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

Momar Talla Gning¹, Ibrahima Sakho^1*, Maurice Faye¹, Malick Sow², Babou Diop², Jean Kouhissoré Badiane³, Diouldé Ba³, Abdourahmane Diallo³

¹Department of Physics Chemistry, UFR Sciences and Technologies, University of Thies, Thies, Senegal.
²Department of Physics, Faculty of Sciences and Techniques, University Cheikh Anta Diop of Dakar, Dakar, Senegal.
³Department of Physics, UFR Sciences and Technologies, University Assane Seck of Ziguinchor, Ziguinchor, Senegal.

DOI: 10.4236/jmp.2021.123024                

Abstract

In this paper, resonance energies and excitation energies of doubly 2sns ^1,3S^e, 2snp ^1,3P⁰, 2pnp ^1,3D^e, 2pnd ^1,3F⁰ and 2pnf ^1,3G^e excited states of the helium isoelectronic sequence with Z ≤ 10 are calculated. Calculations are carried out in the framework of the variational procedure of the formalism of the Screening Constant per Unit Nuclear Charge (SCUNC). New correlated wave function of Hylleraas type is used. Precise resonance and excitation energies are tabulated and good agreement is obtained when a comparison is made with available literature values.

Keywords

Doubly Excited States, Helium Isoelectronic Sequence, Screening Constant per Unit Nuclear Charge (SCUNC), Correlated Wave Function, Resonance Energy, Excitation Energies

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 Rodbroet 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 ^1,3D 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 ^1,3S 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'¹L^e resonances (with N = 2, ∙∙∙, 5; n = N, ∙∙∙, 5; l = l' = 0, 1, 2 and L = l + l') for the ions from Z = 2 to Z = 5 in the helium isoelectronic sequence, complex rotation combined with discrete finite base sets to accurately describe doubly excited states [26].

Recently, Gning et al., [27] complex rotation method to determine the resonance parameters of the ((2s²) ¹S^e, (2s2p) ^1,3P⁰) and ((3s²) ¹S^e, (3s3p) ^1,3P⁰) states of helium-like ions with Z ≤ 10 via a Scilab program. The variational method of Hylleraas was used by Dieng et al., [28] to determine the resonance energies of the doubly excited states nln'l and nln'l' (with n £ 3, n' £ 4 and l = l' or l ≠ l') of helium-like ions. Very recently, the resonance parameters of the doubly excited 2sns ^1,3S^e, 2snp ^1,3P⁰, 2pnp ^1,3D^e, 3d4d ^1,3G^e states of helium and heliumlike ions are calculated by Sow et al., [29].

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+1L^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 nlnl' (n = 2 - 4) in heliumlike ions by Sakho [30] using a special Hylleraas-type wavefunction. The goal of this present work is to extend this study to doubly excited states Nlnl' (with l = l' or l ¹l') by reporting precise resonance energies and excitation energies up to n = 10. In this work, calculations are devoted to the doubly 2sns ^1,3S^e, 2snp ^1,3P⁰, 2pnp ^1,3D^e, 2pnd ^1,3F⁰ and 2pnf ^1,3G^e excited states of He-like ions up to Z = 10 via analytical expressions in contrast with ab initio method requiring computational codes to report accurate resonance data.

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.

$\hat{H}\Psi =E\Psi$ (1)

where $\hat{H}$ 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 Δ₁ is the Laplacian with reference to the coordinates of the vector radius r₁ which detect the position of the electron 1. Δ₂ Laplacian defines the coordinates of the vector radius r₂ which detect the position of the electron 2 and r₁₂ 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+1L^π). These wavefunctions are defined as follows:

$\begin{array}{c}\Psi ={\displaystyle \underset{\upsilon =0}{\overset{\upsilon =N-\mathcal{l}-1}{\sum }}{\left(N^2{r}_0^2\right)}^{\upsilon }}{\displaystyle \underset{\upsilon \text{'}=0}{\overset{\upsilon \text{'}=n-\mathcal{l}\text{'}-1}{\sum }}{\left(n^2{r}_0^2\right)}^{\upsilon \text{'}}}\left[1+{\left(-1\right)}^S{C}_{0}Z\left(r_1-r_2\right)\right]\\ \times {\left(r_1+r_2\right)}^j{\left(r_1-r_2\right)}^k{\left|r_1-r_2\right|}^m\text{e}{}^{-\alpha \left(r_1+r_2\right)}\end{array}$ (3)

In this expression, N and n are the principal quantum numbers, l and l' are orbital quantum numbers, r₀ is Bohr radius, S is the total spin of atomic system, α and C₀ are the variational parameters to be determined by minimizing the energy, Z is the nuclear charge number, r₁ and r₂ 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, C₀) is obtained when the minima of the function (d²E(α, C₀)/dαdC₀) 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 αand C₀ 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:

$\begin{array}{c}\Psi ={\displaystyle \underset{\upsilon =0}{\overset{\upsilon =N-\mathcal{l}-1}{\sum }}{\left(N^2{r}_0^2\right)}^{\upsilon }}{\displaystyle \underset{{\upsilon }^{\prime }=0}{\overset{{\upsilon }^{\prime }=n-{\mathcal{l}}^{\prime }-1}{\sum }}{\left(n^2{r}_0^2\right)}^{{\upsilon }^{\prime }}}\left[1+{\left(-1\right)}^S{C}_{0}Z\left(r_1-r_2\right)\right]\\ \times \left(r_1-r_2\right)\left|r_1-r_2\right|\text{e}{}^{-\alpha \left(r_1+r_2\right)}\end{array}$ (6)

In the framework of the Ritz’ variation principle, the energy $E\left(\alpha ,C_0\right)=\langle H\rangle \left(\alpha ,C_0\right)$ is calculated from the relation:

$E\left(\alpha ,C_0\right)=\langle H\rangle \left(\alpha \right)=\frac{\langle \Psi \left(\alpha ,C_0\right)|H|\Psi \left(\alpha ,C_0\right)\rangle }{\langle \Psi \left(\alpha ,C_0\right)|\Psi \left(\alpha ,C_0\right)\rangle }$ (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:

${\displaystyle \iint \text{d}{r}_1^3\text{d}{r}_2^3|r_1,r_2\rangle \langle r_1,r_2|=1}$ (8)

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

$\begin{array}{l}E\left(\alpha ,C_0\right){\displaystyle \iint \text{d}{r}_1^3\text{d}{r}_2^3\langle \Psi \left(\alpha ,C_0\right)||r_1,r_2\rangle \times \langle r_1,r_2||\Psi \left(\alpha ,C_0\right)\rangle }\\ ={\displaystyle \iint \text{d}{r}_1^3\text{d}{r}_2^3}\langle \Psi \left(\alpha ,C_0\right)||r_1,r_2\rangle \hat{H}\langle r_1,r_2||\Psi \left(\alpha ,C_0\right)\rangle \end{array}$ (9)

By developing this expression (9), we find:

$\begin{array}{l}E\left(\alpha ,C_0\right){\displaystyle \iint \text{d}{r}_1^3\text{d}{r}_2^3\Psi \left(\alpha ,C_0\right)\times \Psi \ast \left(\alpha ,C_0\right)}\\ ={\displaystyle \iint \text{d}{r}_1^3\text{d}{r}_2^3}\Psi \left(\alpha ,C_0\right)\hat{H}\Psi \ast \left(\alpha ,C_0\right)\end{array}$ (10)

This means:

$N\ast E\left(\alpha ,C_0\right)={\displaystyle \iint \text{d}{r}_1^3\text{d}{r}_2^3}\Psi \left(\alpha ,C_0\right)\hat{H}\Psi \ast \left(\alpha ,C_0\right)$ (11)

With the normalization constant

$N={\displaystyle \iint \text{d}{r}_1^3\text{d}{r}_2^3}{\left|\Psi \left(\alpha ,C_0\right)\right|}^2$ (12)

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

$s=r_1+r_2;t=r_1-r_2;u=r_{12}$ (13)

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

$\text{d}\tau ={\text{d}}^3{r}_1{\text{d}}^3{r}_2=2{\text{π}}^2\left(s^2-t^2\right)u\text{d}s\text{d}u\text{d}t$ (14)

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

$\begin{array}{l}NE\left(\alpha ,C_0\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}\text{d}s}{\displaystyle \underset{0}{\overset{s}{\int }}\text{d}u}{\displaystyle \underset{0}{\overset{u}{\int }}\text{d}t}\{u\left(s^2-t^2\right)\times \left[{\left(\frac{\partial \Psi }{\partial s}\right)}^2+{\left(\frac{\partial \Psi }{\partial t}\right)}^2+{\left(\frac{\partial \Psi }{\partial u}\right)}^2\right]\\ +2\left(\frac{\partial \Psi }{\partial u}\right)\times \left[s\left(u^2-t^2\right)\times \frac{\partial \Psi }{\partial s}+t\left(s^2-u^2\right)\times \frac{\partial \Psi }{\partial t}-{\Psi }^2\times \left(4Zsu-s^2+t^2\right)\right]\}\end{array}$ (15)

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

$\begin{array}{l}\Psi \left(s,t,u,\alpha ,C_0\right)\\ ={\displaystyle \underset{\upsilon =0}{\overset{\upsilon =N-\mathcal{l}-1}{\sum }}{\left(N^2{r}_0^2\right)}^{\upsilon }}{\displaystyle \underset{{\upsilon }^{\prime }=0}{\overset{{\upsilon }^{\prime }=n-{\mathcal{l}}^{\prime }-1}{\sum }}{\left(n^2{r}_0^2\right)}^{{\upsilon }^{\prime }}}\left[1+{\left(-1\right)}^S{C}_{0}Zt\right]\times tu\exp \left(-\alpha s\right)\end{array}$ (16)

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

$N={\displaystyle \underset{0}{\overset{\infty }{\int }}\text{d}s{\displaystyle \underset{0}{\overset{s}{\int }}\text{d}u}}{\displaystyle \underset{0}{\overset{u}{\int }}\text{d}t}u\left(s^2-t^2\right)\times {\Psi }^2$ (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 (2snl) ^1,3L^π and (2pnl) ^1,3L^π doubly excited states of the helium-isoelectronic up to Z = 10.

In the framework of the Screening Constant by Unit Nuclear Charge (SCUNC) formalism, resonance energies of the (Nlnl', ^2S+1L^π) doubly excited states are expressed in Rydberg (Ry) as below [22] [23] [32]

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

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

where $f_k=f_k\left(N\mathcal{l}n{\mathcal{l}}^{\prime },{}^{2S+1}{L}^{\pi }\right)$ 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 (2snl) ^1,3L^π

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

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

$\beta \left(2pn\mathcal{l},{}^{1,3}{L}^{\pi },Z,\alpha \right)=\frac{\alpha }{Z^2}\left(1+\frac{L-S}{n+S\left(S+1\right)+3}\right)$ (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 (2snl) ^1,3L^π and (2pnl) ^1,3L^π Doubly Excited States of Helium and Heliumlike Ions of Nuclear Charge Z ≤ 10

Using equations (20) and (21), the resonance energies of the doubly excited (2snl) ^1,3L^π and (2pnl) ^1,3L^π states of helium and heliumlike ions of nuclear charge Z ≤ 10 are then expressed as follows in Rydberg (Ry):

· For the doubly excited states (2snl) ^1,3L^π

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

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

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

In these equations, only the parameter α is unknown. Considering the 2p3p ¹D^e level of heliumlike ions (Z = 2 - 10), we calculated the values of the variational parameters αand C₀, the results are presented in Table 1 below.

The Equations (22) and (23) are used to calculate the resonance energies of the (2snl) ^1,3L^π and (2pnl) ^1,3L^π doubly excited states of helium and heliumlike ions of nuclear charge Z ≤ 10 without a complex calculation program.

3. Results and Discussions

The results obtained in the present study for the resonance energies and the excitation energies of the doubly excited 2sns ^1,3S^e, 2snp ^1,3P⁰, 2pnp ^1,3D^e, 2pnd ^1,3F⁰ and 2pnf ^1,3G^e states with n = 2 - 10 of the helium isoelectronic sequence (Z = 2 - 10) are presented in Tables 1-14 where a comparison between our present results and the experimental and theoretical values available in the literature is made.

Table 1 presents the values of the variational parameters α and C₀ 2 ≤ Z ≤ 10. These parameters are calculated by determining the expression of E = f(a, C₀) 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 2sns ^1,3S^e and 2snp^1,3P⁰ doubly excited states of the helium isoelectronic sequence up to Z = 10. In Table 4, Table 5, Table 6, the present results for the resonance energies of the 2pnp ^1,3D^e, 2pnd ^1,3F⁰ and 2pnf ^1,3G^e doubly excited states up to Z = 10 are presented.

In Table 7, the present resonance energies of doubly 2sns ^1,3S^e (n = 3 - 5) excited statesare compared with various calculations. The data of Sow et al., [29] [33] are obtained from variational calculations using wave fubction of Hylleraas type. Ho [18] [19] and Kar and Ho [34] applied the complex coordinate rotation (CCR) method whereas Sakho [24] used the semi-empirical procedure of the SCUNC method. Lipsky et al., [35] used truncated diagonalization method of Ray and Mukherjee [25] applied the Time-Dependent Variation Perturbation Theory (TDVPT) whereas Inanov and Safronova [36] computed double sums over the complete hydrogen spectrum (CHS) to report their data. Roy et al., [15] used the Density-Functional Theory (DFT) formalism and Macias and Riera [21] applied the discretization method. Comparison shows that the present results are generally in good agreement with all the citred literature data up to Z = 10. These agreements point out the validy of the new correlated wave functions used in the present calculations.

Table 1. Values of variational parameters α and C₀ of Helium-like ions (Z = 2 - 10).

Table 2. Energy resonances (−E) of doubly excited 2sns^1,3S^e (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 3. Energy resonances (−E) of doubly excited 2snp ^1,3P⁰ (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 2pnp ^1,3D^e (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 5. Energy resonances (−E) of doubly excited 2pnd ^1,3F⁰ (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 2pnf ^1,3G^e (n = 4 - 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 7. Comparison of the present calculations on resonance energies for the doubly 2sns ^1,3S^e (n = 3 - 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); −E^a: Sow et al. [33]; −E^b: Ho [18]; −E^c: Sakho [24]; −E^d: Lipsky et al. [35]; −E^e: Ray and Mukherjee [25]; −E^f: Kar [34]; −E^g: Roy et al. [15]; −E^h: Ho [19]; −Eⁱ: Ivanov and Safronova [36]; −E^j: Macias and Riera [21]; −E^k: Sow et al. [29].

Table 8. Comparison of the present calculations on resonance energies for the doubly 2snp ^1,3P⁰ (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); −E^a: Ho [17]; −E^b: Ivanov and Safronova [36]; −E^c: Drake and Dalgarno [2]; −E^d: Experimental data, Diehl et al. [37]; −E^e: Experimental data, Kossmann et al. [40]; −E^f: Sakho et al. [22]; −E^g: Biaye et al. [38] [39]; −E^h: Gninget al. [27]; −Eⁱ: Sow et al. [29]; −E^j: Sakho et al. [23]; −E^k: Ho [18]; −E^l: Lipsky et al. [35].

Table 9. Comparison of the present calculations on resonance energies for the doubly 2pnp ^1,3D^e (n = 3 - 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 (23); −E^a: Sow et al. [33]; −E^b: Sakho [22]; −E^c: Roy et al. [15]; −E^d: Ray and Mukherjee [25]; −E^e: Ho and Bathia [12]; −E^f: Lipsky et al. [35]; −E^g: Bhatia [41]; −E^h: Herrick and Sinanoglu [42]; −Eⁱ: Sow et al. [29].

Table 8 shows a comparison of the present SCUNC results of resonance energies of the doubly 2snp ^1,3P⁰ (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 2pnd ^1,3F⁰ (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); −E^a: Roy et al. [15]; −E^b: Lipsky et al. [35].

Table 11. Comparison of the present calculations on the variational calculation of the excitation energies of the doubly excited states 2sns ^1,3S^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; E^a: Burgers et al. [44]; E^b: Oza [33]; E^c: Koyama et al. [46]; E^d: Experimental value of Hicks and Comer [7]; E^e: Roy et al. [15]; E^f: Ray and Mukherjee [25]; E^g: Lipsky et al. [35]; E^h: Macias and Riera [21]; Eⁱ: Conneely and Lipsky [20]; E^j: Bhatia [41]; E^k: Ho [18]; E^l: Sakho [24]; E^m: Ivanov and Safronova [36].

Table 12. Comparison of the present calculations on the variational calculation of the excitation energies of the doubly excited states 2snp ^1,3P⁰ (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; E^a: Sakho [23]; E^b: Roy et al. [15]; E^c: Koyama et al. [46]; E^d: Kar and Ho [34]; E^e: Lipsky et al. [35].

Table 13. Comparison of the present calculations on the variational calculation of the excitation energies of the doubly excited states 2pnp ^1,3D^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; E^a: Roy et al. [15]; E^b: Lindroth [26]; E^c: Oza [45]; E^d: Ivanov and Safronova [36]; E^e: Ho and Bhatia [12]; E^f: Bhatia [41]; E^g: Macias et al. [48]; E^h: Sakho [24]; Eⁱ: Conneely and Lipsky [20]; E^j: Ray and Mukherjee [25]; E^k: Bhatia and Temkin [49]; E^l: Lipsky et al. [35].

Table 14. Comparison of the present calculations on the variational calculation of the excitation energies of the doubly excited states 2pnd ^1,3F⁰ (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; E^a: Roy et al. [15]; E^b: Lindroth [26]; E^c: Conneely and Lipsky [20]; E^d: Lipsky et al. [36]; E^e: Ray et al. [25].

Table 9 compares the results for resonance energies of the doubly 2p3p ^1,3D^e, 2p4p ^1,3D^e and 2p5p ¹D^e excited states with the theoretical results of Sow et al., [29] [33], Ho and Bathia [12] using the complex roration method, Sakho [24], Lipsky et al., [35], Roy et al., [15], Ray and Mukherjee [25], Bhatia [41] who employed the Feshbach projection Operator (FPO) formalism, Herrick and Sinanoglu [42]. In general comparison show satisfactory agreement between the calculations.

Table 10 lists the present resonance energies of the doubly 2pnd ^1,3F⁰ (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 2pnf ^1,3G^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 2sns ^1,3S^e excited states with n = 2 - 5 and Z = 2 - 5 of He-like systems are listed. The excitation energies are evaluated with respect to the ground state of Frankowski and Pekeris [43]. Comparisons are made with the results from the CCR results [19] and of Burgers et al., [44], the values of the variational algebraic (VA) method of Oza [45], the values of Koyama et al., [46] obtained from the the hyperspherical coordinate approaches, the experimental data of Hicks and Comer [7], the DFT results of Roy et al., [15], the TDVPT results of Ray and Mukherjee [25], the TDM data of Lipsky et al., [35] and of Macias and Riera [21], the diagonalization values of Conneely and Lipsky [20], the FPO values of Bhatia [47], the CCR calculations of Ho [18], the SCUNC method data of Sakho [24], the CHS results of Ivanov and Safronova [36]. Comparisons indicate that the current SCUNC calculations agree very well with all the literature data. This demonstrates again the validity of the present SCUNC variational procedure.

Table 12 shows the present excitation energies of the doubly 2snp ^1,3P⁰ 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 2pnp ^1,3D^e (n = 4 - 5) of helium and helium-like ions of nuclear charge Z ≤ 5. Comparison is done with the DFT data of Roy et al., [15], the CCR results of Lindroth [26], the VA calculations of Oza [45], the CHS results of Ivanov and Safronova [36], the CCR results of Ho and Bathia [12] and of Bhatia [41], the discretization calculations of Macias et al., [48], the semi-empirical SCUNC data of Sakho [24], the TDM values of Conneely and Lipsky [20] and of Lipsky et al., [35], the TDVPT data of Ray and Mukherjee [25], to the results of Bhatia and Temkin [49] and the TDM. Overall, good agreement is obtained.

In Table 14, we have listed our results on the calculation of the excitation energies of the doubly excited 2pnd ^1,3F⁰ (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+1L^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 (Nlnl',^2S+1L^π) excited states of He-like systems. The new results obtained for 2pnf ^1,3G^e doubly excited states may be benchmaked data for theorists focusing their study on the DES of He-lile systems. It should be underlined that, until this date the variational procedure of the SCUNC formalism has not been applied to atomic systems containing more than two electrons such as lithium isoelectronic sequence. The good agreement obtained in this paper open the way for applying the variational procedure of the SCUNC formalism to investigate the properties of complex atomic systems. Study is in such direction.

4. Conclusion

In this work, resonance energies and excitation energies of the doubly 2sns ^1,3S^e, 2snp ^1,3P⁰, 2pnp ^1,3D^e, 2pnd ^1,3F⁰ and 2pnf ^1,3G^e excited states of He-like systems are reported up to Z = 10. Calculations are made in the framework of the variational procedure of the Screening Constant by Unit Nuclear Charge (SCUNC) formalism. New correlated wavefunction of Hylleraas-type adapted to the correct description of electron-electron correlation phenomena in the (Nlnl',^2S+1L^π) doubly excited states of two-electron systems are constructed. Overall, good agreement is obtained with various theoretical and experimental literature data. The adequacy of the present theory in the treatment of the properties of two-electron systems demonstrates the possibilities to extend the variational procedure of the SCUNC formalism to investigate the properties of complex atomic systems beginning by the lithium isoelectronic sequence.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References

[1]	Doscheek, G.A., Meekins, P., Kerptin, R.W., Chubb, T.A. and Friedman, H. (1971) The Astrophysical Journal, 164, 165. https://doi.org/10.1086/150827
[2]	Drake, G.W.F. and Dalgarno, A. (1971) Proceedings of the Royal Society of London A, 320, 549-560. https://doi.org/10.1098/rspa.1971.0009
[3]	Madden, R.P. and Codling, K. (1963) Physical Review Letters, 10, 516. https://doi.org/10.1103/PhysRevLett.10.516
[4]	Madden, R.P. and Codling, K. (1965) Astrophysical Journal, 141, 364. https://doi.org/10.1086/148132
[5]	Hansen, J.E., et al. (1991) Zeitschrift für Physik D, 21, 587. https://doi.org/10.1007/978-3-322-84218-3_3
[6]	Oda, N., Nishimura, F. and Tashira, S. (1970) Physical Review Letters, 24, 42. https://doi.org/10.1103/PhysRevLett.24.42
[7]	Hicks, P.J. and Comer, J. (1975) Journal of Physics B: Atomic and Molecular Physics, 8, 1866. https://doi.org/10.1088/0022-3700/8/11/022
[8]	Rudd, M.E. (1965) Physical Review Letters, 15, 580. https://doi.org/10.1103/PhysRevLett.15.580
[9]	Bordenave-Montesquieu, A., Gleizes, A., Rodiere, M. and Benoit-Cattin, P. (1973) Journal of Physics B: Atomic and Molecular Physics, 6, 1997. https://doi.org/10.1088/0022-3700/6/10/016
[10]	Gelebart, F., Tweed, R.J. and Peresse, J. (1976) Journal of Physics B: Atomic and Molecular Physics, 9, 1739. https://doi.org/10.1088/0022-3700/9/10/018
[11]	Rodbro, M., Bruch, R. and Bisgaard, P. (1979) Journal of Physics B: Atomic and Molecular Physics, 12, 2413. https://doi.org/10.1088/0022-3700/12/15/009
[12]	Ho, Y.K. and Bhatia, A.K. (1991) Physical Review A, 44, 2895. https://doi.org/10.1103/PhysRevA.44.2895
[13]	Hylleraas, E.A. (1928) Zeitschrift für Physik, 48, 469-494. https://doi.org/10.1007/BF01340013
[14]	Hylleraas, E.A. (1929) Zeitschrift für Physik, 54, 347-366. https://doi.org/10.1007/BF01375457
[15]	Roy, K.A., Singh, R. and Deb, M.B. (1997) Journal of Physics B: Atomic, Molecular and Optical Physics, 30, 4763. https://doi.org/10.1088/0953-4075/30/21/014
[16]	Bachau, H., Martin, F., Riera, A. and Yanez, M. (1991) Atomic Data Nuclear Data Tables, 48, 167-212. https://doi.org/10.1016/0092-640X(91)90006-P
[17]	Ho, Y.K. (1980) Physics Letters. A, 79, 44. https://doi.org/10.1016/0375-9601(80)90313-8
[18]	Ho, Y.K. (1981) Physical Review A, 23, 2137. https://doi.org/10.1103/PhysRevA.23.2137
[19]	Ho, Y.K. (1986) Physical Review A, 34, 4402. https://doi.org/10.1103/PhysRevA.34.4402
[20]	Conneely, M.J. and Lipsky, L. (1978) Journal of Physics B: Atomic and Molecular Physics, 24, 4135. https://doi.org/10.1088/0022-3700/11/24/008
[21]	Macias, A. and Riera, A. (1986) Physics Letters A, 119, 28-32. https://doi.org/10.1016/0375-9601(86)90639-0
[22]	Sakho, I., Ndao, A.S., Biaye, M. and Wagué, A. (2008) European Physical Journal D, 47, 37-44. https://doi.org/10.1140/epjd/e2008-00018-2
[23]	Sakho, I., Ndao, A.S., Biaye, M. and Wagué, A. (2008) Physica Scripta, 77, Article ID: 055303. https://doi.org/10.1088/0031-8949/77/05/055303
[24]	Sakho, I. (2011) European Physical Journal D, 61, 267-283. https://doi.org/10.1140/epjd/e2010-10294-8
[25]	Ray, D. and Mukherjee, P.K. (1991) Journal of Physics B: Atomic, Molecular and Optical Physics, 24, 1241. https://doi.org/10.1088/0953-4075/24/6/013
[26]	Lindroth, E. (1994) Physical Review A, 49, 4473. https://doi.org/10.1103/PhysRevA.49.4473
[27]	Gning, et al. (2015) Radiation Physics and Chemistry, 106, 1-6.
[28]	Dieng, M., Biaye, M., Gning, Y. and Wagué, A. (2013) Chinese Journal of Physics, 51, 674-691. http://PSROC.phys.ntu.edu.tw/cjp
[29]	Sow, B., Diouf, A., Gning, Y., Diop, B., Sow, M. and Wague, A. (2018) International Journal of Physics, 6, 174-180.
[30]	Sakho, I. (2011) Journal of Atomic and Molecular Sciences, 2, 20-42. https://doi.org/10.4208/jams.062910.072810a
[31]	Hylleraas, E.A. and Undheim, B. (1930) Zeitschrift für Physik, 65, 759-772. https://doi.org/10.1007/BF01397263
[32]	Sakho, I. (2018) The Screening Constant by Unit Nuclear Charge Method. Description & Application to the Photoionization of Atomic Systems. ISTE Science Publishing Ltd., London and John Wiley & Sons, Inc., Hoboken. https://doi.org/10.1002/9781119476948
[33]	Sow, B., et al. (2016) American Journal of Modern Physics, 5, 146-153.
[34]	Kar, S. and Ho., Y.K. (2009) Journal of Physics B: Atomic, Molecular and Optical Physics, 42, Article ID: 044007. https://doi.org/10.1088/0953-4075/42/4/044007
[35]	Lipsky, L., Anania, R. and Conneely, M.J. (1977) Atomic Data Nuclear Data Tables, 20, 127-141. https://doi.org/10.1016/0092-640X(77)90042-0
[36]	Ivanov, A.I. and Safronova, I.U. (1993) Optics and Spectroscopy, 75, 516.
[37]	Diehl, S., Cubaynes, D., Bizau, J.-P., Wuilleumier, F.J., Kenedy, E.T., Mosnier, J.-P. and Morgan, T.J. (1999) Journal of Physics B: Atomic, Molecular and Optical Physics, 32, 4193. https://doi.org/10.1088/0953-4075/32/17/305
[38]	Biaye, M., Konté, A., Ndao, A.S., Faye, N.A.S. and Wagué, A. (2005) Physica Scripta, 71, 39. https://doi.org/10.1088/0031-8949/71/1/006
[39]	Biaye, M., Konté, A., Ndao, A.S. and Wagué, A. (2005) Physica Scripta, 72, 373. https://doi.org/10.1238/Physica.Regular.072a00373
[40]	Kossmann, H., Krassig, B. and Schmidt, V. (1988) Journal of Physics B: Atomic, Molecular and Optical Physics, 21, 1489. https://doi.org/10.1088/0953-4075/21/9/009
[41]	Bhatia, A.K. (1972) Physical Review A, 6, 120. https://doi.org/10.1103/PhysRevA.6.120
[42]	Herrick, D.R. and Sinanoglu, O. (1975) Physical Review A, 11, 97. https://doi.org/10.1103/PhysRevA.11.97
[43]	Frankowski, F. and Pekeris, L.C. (1966) Physical Review, 146, 46. https://doi.org/10.1103/PhysRev.146.46
[44]	Burgers, A., Wintgen, D. and Rest, J.M. (1995) Journal of Physics B: Atomic, Molecular and Optical Physics, 28, 3163. https://doi.org/10.1088/0953-4075/28/15/010
[45]	Oza, D.H. (1986) Physical Review A, 33, 824. https://doi.org/10.1103/PhysRevA.33.824
[46]	Koyama, N., Fukuda, H., Motoyama, T. and Matsuzawa, M. (1986) Journal of Physics B: Atomic and Molecular Physics, 19, L331. https://doi.org/10.1088/0022-3700/19/10/001
[47]	Bhatia, A.K. (1977) Physical Review A, 15, 1315. https://doi.org/10.1103/PhysRevA.15.1315
[48]	Macias, A., Martin F., Riera, A. and Yanez, M. (1987) Physical Review A, 36, 4187. https://doi.org/10.1103/PhysRevA.36.4187
[49]	Bhatia, A.K. and Temkin, A. (1975) Physical Review A, 11, 2018. https://doi.org/10.1103/PhysRevA.11.2018