Variational Calculations of Energies of the (2snl) 1,3Lπ and (2pnl) 1,3 LπDoubly Excited States in Two-Electron Systems Applying the Screening Constant per Unit Nuclear Charge

Momar Talla Gning^{1},
Ibrahima Sakho^{1}^{*},
Maurice Faye^{1},
Malick Sow^{2},
Babou Diop^{2},
Jean Kouhissoré Badiane^{3},
Diouldé Ba^{3},
Abdourahmane Diallo^{3}

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1. Introduction

Study of Doubly Excited States (DES) of He-like ions remains an active field of investigation due to their importance in the interpretation of astrophysical data [1] [2]. These states were first observed by Madden and Codling [3] [4] in photoabsorption experiments on helium using synchrotron radiation, and further experimental studies have shown their presence in highly charged ions [5]. As shown in various studies, electron correlations play an important role in understanding lines atomic species for the diagnosis of astrophysical and laboratory plasma. In addition, DES of the two-electron systems are the most fundamental systems that autoionize. They have been attracting considerable interest, because they are best suited to theoretical study on the resonance phenomena. The understanding of elementary processes in the collisions of electrons with atoms or ions is very important in plasma physics, laser technology, astrophysics and physics of the upper atmosphere.

Experimentally, many of these doubly excited states have been observed in electronic impact experiments by Oda *et al.*, [6] and Hicks and Comer [7]. In their studies, these authors have worked on the energy spectra of ejected electrons from autoionization states in helium excited by electron impact. Other doubly excited states were observed by ion impact by Rudd [8] and by Bordenave-Montesquieu *et al.*, [9]. These DES were also studied by examining the spectra of ejected electrons by Gelabart *et al.*, [10] and by Rodbro*et al.*, [11].

From a theoretical point of view, several ab initio methods have been used. The complex rotation method [12] used in studies of Feshbach-type ^{1,3}*D* resonances in two-electron systems, *Z* = 2 - 10, the variational method [13] [14], the density functional theory [15] was used to calculate the nonrelativistic energies and densities of the doubly excited states of the He-isoelectronics series (*Z* = 2 - 5). The formalism of the Feshbach projection operators [16] was applied for the calculations of energy positions and widths of singlet and triplet (even and odd) resonances of the heliumlike (*Z* = 2 - 10) systems lying between the *n* = 2 and *n* = 3 thresholds, the complex rotation method [17] [18] [19]. The truncated diagonalization method used for calculations of widths for doubly excited states of two-electron systems [20]. The discretization technique [21] applied to the calculation of energies and widths of ^{1,3}*S* resonances of the He isoelectronic series, the semi-empirical procedure of the Screening Constant by Unit Nuclear Charge (SCUNC) method [22] [23] [24]. The time-dependent variation perturbation theory (TDVPT) [25] is employed to study the *Nlnl'*^{1}*L ^{e}* resonances (with

Recently, Gning *et al.*, [27] complex rotation method to determine the resonance parameters of the ((2*s*^{2}) ^{1}*S ^{e}*, (2

In general Most of the theoretical methods mentioned above are based on calculation codes or on tedious and complex mathematical calculation programs and in some cases require very powerful computers. In contrast to these methods, the Screening Constant by Nuclear Unit of Charge (SCUNC) method is a very flexible method and has the advantage of providing very precise resonance energies and excitation energies for very high *n* = 10 of the doubly excited states (*Nlnl'*, ^{2S+1}*L ^{π}*) without complex mathematical programs or calculation codes. In addition, in the recent past, the variational procedure of the SCUNC method has been successfully applied to calculations of resonance energies of doubly excited states

Section 2 gives the procedure of the construction of the correlated wavefunctions used along with a brief overview of the establishment of the analytical expressions used in the calculations. Section 3 gives the presentation and the discussion of the results obtained compared to available theoretical and experimental data.

2. Theory

2.1. Hamiltonian and Hylleraas—Type Wavefunctions

The description of the properties of matter at the atomic scale is in principle based on the solution of the time independent Schrödinger equation.

$\stackrel{^}{H}\Psi =E\Psi $ (1)

where
$\stackrel{^}{H}$ represents the Hamiltonian operator of the considered system (atom, molecule, solid), Ψ the trial wavefunction and *E* the associated energy.

The Hamiltonian *H* of the helium isoelectronic series in given by (in atomic units)

$H=-\frac{1}{2}{\Delta}_{1}-\frac{1}{2}{\Delta}_{2}-\frac{Z}{{r}_{1}}-\frac{Z}{{r}_{2}}+\frac{1}{{r}_{12}}$ (2)

In this equation, *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.

The exact resolution of Equation (1) is usually far too complicated because of the term ${r}_{12}=u=\left|{r}_{1}-{r}_{2}\right|$. It is therefore necessary to implement a rough calculation method using a correlated wavefunction.

In this previous work, Sakho [30] used a special-form Hylleraas correlated wavefunction to calculate the energies of the doubly excited states *nlnl'* (*n* = 2 - 4) of heliumlike ions. In the present study, we have made modifications to these wavefunctions to extend these calculations to the doubly excited states (*Nlnl' *^{2S+1}*L** ^{π}*). These wavefunctions are defined as follows:

$\begin{array}{c}\Psi ={\displaystyle \underset{\upsilon \text{\hspace{0.17em}}=\text{\hspace{0.17em}}0}{\overset{\upsilon \text{\hspace{0.17em}}=\text{\hspace{0.17em}}N-\text{\hspace{0.17em}}\mathcal{l}-\text{\hspace{0.17em}}1}{\sum}}{\left({N}^{2}{r}_{0}^{2}\right)}^{\text{\hspace{0.17em}}\upsilon}}\text{\hspace{0.17em}}{\displaystyle \underset{\upsilon \text{'}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0}{\overset{\upsilon \text{'}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}n\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\mathcal{l}\text{'}-\text{\hspace{0.17em}}1}{\sum}}{\left({n}^{2}{r}_{0}^{2}\right)}^{\text{\hspace{0.17em}}\upsilon \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{'}}}\left[1+{\left(-1\right)}^{\text{\hspace{0.17em}}S}{C}_{0}Z\text{\hspace{0.17em}}\left({r}_{1}-{r}_{2}\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\times {\left({r}_{1}+{r}_{2}\right)}^{j}{\left({r}_{1}-{r}_{2}\right)}^{k}{\left|{r}_{1}-{r}_{2}\right|}^{m}\text{\hspace{0.17em}}\text{e}{\text{\hspace{0.17em}}}^{-\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\left({r}_{1}+{r}_{2}\right)}\end{array}$ (3)

In this expression, *N* and *n* are the principal quantum numbers, *l* and *l'* are orbital quantum numbers, *r*_{0} is Bohr radius, *S* is the total spin of atomic system, *α* and *C*_{0} are the variational parameters to be determined by minimizing the energy, Z is the nuclear charge number,* r*_{1} and *r*_{2} are the coordinates of electrons with respect to the nucleus, *j*, *k*, *m *are Hylleraas parameters satisfying the double condition (*j*, *k*, *m *≥ 0) and *j *+ *k *+ *m *≤ 3. The set of the parameters (*j*, *k*, *m*) define the basis states and then give their dimension *D*. From the theoretical viewpoint, the Hylleraas variational method is based on the Hylleraas and Undheim theorem [31] according to which, a good approximation of the energy eigenvalue *E *(*α*, *C*_{0}) is obtained when the minima of the function (*d*^{2}*E*(*α*, *C*_{0})/*d**α**dC*_{0}) converge with increasing values of the dimension *D *of the basis states and when the function exhibit a plateau.

Using this theorem, the values of the varitional parameters *α*and *C*_{0} can be determined by the following conditions:

$\frac{\partial E\left(\alpha ,{C}_{0}\right)}{\partial {C}_{0}}=0$ (4)

and

$\frac{\partial E\left(\alpha ,{C}_{0}\right)}{\partial \alpha}=0$ (5)

For all calculations, we fixed the value of *j =* 0 and *k* = *m* = 1 and this choice has allowed us to obtain:

$\begin{array}{c}\Psi ={\displaystyle \underset{\upsilon \text{\hspace{0.17em}}=\text{\hspace{0.17em}}0}{\overset{\upsilon \text{\hspace{0.17em}}=\text{\hspace{0.17em}}N-\text{\hspace{0.17em}}\mathcal{l}-\text{\hspace{0.17em}}1}{\sum}}{\left({N}^{2}{r}_{0}^{2}\right)}^{\text{\hspace{0.17em}}\upsilon}}\text{\hspace{0.17em}}{\displaystyle \underset{{\upsilon}^{\prime}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0}{\overset{{\upsilon}^{\prime}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}n\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\mathcal{l}}^{\prime}-\text{\hspace{0.17em}}1}{\sum}}{\left({n}^{2}{r}_{0}^{2}\right)}^{{\upsilon}^{\prime}}}\left[1+{\left(-1\right)}^{\text{\hspace{0.17em}}S}{C}_{0}Z\text{\hspace{0.17em}}\left({r}_{1}-{r}_{2}\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\times \left({r}_{1}-{r}_{2}\right)\text{\hspace{0.17em}}\left|{r}_{1}-{r}_{2}\right|\text{\hspace{0.17em}}\text{e}{\text{\hspace{0.17em}}}^{-\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\left({r}_{1}+{r}_{2}\right)}\end{array}$ (6)

In the framework of the Ritz’ variation principle, the energy $E\text{\hspace{0.17em}}\left(\alpha ,{C}_{0}\right)=\langle H\rangle \text{\hspace{0.17em}}\left(\alpha ,{C}_{0}\right)$ is calculated from the relation:

$E\text{\hspace{0.17em}}\left(\alpha ,{C}_{0}\right)=\langle H\rangle \text{\hspace{0.17em}}\left(\alpha \right)\text{\hspace{0.17em}}=\frac{\langle \Psi \text{\hspace{0.17em}}\left(\alpha ,{C}_{0}\right)\text{\hspace{0.17em}}|H|\text{\hspace{0.17em}}\Psi \text{\hspace{0.17em}}\left(\alpha ,{C}_{0}\right)\text{\hspace{0.17em}}\rangle}{\langle \Psi \text{\hspace{0.17em}}\left(\alpha ,{C}_{0}\right)\text{\hspace{0.17em}}|\text{\hspace{0.17em}}\Psi \text{\hspace{0.17em}}\left(\alpha ,{C}_{0}\right)\text{\hspace{0.17em}}\rangle}$ (7)

In this equation, the correlated wavefunctions are given by (6) and the Hamiltonian H of the helium isoelectronic series in given by (2) in atomic units.

Furthermore, the closure relation represents the fact that $|{r}_{1},{r}_{2}\rangle $ are continuous bases in the space of the two-electron space, written as follow:

$\iint \text{d}{r}_{1}^{3}\text{d}{r}_{2}^{3}|{r}_{1},{r}_{2}\rangle \text{\hspace{0.17em}}\langle {r}_{1},{r}_{2}|=1$ (8)

Using this relation, according to (7), we obtain:

$\begin{array}{l}E\left(\alpha ,{C}_{0}\right){\displaystyle \iint \text{d}{r}_{1}^{3}\text{d}{r}_{2}^{3}\langle \Psi \left(\alpha ,{C}_{0}\right)||{r}_{1},{r}_{2}\rangle \text{\hspace{0.17em}}\times \langle {r}_{1},{r}_{2}||\Psi \left(\alpha ,{C}_{0}\right)\rangle}\\ ={\displaystyle \iint \text{d}{r}_{1}^{3}\text{d}{r}_{2}^{3}}\langle \Psi \left(\alpha ,{C}_{0}\right)||{r}_{1},{r}_{2}\rangle \text{\hspace{0.17em}}\stackrel{^}{H}\langle {r}_{1},{r}_{2}||\Psi \left(\alpha ,{C}_{0}\right)\rangle \end{array}$ (9)

By developing this expression (9), we find:

$\begin{array}{l}E\left(\alpha ,{C}_{0}\right){\displaystyle \iint \text{d}{r}_{1}^{3}\text{d}{r}_{2}^{3}\Psi \left(\alpha ,{C}_{0}\right)\times \Psi \ast \left(\alpha ,{C}_{0}\right)}\\ ={\displaystyle \iint \text{d}{r}_{1}^{3}\text{d}{r}_{2}^{3}}\Psi \left(\alpha ,{C}_{0}\right)\stackrel{^}{H}\Psi \ast \left(\alpha ,{C}_{0}\right)\end{array}$ (10)

This means:

$N\ast E\left(\alpha ,{C}_{0}\right)={\displaystyle \iint \text{d}{r}_{1}^{3}\text{d}{r}_{2}^{3}}\Psi \left(\alpha ,{C}_{0}\right)\stackrel{^}{H}\Psi \ast \left(\alpha ,{C}_{0}\right)$ (11)

With the normalization constant

$N={\displaystyle \iint \text{d}{r}_{1}^{3}\text{d}{r}_{2}^{3}}{\left|\Psi \text{\hspace{0.17em}}\left(\alpha ,{C}_{0}\right)\right|}^{2}$ (12)

To make it easier to integrate Equation (11), we operate the variable changes in elliptic coordinates by:

$s={r}_{1}+{r}_{2};\text{\hspace{0.17em}}\text{\hspace{0.17em}}t={r}_{1}-{r}_{2};\text{\hspace{0.17em}}\text{\hspace{0.17em}}u={r}_{12}$ (13)

On the basis of these variable changes, the elementary volume element

$\text{d}\tau ={\text{d}}^{3}{r}_{1}{\text{d}}^{3}{r}_{2}=2{\text{\pi}}^{2}\text{\hspace{0.17em}}\left({s}^{2}-{t}^{2}\right)\text{\hspace{0.17em}}u\text{\hspace{0.17em}}\text{d}s\text{\hspace{0.17em}}\text{d}u\text{\hspace{0.17em}}\text{d}t\text{\hspace{0.17em}}$ (14)

Using these elliptical coordinates, Equation (11) is written as follows

$\begin{array}{l}\text{\hspace{0.17em}}NE\text{\hspace{0.17em}}\left(\alpha ,{C}_{0}\right)={\displaystyle \underset{0}{\overset{\infty}{\int}}\text{d}s}{\displaystyle \underset{0}{\overset{s}{\int}}\text{d}u}{\displaystyle \underset{0}{\overset{u}{\int}}\text{d}t}\{u\text{\hspace{0.17em}}\left({s}^{2}-{t}^{2}\right)\times \left[{\left(\frac{\partial \Psi}{\partial s}\right)}^{2}+{\left(\frac{\partial \Psi}{\partial t}\right)}^{2}+{\left(\frac{\partial \Psi}{\partial u}\right)}^{2}\right]\\ \text{\hspace{0.17em}}+2\text{\hspace{0.17em}}\left(\frac{\partial \Psi}{\partial u}\right)\times \left[s\text{\hspace{0.17em}}\left({u}^{2}-{t}^{2}\right)\times \frac{\partial \Psi}{\partial s}+t\text{\hspace{0.17em}}\left({s}^{2}-{u}^{2}\right)\times \frac{\partial \Psi}{\partial t}-{\Psi}^{2}\times \left(4Zsu-{s}^{2}+{t}^{2}\right)\right]\}\end{array}$ (15)

With respect to the correlated wave functions given by expression (6), it is expressed as follows

$\begin{array}{l}\Psi \left(s,t,u,\alpha ,{C}_{0}\right)\\ ={\displaystyle \underset{\upsilon \text{\hspace{0.17em}}=\text{\hspace{0.17em}}0}{\overset{\upsilon \text{\hspace{0.17em}}=\text{\hspace{0.17em}}N-\text{\hspace{0.17em}}\mathcal{l}-\text{\hspace{0.17em}}1}{\sum}}{\left({N}^{2}{r}_{0}^{2}\right)}^{\upsilon}}\text{\hspace{0.17em}}{\displaystyle \underset{{\upsilon}^{\prime}=\text{\hspace{0.17em}}0}{\overset{{\upsilon}^{\prime}=\text{\hspace{0.17em}}n\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\mathcal{l}}^{\prime}-\text{\hspace{0.17em}}1}{\sum}}{\left({n}^{2}{r}_{0}^{2}\right)}^{{\upsilon}^{\prime}}}\left[\text{\hspace{0.17em}}1+{\left(-1\right)}^{S}{C}_{0}Zt\right]\times t\text{\hspace{0.17em}}u\text{\hspace{0.17em}}\mathrm{exp}\text{\hspace{0.17em}}\left(-\alpha s\right)\end{array}$ (16)

Furthermore, according to (12), the normalization constant is written in elliptic coordinates as:

$N={\displaystyle \underset{0}{\overset{\infty}{\int}}\text{d}s\text{\hspace{0.17em}}{\displaystyle \underset{0}{\overset{s}{\int}}\text{d}u}}\text{\hspace{0.17em}}{\displaystyle \underset{0}{\overset{u}{\int}}\text{d}t}\text{\hspace{0.17em}}u\text{\hspace{0.17em}}\left({s}^{2}-{t}^{2}\right)\text{\hspace{0.17em}}\times {\Psi}^{2}$ (17)

2.2. General Formalism of the SCUNC Method

The Screening Constant by Unit Nuclear Charge (SCUNC) formalism is used in this work to calculate the resonance energies and the excitation energies of the (2*snl*) ^{1,3}*L ^{π}* and (2

In the framework of the Screening Constant by Unit Nuclear Charge (SCUNC) formalism, resonance energies of the (*Nlnl'*, ^{2S+1}*L** ^{π}*) doubly excited states are expressed in Rydberg (

$E\text{\hspace{0.17em}}\left(N\mathcal{l}n{\mathcal{l}}^{\prime},{\text{\hspace{0.17em}}}^{2S+1}{L}^{\pi}\right)=-{Z}^{2\text{\hspace{0.17em}}}\left(\frac{1}{{N}^{2}}+\frac{1}{{n}^{2}}\text{\hspace{0.17em}}{\left[\text{\hspace{0.17em}}1-\beta \text{\hspace{0.17em}}\left(N\mathcal{l}n{\mathcal{l}}^{\prime}{,}^{2S+1}{L}^{\pi},Z\right)\right]}^{\text{\hspace{0.17em}}2}\right)Ry$ (18)

In this equation, the principal quantum numbers *N *and *n*, are respectively for the inner and the outer electron of the He-isoelectronic series. In this equation, the *β*-parameters are screening constant by unit nuclear charge expanded in inverse powers of *Z *and given by

$\beta \text{\hspace{0.17em}}\left(N\mathcal{l}n{\mathcal{l}}^{\prime},{\text{\hspace{0.17em}}}^{2S+1}{L}^{\pi},Z\right)=\text{\hspace{0.17em}}{\displaystyle \underset{k\text{\hspace{0.17em}}=1}{\overset{q}{\sum}}{f}_{k}}\text{\hspace{0.17em}}{\left(\frac{1}{Z}\right)}^{k}$ (19)

where ${f}_{k}={f}_{k}\text{\hspace{0.17em}}\left(N\mathcal{l}n{\mathcal{l}}^{\prime},{\text{\hspace{0.17em}}}^{2S+1}{L}^{\pi}\right)$ are screening constants to be evaluated based on variational predictable using a wavefunction.

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 follows:

· For the doubly excited states (2s*nl*) ^{1,3}*L*^{π}

$\beta \text{\hspace{0.17em}}\left(2sn\mathcal{l},{\text{\hspace{0.17em}}}^{1,3}{L}^{\pi},Z,\alpha \right)=\text{\hspace{0.17em}}\frac{\alpha}{{Z}^{2}}\text{\hspace{0.17em}}\left(1+\frac{L-S+1}{2n+8}\right)$ (20)

· For the doubly excited states (2*pnl*) ^{1,3}*L*^{π}

$\beta \text{\hspace{0.17em}}\left(2pn\mathcal{l},{\text{\hspace{0.17em}}}^{1,3}{L}^{\pi},Z,\alpha \right)=\text{\hspace{0.17em}}\frac{\alpha}{{Z}^{2}}\text{\hspace{0.17em}}\left(1+\frac{L-S}{n+S\text{\hspace{0.17em}}\left(S+1\right)+3}\right)$ (21)

In these expressions, *N* and *n*, are respectively the principal quantum numbers for the inner and outer electron, *L* characterizes the quantum state under consideration (*S*,*P*,*D*,*F*, etc.), S is the total spin of the atomic system and a is the variational parameter.

2.3. Energy Resonances of the (2*snl*) ^{1,3}*L ^{π}* and (2

Using equations (20) and (21), the resonance energies of the doubly excited (2s*nl*) ^{1,3}*L ^{π} *and (2

· For the doubly excited states (2s*nl*) ^{1,3}*L ^{π}*

$E\text{\hspace{0.17em}}\left(2sn\mathcal{l},{\text{\hspace{0.17em}}}^{1,3}{L}^{\pi},Z\right)=-{Z}^{2\text{\hspace{0.17em}}}\left(\frac{1}{{N}^{2}}+\frac{1}{{n}^{2}}\text{\hspace{0.17em}}{\left[\text{\hspace{0.17em}}1-\frac{\alpha}{{Z}^{2}}\text{\hspace{0.17em}}\left(1+\frac{L-S+1}{2n+8}\right)\right]}^{2}\right)Ry$ (22)

· For the doubly excited states (2*pnl*) ^{1,3}*L ^{π}*

$E\text{\hspace{0.17em}}\left(2pn\mathcal{l},{\text{\hspace{0.17em}}}^{1,3}{L}^{\pi},Z\right)=-{Z}^{2\text{\hspace{0.17em}}}\left(\frac{1}{{N}^{2}}+\frac{1}{{n}^{2}}\text{\hspace{0.17em}}{\left[\text{\hspace{0.17em}}1-\frac{\alpha}{{Z}^{2}}\text{\hspace{0.17em}}\left(1+\frac{L-S}{n+S\text{\hspace{0.17em}}\left(S+1\right)+3}\right)\right]}^{2}\right)Ry$ (23)

In these equations, only the parameter *α* is unknown. Considering the 2*p*3*p* ^{1}*D ^{e}* level of heliumlike ions (

The Equations (22) and (23) are used to calculate the resonance energies of the (2*snl*) ^{1,3}*L ^{π}* and (2

3. Results and Discussions

The results obtained in the present study for the resonance energies and the excitation energies of the doubly excited 2*sns* ^{1,3}*S ^{e}*, 2

Table 1 presents the values of the variational parameters *α* and *C*_{0} 2 ≤ *Z* ≤ 10. These parameters are calculated by determining the expression of *E = f*(*a*, *C*_{0}) from Equation (15) and wavefunction (16) using conditions (4) and (5). All the calculations are performed using a Maxima computer program.

In Table 2 and Table 3, we have listed resonance energies of the 2*sns* ^{1,3}*S** ^{e} *and 2

In Table 7, the present resonance energies of doubly 2*s**ns *^{1,3}*S ^{e} *(

Table 1. Values of variational parameters *α* and *C*_{0} of Helium-like ions (*Z* = 2 - 10).

Table 2. Energy resonances (−*E*) of doubly excited 2*sns*^{1,3}*S ^{e}* (

Table 3. Energy resonances (−*E*) of doubly excited 2*snp* ^{1,3}*P*^{0} (*n* = 2 - 10) states of He-like systems (*Z* = 2 - 10). The results are expressed in atomic units. 1 a.u. = 2 *Ry* = 27.211396 eV.

Table 4. Energy resonances (−*E*) of doubly excited 2*pnp* ^{1,3}*D ^{e}* (

Table 5. Energy resonances (−*E*) of doubly excited 2*pnd* ^{1,3}*F*^{0} (*n* = 3 - 10) states of He-like systems (*Z* = 2 - 10). The results are expressed in atomic units. 1 a.u. = 2 *Ry* = 27.211396 eV.

Table 6. Energy resonances (−*E*) of doubly excited 2*pnf* ^{1,3}*G ^{e}* (

Table 7. Comparison of the present calculations on resonance energies for the doubly 2*sns* ^{1,3}*S ^{e}* (

−*E ^{p}*: Present work, values calculated from Equation (22); −

Table 8. Comparison of the present calculations on resonance energies for the doubly 2s*n*p ^{1,3}*P*^{0} (*n* = 2 - 5) excited states of He-like ions up to *Z* = 10 with available literature values. All energies are given in atomic units. 1 a.u. = 2 *Ry* = 27.211396 eV.

*−**E ^{p}*: Present work, values calculated from Equation (22);

Table 9. Comparison of the present calculations on resonance energies for the doubly 2*pnp* ^{1,3}*D ^{e}* (

−*E ^{p}*: Present work, values calculated from Equation (23); −

Table 8 shows a comparison of the present SCUNC results of resonance energies of the doubly 2*snp* ^{1,3}*P*^{0} (*n *= 2 - 5) excited states of He-like systems up to *Z* = 10 with the results of Ho [17] who used the complex rotation method, Ivanov and Safronova [36], the results of Drake and Dalgarno [2] from the 1/*Z *expansion perturbation theory, the values of Sakho *et al.*, [22] [23] obtained from the semi-empirical procedure of the SCUNC formalism. Comparison is also done with the experimental data of Diehl *et al.*, [37], the theoretical results of Lipsky *et al.*, [35], Gning *et al.*, [27], Sow *et al.*, [29], Biaye *et al.*, [38] [39] who performed their calculations in the framework of a variational calculations using wave function of Hylleraas type, and with the experimental data of Kossmann *et al.*, [40]. In general, very good agreement is obtained between the present calculations and those of the above-mentioned works for all the states studied for *Z* = 2 - 10. As underlined above, the present goog agreements between theory and experiments demonstrate the validity of the new correlated wave functions constructed in this work.

Table 10. Comparison of the present calculations on resonance energies for the doubly 2*pnd* ^{1,3}*F*^{0} (*n* = 3 - 5) excited states of He-like ions up to *Z* = 5 with available literature values. All energies are given in atomic units. 1 a.u. = 2 *Ry* = 27.211396 eV.

*−E ^{p}*: Present work, values calculated from Equation (23);

Table 11*.** *Comparison of the present calculations on the variational calculation of the excitation energies of the doubly excited states 2s*n*s ^{1,3}*S*^{e} (*n* = 3 - 4) of He-like systems with some theoretical results available in the literature consulted for *Z* = 2 - 5. All the results are expressed in atomic units: 1 a.u. = 2 *Ry* = 27.211396 eV.

*E ^{p}*: Present work;

Table 12*.** *Comparison of the present calculations on the variational calculation of the excitation energies of the doubly excited states 2*snp* ^{1,3}*P*^{0} (*n* = 3 - 5) of He-like systems with some theoretical results available in the literature consulted for *Z* = 2 - 5. All the results are expressed in atomic units: 1 a.u. = 2 *Ry* = 27.211396 eV.

*E ^{p}*: Present work;

Table 13. Comparison of the present calculations on the variational calculation of the excitation energies of the doubly excited states 2*pnp* ^{1,3}*D ^{e}* (

*E ^{p}*: Present work;

Table 14. Comparison of the present calculations on the variational calculation of the excitation energies of the doubly excited states* *2*pnd* ^{1,3}*F*^{0} (*n* = 3 - 5) of He-like systems with some theoretical results available in the literature consulted for *Z* = 2 - 5. All the results are expressed in atomic units: 1 a.u. = 2 *Ry* = 27.211396 eV.

*E ^{p}*: Present work;

Table 9 compares the results for resonance energies of the doubly 2*p*3*p* ^{1,3}*D ^{e}*, 2

Table 10 lists the present resonance energies of the doubly 2*pnd *^{1,3}*F*^{0} (*n* = 3 - 5) excited states. For these levels, literature data are very scarce. Comparisons show a good agreement between the present calculations and the theoretical results of Roy *et al*., [15] and of Lipsky *et al.*, [35]. For the 2*pnf* ^{1,3}*G ^{e}* doubly excited states, no literature data were found for comparison. The SCUNC data quoted may be good reference for these doubly excited states.

In Table 11, the present SCUNC results for excitation energies of the doubly 2*sns* ^{1,3}*S ^{e}* excited states with

Table 12 shows the present excitation energies of the doubly 2*snp* ^{1,3}*P*^{0} excited states (*n* = 2 – 5) of He-like systems. Good agreement is obtained when comparing the SCUNC results to the results of Sakho [23], Roy *et al.*, [15], Koyama *et al.*, [46], Kar and Ho [34], Lipsky *et al.*, [35].

Table 13 presents the SCUNC values for excitation energies of the doubly 2*pnp* ^{1,3}*D ^{e}* (

In Table 14, we have listed our results on the calculation of the excitation energies of the doubly excited 2*pnd* ^{1,3}*F*^{0} (*n* = 2 - 5) states of He-like systems with *Z* ≤ 5. The SCUNC calculations are seen to agree well with the results of Roy *et al.*, [15], Lindroth [26], Conneely and Lipsky, Lipsky *et al.*, [35] and of Ray *et al.*, [25].

Overall, the good agreements between our present calculations and the various experimental and theoretical literature results justify the possibility to use the variationnal procedure of the Screening Constant by Unit Nuclear Charge formalism to calculate precise resonance energies of doubly (*Nlnl'*,^{2S+1}*L** ^{π}*) excited states of two electrons systems. It should be mentioned that the present results are obtained from analytical formulae without any code of calculations or a super-powerful computer. In this work, it has been demonstrated that the variational procedure of the Screening Constant by Unit Nuclear Charge (SCUNC) method can be used to compute precise resonance energies and excitation energies of doubly (

4. Conclusion

In this work, resonance energies and excitation energies of the doubly 2*sns *^{1,3}*S ^{e}*, 2

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