nuclear Charge (SCUNC). These calculations are performed using a new wavefunction correlated to Hylleraas-type. The possibility of using the SCUNC method in the investigation of high-lying Doubly Excited States(DES) in two-electron systems is demonstrated in the present work in the case of the (nl) 1Lπ doubly excited states, where accurate total energies are tabulated up to n = 20. All the results obtained in this paper are in agreement with the values of the available literature and may be useful for future experimental and theoretical studies on the doubly excited (nl) 1Lπ states of two-electron systems.


Introduction
Studies of doubly-excited states of helium-like systems remain an active field of research since the early experiments of Madden and Codling [1] [2] concerning the observation of resonant structures in the absorption spectrum of helium using synchrotron radiation. The strong correlation between electrons in the 2. Theory and Calculations

Hamiltonian and Hylleraas-Type Wave Functions
The time independent Schrödinger equation for the Helium atom, or the positive ions of its isoelectronic sequence, or of the negative Hydrogen ion, is where Ĥ represents the Hamiltonian operator of the considered system, Ψ the trial wave function and E the associated energy.
The Hamiltonian H of the helium isoelectronic series in given by (in atomic units) In this equation, r 1 and r 2 denote the position of the two electrons from the nucleus, Z is the nuclear charge, Δ 1 is the Laplacian with reference to the coordinates of the vector radius r 1 which detect the position of the electron 1. Δ 2 Laplacian defines the coordinates of the vector radius r 2 which detect the position of the electron 2 and r 12 inter-electronic distance.
Solving Equation (1) is very difficult because of the term 12 1 2 r u r r = = − representing electron-electron repulsion. It is therefore necessary to implement a rough calculation method using a correlated wavefunction.
The groundbreaking work in this area was conducted by Hylleraas [28] [29] [30]. The simplest Hylleraas wave function is written as follow: ( ) ( ) ( ) 1 Since Hyllerass' original work, tremendous efforts have been made to improve upon that work, using larger and larger expansions, adding more complicated terms. In this present work, we have modified this Hylleraas wavefunction in order to adapt it to the study of symmetrical (nl 2 ) 1 L π doubly excited states in two-electron atomic systems. These wave functions are defined as follow: In this expression, n is the principal quantum number; ℓ is orbital quantum number, r 0 is Bohr radius, C 0 and α are the variational parameters to be determined by minimizing the energy, Z is the nuclear charge number, r 12 represents the term electron-electron repulsion r 1  , , In this equation, the correlated wave functions are given by (4) and the Hamiltonian H of the helium isoelectronic series in given by (2) (in atomic units).
Furthermore, the closure relation represents the fact that 1 2 , r r are continuous bases in the space of the two-electron space, written as follow: Using this relation, according to (7), we obtain: 3  0  1  2  0  1 2  1 2  0   3  3  1  2  0  1 2  1 2  0   ,  d d  ,  ,  ,  ,   d d  ,  , , , ∫∫ ∫∫ r r r r r r r r (9) By developing this expression (9), we find: This means: with the normalization constant ( ) To make it easier to integrate Equation (11), we operate the variable changes in elliptic coordinates by: ; ; s r r t r r u r = + = − = (13) On the basis of these variable changes, the elementary volume element ( ) with respect to the correlated wave functions given by expression (4), it is ex- Furthermore, according to (12), the normalization constant is written in elliptic coordinates as:

General Formalism of the SCUNC Method
The Screening Constant by Unit Nuclear Charge (SCUNC) formalism is used in this work to calculate the total energies of the symmetrical (nl 2 ) 1 L π doubly excited states of the helium-isoelectronic up to Z = 10. In In this equation, the principal quantum numbers N and n, are respectively the inner and the outer electron of the helium-isoelectronic series. In this equation, the β-parameters are screening constant by unit nuclear charge expanded in inverse powers of Z and given by ( ) are screening constants to be evaluated based on variational predicable using a wavefunction.
For the states (nl 2 ) 1 L π , N = n and l = l'. Hence, the total energy is written as follow: Furthermore, in the framework of the screening constant by unit nuclear charge formalism, the β-screening constant is expressed in terms of the variational α-parameter as follow ( ) In this expression, n denotes the principal quantum number, L characterizes the considered quantum state (S, P, D, F etc.) and α is the variational parameter. Then, using Equation (21), the total energies of the symmetrical (nl 2 ) 1 L π doubly excited states in the helium isoelectronic series is expressed in Rydberg (Ry) as below: ( ) In this equation, only the parameter α is unknown. Considering the (2s 2 ) 1 S e state of Helium-like ions (Z = 2 -10), we calculated the values of the variational parameters α and C 0 , the results of which are presented in Table 1.
The Equation (22) is used to calculate the total energies of the (nl 2 ) 1 L π doubly excited states of helium-like ions without a complex calculation program.

Results and Discussions
The results obtained in the present study for (ns 2 ) 1 S e , (np 2 ) 1 D e , (nd 2 ) 1 G e , (nf 2 ) 1 I e , (ng 2 ) 1 K e , and (nh 2 ) 1 M e with n ≤ 20 in the helium-like ions up to Z = 10 are listed in Tables 1-16 and compared to various other calculations. Table 1 presents our results on the calculation of the variational parameters α and C 0 . These variational parameters are calculated by determining the expression of E = f(a, C 0 ) from the expression (15) and the wavefunction (16), then according to conditions (5) and (6) we obtained a system of equations whose resolution to give the values of the variational parameters α and C 0 with 2 ≤ Z ≤ 10. All calculations in this work were performed with the calculation program MAXIMA. In Tables 2-7 we have listed our present results E on the calculation of the total energies of the (nl 2 ) 1 L π doubly excited states of the helium isoelectronic sequence with 2 ≤ Z ≤10 and 2 ≤ n ≤ 20 obtained using Equation (22). Table 2 shows our present results of the (ns 2 ) 1 S e (n = 2 -20) doubly excited states of helium -like systems (Z = 2 -10). Table 3 shows our present results of the (np 2 ) 1 D e (n = 2 -20) doubly excited states of helium-like systems (Z = 2 -10). Table 4 shows our present results of the (nd 2 ) 1 G e (n = 3 -20) doubly excited states of helium-like systems (Z = 2 -10). Table 5 shows our present results of the (nf 2 ) 1 I e (n = 4 -20) doubly excited states of helium-like systems (Z = 2 -10). Table 6 shows our present results of the (ng 2 ) 1 K e (n = 5 -20) doubly excited states of helium-like systems (Z = 2 -10). Table 7 shows our present results of the (nh 2 ) 1 M e (n = 6 -20) doubly excited states of helium-like systems (Z = 2 -10). Table 8 shows a comparison of the present calculations for the (ns 2 ) 1 S e states with the results of the semi-empirical procedure of the screening constant by      unit nuclear charge method of Sakho et al. [24] [32], with the results of the complex rotation of Ho [33] [34] [35], with the results of Sow et al. [36] who used the variational method of the SCUNC formalism, with the complex rotation values of Gning et al. [37], with the Konté et al. [38] data, with the data from the time-dependent variation perturbation theory of Ray et al. [23], with the results of Diouf et al. [39] and finally with the data from the modified slater theory of Sakho [40]. The observation of our results shown in this table shows that our present calculations are generally in good agreement with the results of the cited authors up to Z = 10.
In Table 9, we compare our calculations for (np 2 ) 1 D e states with the results of the calculations of Badiane et al. [41], with the data of Sakho [24] [32] [40], with the results of the complex rotation calculations (CRC) of Ho and Bhatia [13], with the values of Ivanov and Safronova [17], with the results of the variational method calculations of Hylleraas de Biaye et al. [16] and finally with those obtained by Roy et al. [18] who applied the functional density theory (FDT). Here, the agreements between the calculations are considered good. Table 10 compares our results for the (nd 2 ) 1 G e (n = 3 -10) states with those obtained by Badiane et al. [41], sakho [32] [40], Bachau et al. [19], Biaye et al. [16], Ivanov and Safronova [17], Diouf et al. [39], Ray et al. [23] and Roy et al. [18]. As regards the (nd 2 ) 1 G e -levels, comparison shows also a good agreement up to Z = 10. Table 11 shows the results of our present calculations of the total energies of the doubly excited (nf 2 ) 1 I e (n = 4 -10) states of helium-like systems up to Z = 10, which we compare with those obtained by Biaye et al. [16], Badiane et al. [41], Sakho et al. [32] [40], Ho [20], Diouf et al. [39], and Sow et al. [36]. A comparative Journal of Modern Physics   [39], g Ray et al. [23], h Roy et al. [18], i Sakho [40]. Journal of Modern Physics Table 11. Comparison of the present calculations on total energies of the doubly (nf 2 ) 1 I e (n = 4 -10) excited states of helium-like systems (Z = 2 -10) with available literature values. All results are expressed in eV. should be noted that comparison with the results of Biaye et al. [16] indicates satisfactory agreement for Z = 2 -10.
In Table 12 and Table 13, we compare the results of our calculations of the total energies of the doubly excited states (ng 2 ) 1 K e and (nh 2 ) 1 M e with those of Sakho [40] and Diouf et al. [39]. The agreements between the calculations are seen to be generally good. It is worth mentioning that there are not many results M. T. Gning et al.  on the states and the only ones available to our knowledge are those of the authors Sakho [40] and Diouf et al. [39]. Moreover for n > 7 there are no results available so we think that the results cited up to n = 20 in this work may be interesting for future experimental and theoretical studies on these states.
In Tables 14-16    G e , (nf 2 ) 1 I e , (ng 2 ) 1 K e , and (nh 2 ) 1 M e of the He-like ions up to Z = 10. This very good agreement sufficiently justifies the validity of the variational procedure of the SCUNC method to give the precise values obtained directly from an analytical expression, unlike all the ab-initio methods cited in this document. Furthermore, the results quoted up to n = 20 in this work may be interesting for future experimental and theoretical studies in the doubly excited states (nl 2 ) 1 L π . In summary, the manuscript reports on new calculations for key atomic-structure parameters of important fundamental few-body systems (helium and helium-like ions). While not allowing more precision tests of physics due to the neglect of relativistic, spin, and QED effects, such results can still be helpful in the future development of theories to describe more complex atoms, or may be further developed to study the time-dependent evolution of atoms in external (e.g. laser) fields.

Conclusion
In this paper, the total energies and excitation energies of the doubly excited (ns 2 ) 1 S e , (np 2 ) 1 D e , (nd 2 ) 1 G e , (nf 2 ) 1 I e , (ng 2 ) 1 K e , and (nh 2 ) 1 M e states of helium-like ions up to Z =10 are reported. These energies are calculated in the framework of the variationnal procedure of the Screening Constant by Unit Nuclear Charge (SCUNC) formalism. In this present work, a new wavefunction correlated to Hylleraas-type adapted to the correct description of electron-electron correlation phenomena in the (nl 2 ) doubly excited states of helium-like systems has been constructed. Our results for total energies and the excitation energies are in good agreement with the values cited in the experimental and theoretical literature.
Furthermore, for n > 10, no theoretical and experimental values from the literature are available for direct comparison. The good precision obtained in this work underlines that the results quoted up to n = 20 in this work may be interesting for future experimental and theoretical studies in the doubly excited states (nl 2 ) 1 L π . The results presented in this paper show that it is therefore possible to perform an analytical calculation of the total energies of the (nl 2 ) doubly excited states for helium-like ions, without having to resort to excessively complicated calculations or a computer program.