JMPJournal of Modern Physics2153-1196Scientific Research Publishing10.4236/jmp.2021.123024JMP-107448ArticlesPhysics&Mathematics Variational Calculations of Energies of the (2<i>snl</i>) <sup>1,3</sup><i>L<sup>π</sup></i> and (2<i>pnl</i>) <sup>1,3 </sup><i>L<sup>π</sup></i>Doubly Excited States in Two-Electron Systems Applying the Screening Constant per Unit Nuclear Charge MomarTalla Gning1IbrahimaSakho1MauriceFaye1MalickSow2BabouDiop2JeanKouhissoré Badiane3DiouldéBa3AbdourahmaneDiallo3Department of Physics Chemistry, UFR Sciences and Technologies, University of Thies, Thies, SenegalDepartment of Physics, UFR Sciences and Technologies, University Assane Seck of Ziguinchor, Ziguinchor, SenegalDepartment of Physics, Faculty of Sciences and Techniques, University Cheikh Anta Diop of Dakar, Dakar, Senegal0702202112033283525, January 202123, February 2021 26, February 2021© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

In this paper, resonance energies and excitation energies of doubly 2 sns 1,3 S e, 2 snp 1,3 P 0, 2 pnp 1,3 D e, 2 pnd 1,3 F 0 and 2 pnf 1,3 G 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.

Doubly Excited States Helium Isoelectronic Sequence Screening Constant per Unit Nuclear Charge (SCUNC) Correlated Wave Function Resonance Energy Excitation Energies1. 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'1Le 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) 1Se, (2s2p) 1,3P0) and ((3s2) 1Se, (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 1,3Se, 2snp 1,3P0, 2pnp 1,3De, 3d4d 1,3Ge 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+1Lπ) 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 1,3Se, 2snp 1,3P0, 2pnp 1,3De, 2pnd 1,3F0 and 2pnf 1,3Ge 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. Theory2.1. 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.

H ^ Ψ = E Ψ (1)

where 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 = − 1 2 Δ 1 − 1 2 Δ 2 − Z r 1 − Z r 2 + 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 = | r 1 − r 2 | . 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+1Lπ). These wavefunctions are defined as follows:

Ψ = ∑ υ   =   0 υ   =   N −   l −   1 ( N 2 r 0 2 )   υ   ∑ υ '   =   0 υ '   =   n   −   l ' −   1 ( n 2 r 0 2 )   υ     ' [ 1 + ( − 1 )   S C 0 Z   ( r 1 − r 2 ) ]     × ( r 1 + r 2 ) j ( r 1 − r 2 ) k | r 1 − r 2 | m   e   −   α   ( r 1 + r 2 ) (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:

∂ E ( α , C 0 ) ∂ C 0 = 0 (4)

and

∂ E ( α , C 0 ) ∂ α = 0 (5)

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

Ψ = ∑ υ   =   0 υ   =   N −   l −   1 ( N 2 r 0 2 )   υ   ∑ υ ′   =   0 υ ′   =   n   −   l ′ −   1 ( n 2 r 0 2 ) υ ′ [ 1 + ( − 1 )   S C 0 Z   ( r 1 − r 2 ) ]     × ( r 1 − r 2 )   | r 1 − r 2 |   e   −   α   ( r 1 + r 2 ) (6)

In the framework of the Ritz’ variation principle, the energy E   ( α , C 0 ) = 〈 H 〉   ( α , C 0 ) is calculated from the relation:

E   ( α , C 0 ) = 〈 H 〉   ( α )   = 〈 Ψ   ( α , C 0 )   | H |   Ψ   ( α , C 0 )   〉 〈 Ψ   ( α , C 0 )   |   Ψ   ( α , C 0 )   〉 (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 〉 are continuous bases in the space of the two-electron space, written as follow:

∬ d r 1 3 d r 2 3 | r 1 , r 2 〉   〈 r 1 , r 2 | = 1 (8)

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

E ( α , C 0 ) ∬ d r 1 3 d r 2 3 〈 Ψ ( α , C 0 ) | | r 1 , r 2 〉   × 〈 r 1 , r 2 | | Ψ ( α , C 0 ) 〉 = ∬ d r 1 3 d r 2 3 〈 Ψ ( α , C 0 ) | | r 1 , r 2 〉   H ^ 〈 r 1 , r 2 | | Ψ ( α , C 0 ) 〉 (9)

By developing this expression (9), we find:

E ( α , C 0 ) ∬ d r 1 3 d r 2 3 Ψ ( α , C 0 ) × Ψ ∗ ( α , C 0 ) = ∬ d r 1 3 d r 2 3 Ψ ( α , C 0 ) H ^ Ψ ∗ ( α , C 0 ) (10)

This means:

N ∗ E ( α , C 0 ) = ∬ d r 1 3 d r 2 3 Ψ ( α , C 0 ) H ^ Ψ ∗ ( α , C 0 ) (11)

With the normalization constant

N = ∬ d r 1 3 d r 2 3 | Ψ   ( α , C 0 ) | 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

d τ = d 3 r 1 d 3 r 2 = 2 π 2   ( s 2 − t 2 )   u   d s   d u   d t   (14)

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

  N E   ( α , C 0 ) = ∫ 0 ∞ d s ∫ 0 s d u ∫ 0 u d t { u   ( s 2 − t 2 ) × [ ( ∂ Ψ ∂ s ) 2 + ( ∂ Ψ ∂ t ) 2 + ( ∂ Ψ ∂ u ) 2 ]   + 2   ( ∂ Ψ ∂ u ) × [ s   ( u 2 − t 2 ) × ∂ Ψ ∂ s + t   ( s 2 − u 2 ) × ∂ Ψ ∂ t − Ψ 2 × ( 4 Z s u − s 2 + t 2 ) ] } (15)

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

Ψ ( s , t , u , α , C 0 ) = ∑ υ   =   0 υ   =   N −   l −   1 ( N 2 r 0 2 ) υ   ∑ υ ′ =   0 υ ′ =   n   −   l ′ −   1 ( n 2 r 0 2 ) υ ′ [   1 + ( − 1 ) S C 0 Z t ] × t   u   exp   ( − α s ) (16)

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

N = ∫ 0 ∞ d s   ∫ 0 s d u   ∫ 0 u d t   u   ( s 2 − t 2 )   × Ψ 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) 1,3Lπ and (2pnl) 1,3Lπ 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+1Lπ) doubly excited states are expressed in Rydberg (Ry) as below [22] [23] [32]

E   ( N l n l ′ ,   2 S + 1 L π ) = − Z 2   ( 1 N 2 + 1 n 2   [   1 − β   ( N l n l ′ , 2 S + 1 L π , Z ) ]   2 ) R y (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

β   ( N l n l ′ ,   2 S + 1 L π , Z ) =   ∑ k   = 1 q f k   ( 1 Z ) k (19)

where f k = f k   ( N l n l ′ ,   2 S + 1 L π ) 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) 1,3Lπ

β   ( 2 s n l ,   1 , 3 L π , Z , α ) =   α Z 2   ( 1 + L − S + 1 2 n + 8 ) (20)

· For the doubly excited states (2pnl) 1,3Lπ

β   ( 2 p n l ,   1 , 3 L π , Z , α ) =   α Z 2   ( 1 + L − S n + S   ( S + 1 ) + 3 ) (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) <sup>1,3</sup>L<sup>π</sup> and (2pnl) <sup>1,3</sup>L<sup>π</sup> 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) 1,3Lπ and (2pnl) 1,3Lπ 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) 1,3Lπ

E   ( 2 s n l ,   1 , 3 L π , Z ) = − Z 2   ( 1 N 2 + 1 n 2   [   1 − α Z 2   ( 1 + L − S + 1 2 n + 8 ) ] 2 ) R y (22)

· For the doubly excited states (2pnl) 1,3Lπ

E   ( 2 p n l ,   1 , 3 L π , Z ) = − Z 2   ( 1 N 2 + 1 n 2   [   1 − α Z 2   ( 1 + L − S n + S   ( S + 1 ) + 3 ) ] 2 ) R y (23)

In these equations, only the parameter α is unknown. Considering the 2p3p 1De 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) 1,3Lπ and (2pnl) 1,3Lπ 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 1,3Se, 2snp 1,3P0, 2pnp 1,3De, 2pnd 1,3F0 and 2pnf 1,3Ge 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 1,3Se and 2snp1,3P0 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 1,3De, 2pnd 1,3F0 and 2pnf 1,3Ge doubly excited states up to Z = 10 are presented.

In Table 7, the present resonance energies of doubly 2sns 1,3Se (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.

Values of variational parameters α and C<sub>0</sub> of Helium-like ions (Z = 2 - 10)
Z2345678910
α1.29962.00742.71573.42424.13274.84145.55006.25876.9674
C0 0.44330.31060.26300.23850.22360.21350.20630.20030.1966
 Energy resonances (−E) of doubly excited 2sns<sup>1,3</sup>S<sup>e</sup> (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
2sns 1Se
Z2s3s 1Se2s4s 1Se2s5s 1Se2s6s 1Se2s7s 1Se2s8s 1Se2s9s 1Se2s10s 1Se
−E−E−E−E−E−E−E−E
20.594500.553630.534570.524140.517820.513690.510860.50882
31.414651.288791.230261.198341.179031.166471.157831.15164
42.595112.336012.215682.150142.110522.084762.067072.05440
54.136273.695503.490963.379613.312353.268633.238623.21712
66.038485.367465.056224.886854.784584.718124.672514.63985
78.301607.351816.911416.671836.527196.433226.368736.32257
810.925789.648649.056598.734588.540208.413938.327308.26528
913.9111412.2580111.4918011.0751210.8236410.6602910.5482210.46801
1017.2574915.1798114.2169713.6934213.3774713.1722613.0314812.93073
2sns 3Se
Z2s3s 3Se2s4s 3Se2s5s 3Se2s6s 3Se2s7s 3Se2s8s 3Se2s9s 3Se2s10s 3Se
−E−E−E−E−E−E−E−E
20.601340.557010.536490.525340.518620.514260.511270.50913
31.426901.294831.233701.200491.180471.167481.158571.15219
42.612872.344762.220662.153252.112612.086222.068142.05520
54.159593.706993.497503.383703.315083.270553.240023.21818
66.067375.381695.064324.891924.787974.720504.674244.64115
78.336097.368806.921086.677876.531236.436056.370806.32412
810.965879.668389.067838.741608.544908.417238.329708.26709
913.9568412.2805111.5046111.0831310.8289910.6640510.5509610.47007
1017.3087915.2050714.2313413.7024113.3834713.1764813.0345612.93304
 Energy resonances (−E) of doubly excited 2snp <sup>1,3</sup>P<sup>0</sup> (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
2snp 1P0
Z2s2p 1P02s3p 1P02s4p 1P02s5p 1P02s6p 1P02s7p 1P02s8p 1P02s9p 1P0
−E−E−E−E−E−E−E−E
20.692910.587890.550360.532690.522960.517030.513140.51045
31.740811.402651.282861.226881.196221.177621.165471.15710
43.286582.577612.327372.210762.147052.108462.083312.06601
55.331244.113213.684133.484483.375563.309643.266733.23723
67.875706.009855.353345.048184.881824.781224.715764.67078
710.919668.267387.334956.901816.665826.523186.430406.36667
814.4634610.885969.629019.045428.727598.535538.410658.32490
918.5074213.8657212.2356311.4790611.0671510.8183210.6565510.54549
1023.0510517.2064615.1546714.2026613.6844713.3714913.1680613.02842
2snp 3P0
Z2s3p 3P02s4p 3P02s5p 3P02s6p 3P02s7p 3P02s8p 3P02s9p 3P02s10p 3P0
−E−E−E−E−E−E−E−E
20.594500.553630.534570.524140.517820.513690.510860.50882
31.414651.288791.230261.198341.179031.166471.157831.15164
42.595112.336012.215682.150142.110522.084762.067072.05440
54.136273.695503.490963.379613.312353.268633.238623.21712
66.038485.367465.056224.886854.784584.718124.672514.63985
78.301607.351816.911416.671836.527196.433226.368736.32257
810.925789.648649.056598.734588.540208.413938.327308.26528
913.9111412.2580111.4918011.0751210.8236410.6602910.5482210.46801
1017.2574915.1798114.2169713.6934213.3774713.1722613.0314812.93073
 Energy resonances (−E) of doubly excited 2pnp <sup>1,3</sup>D<sup>e</sup> (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
2pnp 1De
Z2p3p 1De2p4p 1De2p5p 1De2p6p 1De2p7p 1De2p8p 1De2p9p 1De2p10p 1De
−E−E−E−E−E−E−E−E
20.571450.542420.528240.520220.515210.511880.509540.50783
31.371901.268121.218661.191171.174291.163161.155431.14985
42.532222.305682.198702.139652.103592.079942.063582.05179
54.053043.655413.468533.365783.303213.262273.234023.21369
65.934825.317585.028334.869664.773234.710234.666804.63558
78.177467.292116.878056.651276.513626.423786.361906.31747
810.781119.579099.017748.710648.524408.402958.319358.25935
913.7459512.1786211.4474611.0478110.8056210.6477610.5391610.46124
1017.0717315.0905614.1671313.6627313.3572213.1581913.0213012.92313
2pnp 3De
Z2p3p 3De2p4p 3De2p5p 3De2p6p 3De2p7p 3De2p8p 3De2p9p 3De2p10p 3De
−E−E−E−E−E−E−E−E
20.589520.551080.533060.523180.517160.513230.510510.50856
31.405631.284171.227551.196611.177851.165621.157211.15116
42.581962.329282.211742.147612.108802.083542.066162.05371
54.118953.686653.485773.376293.310093.267023.237423.21622
66.016985.356475.049784.882744.781784.716124.671034.63873
78.275917.338686.903726.666916.523846.430836.366976.32122
810.895899.633379.047658.728868.536318.411168.325248.26372
913.8770512.2405911.4816011.0686010.8192010.6571310.5458810.46623
1017.2191915.1602414.2055113.6860913.3724913.1687113.0288612.92873
 Energy resonances (−E) of doubly excited 2pnd <sup>1,3</sup>F<sup>0</sup> (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
2pnd 1F0
Z2p3d 1F02p4d 1F02p5d 1F02p6d 1F02p7d 1F02p8d 1F02p9d 1F02p10d 1F0
−E−E−E−E−E−E−E−E
20.558470.535930.524520.517870.513640.510770.508730.50722
31.346481.255621.211571.186741.171331.161091.153931.14871
42.494032.287022.188142.133092.099232.076882.061362.05012
54.001933.630523.454493.357063.297423.258233.231083.21148
65.870755.286445.010784.858784.766014.705194.663134.63284
78.100357.254686.856986.638226.504966.417746.357526.31418
810.690959.535378.993158.695418.514318.395918.314248.25552
913.6427412.1286011.4193511.0304110.7940810.6397210.5333210.45687
1016.9554415.0342414.1354813.6431413.3442413.1491413.0147412.91822
2pnd 3F°
Z2p3d 3F02p4d 3F02p5d 3F02p6d 3F02p7d 3F02p8d 3F02p9d 3F02p10d 3F0
−E−E−E−E−E−E−E−E
20.578440.545480.529810.521110.515760.512230.509780.50800
31.385131.273861.221581.192821.175291.163811.155871.15015
42.551852.314162.203002.142082.105072.080892.064222.05223
54.079133.666673.474233.368993.305163.263533.234863.21428
65.967415.331625.035434.873664.775664.711794.667854.63632
78.216567.308946.886566.656066.516536.425656.363166.31835
810.826749.598729.027668.716228.527798.405138.320828.26037
913.7981112.2010511.4587911.0541810.8094910.6502510.5408410.46241
1017.1304315.1157914.1798813.6698913.3615713.1609813.0231912.92444
 Energy resonances (−E) of doubly excited 2pnf <sup>1,3</sup>G<sup>e</sup> (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
2pnf 1Ge
Z2p4f 1Ge2p5f 1Ge2p6f 1Ge2p7f 1Ge2p8f 1Ge2p9f 1Ge2p10f 1Ge
−E−E−E−E−E−E−E
20.535930.524520.517870.513640.510770.508730.50722
31.255621.211571.186741.171331.161091.153931.14871
42.287022.188142.133092.099232.076882.061362.05012
53.630523.454493.357063.297423.258233.231083.21148
65.286445.010784.858784.766014.705194.663134.63284
77.254686.856986.638226.504966.417746.357526.31418
89.535378.993158.695418.514318.395918.314248.25552
912.1286011.4193511.0304110.7940810.6397210.5333210.45687
1015.0342414.1354813.6431413.3442413.1491413.0147412.91822
2pnf 3Ge
Z2p4f 3Ge2p5f 3Ge2p6f 3Ge2p7f 3Ge2p8f 3Ge2p9f 3Ge2p10f 3Ge
−E−E−E−E−E−E−E
20.540200.526720.519130.514420.511270.509070.50746
31.263891.215791.189141.172801.162041.154571.14916
42.299402.194442.136652.101402.078292.062302.05078
53.647053.462883.361803.300313.260093.232333.21236
65.307135.021284.864704.769614.707514.664704.63393
77.279576.869596.645326.509286.420526.359396.31549
89.564459.007878.703708.519348.399158.316438.25705
912.1618811.4361811.0398810.7998410.6434210.5358210.45862
1015.0717214.1544413.6538013.3507113.1533113.0175512.92018
 Comparison of the present calculations on resonance energies for the doubly 2sns <sup>1,3</sup>S<sup>e</sup> (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
States
2sns 1,3Se Z2345678910
2s3s 1Se−Ep 0.594501.414652.595114.136276.038488.3016010.9257813.9111417.25749
−Ea 0.599241.423382.596444.141216.053488.3262010.9597313.9457617.30205
−Eb 0.589931.415572.602054.149506.057998.3275010.9582013.9498917.30269
−Ec 0.59031 1.41479 2.600734.147936.056318.3258410.9565113.9482917.30121
−Ed 0.611641.411692.596644.14303
−Ee 0.595221.409992.585344.12279
−Ef 6.0579510.9581417.30265
−Eg0.576981.391252.567214.10436
−Eh1.415572.602054.14950
−Ek0.587431.418722.607714.141546.062298.3290210.9414313.9383717.28455
2s4s 1Se−Ep0.553631.288792.336013.695505.367467.351819.6486412.2580115.17981
−Ea 0.550441.285042.339223.690315.320847.293069.5884712.1914715.11308
−Ei 0.535291.262792.302793.655295.320307.297809.5878012.1899915.10499
−Ec 0.544491.276022.320173.676865.346097.327829.6220712.2288215.14806
−Ed 0.544021.237042.257363.67871
−Eg 0.561791.292092.335143.69129
−Ef 5.354769.6346315.16445
−Ek0.539281.274972.456223.848775.498947.335839.9004412.7784815.07605
2s5s 1Se−Ep0.534571.230262.215683.490965.056226.911419.0565911.4918014.21697
−Ee0.546821.239812.219563.48607
2s3s 3Se−Ep0.601341.426902.612874.159596.067378.3360910.9658713.9568417.30879
−Ea0.587751.423522.620044.175506.047568.3189310.9933814.0287817.42454
−Ec0.601491.438402.637284.197616.119228.4020411.0460414.0511917.41748
−Ed0.559311.439342.637924.19769
−Ei0.585591.411692.598994.147396.056798.3273910.9589913.9519917.30599
−Eg0.593551.423762.615164.16756
−Eh1.440202.639014.19894
−Ek0.611361.442962.622784.199096.144738.4185911.0339814.0654717.36901
2s4s 3Se−Ep0.557011.294832.344763.706995.381697.368809.6683812.2805115.20507
−Ea0.554851.265472.308843.694715.395027.408359.6608712.2593115.20525
−Ec0.549091.285592.334793.696695.371197.358099.6575912.2699615.19419
−Ek0.544911.279862.311083.697555.349817.355299.6305412.2917815.13992

−Ep: Present work, values calculated from Equation (22); −Ea: Sow et al. [33]; −Eb: Ho [18]; −Ec: Sakho [24]; −Ed: Lipsky et al. [35]; −Ee: Ray and Mukherjee [25]; −Ef: Kar [34]; −Eg: Roy et al. [15]; −Eh: Ho [19]; −Ei: Ivanov and Safronova [36]; −Ej: Macias and Riera [21]; −Ek: Sow et al. [29].

Comparison of the present calculations on resonance energies for the doubly 2snp <sup>1,3</sup>P<sup>0</sup> (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
States
2snp 1,3P0Z2345678910
2s2p 1P0−Ep 0.692911.740813.286585.331247.8757010.9196614.4634618.5074223.05105
−Ea0.693091.757353.319575.380107.9404210.9998014.5593418.6186723.17779
−Eb0.713671.772423.330605.389657.9481411.0067914.5652218.6238123.18293
−Ec0.693461.758093.319935.380837.9411611.0005414.5600818.6025023.17852
−Ed,e0.69383d 1.75588e
−Ef 0.693831.755153.316265.377897.9389511.0001714.5615518.6230823.18440
−Eg0.713621.774893.368395.427607.9815111.0900514.6733418.7325623.26332
−Eh0.692981.756963.318095.386087.9440310.9748414.5618918.5769123.15271
−Ei0.692261.764713.317575.374697.9360210.9899114.5617518.6229023.17804
2s3p 1P0−Ep0.587891.402652.577614.113216.009858.2673810.8859613.8657217.20646
−Eb 0.586151.396112.567314.099395.992718.2469110.8627313.8390517.17699
−Ef0.586151.404932.584954.126216.028368.2917510.9163813.9019017.24865
−Ei0.564311.359362.513074.042895.916188.1597910.7506913.7170017.04579
2s4p 1P0−Ep0.550361.282862.327373.684135.353347.334959.6290112.2356315.15467
−Eb 0.540581.267122.305653.656925.320937.297319.5860612.1871715.10103
−Ef0.548301.282552.329173.688165.359897.343989.6404512.2492815.17122
2s5p 1P0−Ep0.532691.226882.210763.484485.048186.901819.0454211.4790614.20266
−Ef0.531031.225962.210473.485305.050096.905209.0498811.4845314.20949
2s3p 3P0−Ep0.594501.414652.595114.136276.038488.3016010.9257813.9111417.25749
−Ej0.586011.404832.584774.125836.028008.2912910.9836913.9012417.24789
−Ek0.579421.393542.579314.121216.024228.2883510.9135913.8999417.24740
−Ei0.578801.395172.582194.119056.022678.2791610.9112713.8361417.21117
−El0.583351.402952.584334.12734
2s4p 3P0−Ep0.553631.288792.336013.695505.367467.351819.6486412.2580115.17981
−Ej0.543371.273262.315663.670575.3379787.3178979.61032312.2152615.13269
−El0.542201.272612.315913.64826
2s5p 3P0−Ep0.534571.230262.215683.490965.056226.911419.0565911.4918014.21697
−Ej0.524731.214222.193713.463215.022726.872239.0117511.4412714.16080

−Ep: Present work, values calculated from Equation (22); −Ea: Ho [17]; −Eb: Ivanov and Safronova [36]; −Ec: Drake and Dalgarno [2]; −Ed: Experimental data, Diehl et al. [37]; −Ee: Experimental data, Kossmann et al. [40]; −Ef: Sakho et al. [22]; −Eg: Biaye et al. [38] [39]; −Eh: Gninget al. [27]; −Ei: Sow et al. [29]; −Ej: Sakho et al. [23]; −Ek: Ho [18]; −El: Lipsky et al. [35].

Comparison of the present calculations on resonance energies for the doubly 2pnp <sup>1,3</sup>D<sup>e</sup> (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
States
2pnp 1,3De Z2345678910
2p3p 1De−Ep 0.571451.371902.532224.053045.934828.1774610.7811113.7459517.07173
−Ea 0.579401.391152.542164.031865.915648.1548610.7489413.6974617.02625
−Eb 0.56907 1.37390 2.539924.067085.955378.2047810.8153113.7869417.11970
−Ec 0.565041.365272.527494.05099
−Ed 0.583901.380202.537244.05599
−Ee 1.346172.540294.067775.871738.1011410.6913813.6425416.95466
−Ef 0.556121.344132.490023.99582
−Ei0.568591.378682.546414.062766.004188.2077110.8456913.8192317.09193
2p4p 1De−Ep 0.542421.268122.305683.655415.317587.292119.5790912.1786215.09056
−Ea 0.559381.276842.328783.640615.290867.267539.5534712.1776715.05129
−Eb 0.536451.260322.296723.645625.307037.280959.5673612.1662715.07769
−Ed 0.557301.279802.315443.66419
−Ei0.538111.265032.307313.640215.299887.288159.5900212.1470315.04906
2p5p 1De−Ep 0.528241.218662.211743.468535.028336.878059.0177411.4474614.16713
−Ed0.544821.233812.210463.47687
−Eg0.52222
−Eh0.52242
2p3p 3De−Ep 0.589521.405632.581964.118956.016988.2759110.8958913.8770517.21919
−Ea 0.580801.405432.584794.130996.035118.3019010.9252613.9119817.25727
−Eb 0.583371.403532.585414.128646.033098.2987410.9255313.9134517.26251
−Ec0.584981.407432.591174.13589
−Ee0.583781.405572.588344.132146.036998.3029310.9299613.9180817.26731
−Ef 0.583191.404012.585994.12923
−Ei0.585461.405112.588094.133786.031638.2995310.9338013.9211417.26719
2p4p 3De−Ep 0.551081.284172.329283.686655.356477.338689.6333712.2405915.16024
−Ea 0.532751.275842.308423.648925.308307.288589.5798212.1705515.10668
−Eb 0.540751.270672.313553.669115.337267.317979.6111912.2170815.13522
−Ei0.548051.280912.294563.658635.349687.320219.6149212.2195215.33671

−Ep: Present work, values calculated from Equation (23); −Ea: Sow et al. [33]; −Eb: Sakho [22]; −Ec: Roy et al. [15]; −Ed: Ray and Mukherjee [25]; −Ee: Ho and Bathia [12]; −Ef: Lipsky et al. [35]; −Eg: Bhatia [41]; −Eh: 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.

Comparison of the present calculations on resonance energies for the doubly 2pnd <sup>1,3</sup>F<sup>0</sup> (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
States
2pnd 1,3F0 Z2345
2p3d 1F0−Ep 0.558471.346482.494034.00193
−Ea 0.547641.327052.470203.97508
−Eb 0.557951.344152.48636
2p4d 1F0−Ep 0.535931.255622.287023.63052
−Ea 0.533771.252682.284213.62815
−Eb 0.532091.248002.273893.61152
2p5d 1F0−Ep 0.524521.211572.188143.45449
−Ea 0.526071.213042.190053.45691
2p3d 3F0−Ep 0.578441.385132.551854.07913
−Ea 0.569771.382432.557414.09360
−Eb0.565871.373712.54341
2p4d 3F0−Ep 0.545481.273862.314163.66667
−Ea 0.542221.270 952.312 363.66617
−Eb0.535671.260722.298193.64827
2p5d 3F0−Ep 0.529811.221582.203003.47423
−Ea 0.529541.221082.202633.47402

−Ep: Present work, values calculated from Equation (23); −Ea: Roy et al. [15]; −Eb: Lipsky et al. [35].

Comparison of the present calculations on the variational calculation of the excitation energies of the doubly excited states 2sns <sup>1,3</sup>S<sup>e</sup> (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
StatesZ2345
2s3s 1SeEp 2.30925.865311.060517.8947
Ea2.3138
Eb2.3139
Ec2.3194
Ed2.3130
Ee2.32675.888711.088417.9266
Ef2.308511.072117.9131
Eg5.8643
Eh5.864211.053317.8812
Ei5.8682
Ej5.8649
Ek11.053517.8815
2s4s 1SeEp2.35015.991111.319618.3355
Ef2.3419
El2.35926.003911.335418.3541
Em2.36846.017111.352818.3757
2s3s 3SeEp2.30245.853011.042717.8714
Ea2.3011
Ee2.31025.856211.040417.8634
Eg11.019317.8379
Eh5.839711.016617.8320
Ei2.30165.8406
El2.3022
2s4s 3SeEp2.34675.985111.310818.3239
El2.35465.994311.320818.3343
Em2.36256.006411.337318.3343

Ep: Present work; Ea: Burgers et al. [44]; Eb: Oza [33]; Ec: Koyama et al. [46]; Ed: Experimental value of Hicks and Comer [7]; Ee: Roy et al. [15]; Ef: Ray and Mukherjee [25]; Eg: Lipsky et al. [35]; Eh: Macias and Riera [21]; Ei: Conneely and Lipsky [20]; Ej: Bhatia [41]; Ek: Ho [18]; El: Sakho [24]; Em: Ivanov and Safronova [36].

Comparison of the present calculations on the variational calculation of the excitation energies of the doubly excited states 2snp <sup>1,3</sup>P<sup>0</sup> (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
StatesZ2345
2s2p 1P0Ep 2.21085.539110.368916.6997
Ea2.209710.339416.6529
Eb5.5249
Ec5.5238
2s3p 1P0Ep2.31585.877311.077917.9178
Ea2.3174 11.070717.9046
Eb5.8751
2s4p 1P0Ep2.35345.997111.328218.3468
Ea2.3553 11.326518.3427
Eb5.9975
2s5p 1P0Ep2.37106.053011.444818.5465
Ea2.3725 11.445218.5455
Eb6.0541
2s3p 3P0Ep2.30925.865311.060517.8947
Ea2.3178 5.875111.070817.9052
Ed5.873511.066317.8977
Ee11.072917.9083
2s4p 3P0Ep2.35015.991111.319618.3355
Ea2.3604 11.3399 18.3605
Ed6.0048
2s5p 3P0Ep2.36926.0497 11.439918.5400
Ea2.3788 6.065811.461918.5678
Ed6.0625

Ep: Present work; Ea: Sakho [23]; Eb: Roy et al. [15]; Ec: Koyama et al. [46]; Ed: Kar and Ho [34]; Ee: Lipsky et al. [35].

Comparison of the present calculations on the variational calculation of the excitation energies of the doubly excited states 2pnp <sup>1,3</sup>D<sup>e</sup> (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
StatesZ2345
2p3p 1DeEp 2.33235.908011.123317.9779
Ea2.338711.128117.9800
Eb2.3345
Ec2.3350
Ed2.33185.910511.1276417.9834
Ee5.905811.115317.9632
Ef5.9016
Eg5.9080
Eh5.906011.1156417.9639
Ei5.9116
Ej11.120217.9799
2p4p 1DeEp2.36136.011811.349918.3756
Ed2.36926.023411.364518.3929
Eh2.36736.019611.358818.3854
2p3p 3DeEp2.31425.874311.073617.9120
Ea2.31875.872511.064417.8951
Eb 2.3199
Ed11.079217.9142
Ee 5.874311.0672
Ef 5.8747
Eg 5.8752
Eh11.070217.9023
Ei 5.8759
Ek 2.3200
El11.071217.9064
2p4p 3DeEp2.35265.995711.326318.3443
Ed2.36416.011611.346118.3679
Eh2.36296.009211.342018.3619

Ep: Present work; Ea: Roy et al. [15]; Eb: Lindroth [26]; Ec: Oza [45]; Ed: Ivanov and Safronova [36]; Ee: Ho and Bhatia [12]; Ef: Bhatia [41]; Eg: Macias et al. [48]; Eh: Sakho [24]; Ei: Conneely and Lipsky [20]; Ej: Ray and Mukherjee [25]; Ek: Bhatia and Temkin [49]; El: Lipsky et al. [35].

Comparison of the present calculations on the variational calculation of the excitation energies of the doubly excited states 2pnd <sup>1,3</sup>F<sup>0</sup> (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
StatesZ2345
2p3d 1F0Ep 2.34535.933411.161518.0290
Ea2.35615.952911.185418.0559
Eb2.3454
Ec2.3458
Ed5.933811.169218.0433
Ee5.916111.148318.0205
2p4d 1F0Ep2.36786.024311.368518.4005
Ea2.37006.027211.371418.4028
Ec2.3716
Ee2.34896.007111.353618.3882
Ed6.031911.381718.4195
2p5d 1F0Ep2.37926.068311.467418.5765
Ea2.37776.066911.465518.5741
Ec 2.3833
Ed 6.076111.480018.5943
Ee2.36006.049511.452618.5628
2p3d 3F0Ep2.32535.894811.103717.9518
Ea2.33405.897511.098217.9374
Eb2.3375
Ec2.3379
Ed5.906711.113817.9611
2p4d 3F0Ep2.35826.006111.341418.3643
Ea2.36156.009011.343218.3648
Ec2.3681
Ed6.019711.359018.3873
2p5d 3F0Ep2.37396.058311.452618.5567
Ea2.37426.058811.452918.5570
Ec2.3815
Ed6.070111.469218.5800

Ep: Present work; Ea: Roy et al. [15]; Eb: Lindroth [26]; Ec: Conneely and Lipsky [20]; Ed: Lipsky et al. [36]; Ee: Ray et al. [25].

Table 9 compares the results for resonance energies of the doubly 2p3p 1,3De, 2p4p 1,3De and 2p5p 1De 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 1,3Ge 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 1,3Se 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 1,3De (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+1Lπ) 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+1Lπ) excited states of He-like systems. The new results obtained for 2pnf 1,3Ge 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 1,3Se, 2snp 1,3P0, 2pnp 1,3De, 2pnd 1,3F0 and 2pnf 1,3Ge 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+1Lπ) 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.

Cite this paper

Gning, M.T., Sakho, I., Faye, M., Sow, M., Diop, B., Badiane, J.K., Ba, D. and Diallo, A. (2021) Variational Calculations of Energies of the (2snl) 1,3Lπ and (2pnl) 1,3Lπ Doubly Excited States in Two-Electron Systems Applying the Screening Constant per Unit Nuclear Charge. Journal of Modern Physics, 12, 328-352. https://doi.org/10.4236/jmp.2021.123024

ReferencesBhatia, A.K. and Temkin, A. (1975) Physical Review A, 11, 2018. https://doi.org/10.1103/PhysRevA.11.2018Macias, A., Martin F., Riera, A. and Yanez, M. (1987) Physical Review A, 36, 4187. https://doi.org/10.1103/PhysRevA.36.4187Bhatia, A.K. (1977) Physical Review A, 15, 1315. https://doi.org/10.1103/PhysRevA.15.1315Koyama, N., Fukuda, H., Motoyama, T. and Matsuzawa, M. (1986) Journal of Physics B: Atomic and Molecular Physics, 19, L331. https://doi.org/10.1088/0022-3700/19/10/001Oza, D.H. (1986) Physical Review A, 33, 824. https://doi.org/10.1103/PhysRevA.33.824Burgers, A., Wintgen, D. and Rest, J.M. (1995) Journal of Physics B: Atomic, Molecular and Optical Physics, 28, 3163. https://doi.org/10.1088/0953-4075/28/15/010Frankowski, F. and Pekeris, L.C. (1966) Physical Review, 146, 46. https://doi.org/10.1103/PhysRev.146.46Herrick, D.R. and Sinanoglu, O. (1975) Physical Review A, 11, 97. https://doi.org/10.1103/PhysRevA.11.97Bhatia, A.K. (1972) Physical Review A, 6, 120. https://doi.org/10.1103/PhysRevA.6.120Kossmann, H., Krassig, B. and Schmidt, V. (1988) Journal of Physics B: Atomic, Molecular and Optical Physics, 21, 1489. https://doi.org/10.1088/0953-4075/21/9/009Biaye, M., Kont&eacute;, A., Ndao, A.S. and Wagu&eacute;, A. (2005) Physica Scripta, 72, 373. https://doi.org/10.1238/Physica.Regular.072a00373Biaye, M., Kont&eacute;, A., Ndao, A.S., Faye, N.A.S. and Wagu&eacute;, A. (2005) Physica Scripta, 71, 39. https://doi.org/10.1088/0031-8949/71/1/006Diehl, 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/10.1088/0953-4075/32/17/305Ivanov, A.I. and Safronova, I.U. (1993) Optics and Spectroscopy, 75, 516.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)90042-0Hicks, P.J. and Comer, J. (1975) Journal of Physics B: Atomic and Molecular Physics, 8, 1866. https://doi.org/10.1088/0022-3700/8/11/022Oda, N., Nishimura, F. and Tashira, S. (1970) Physical Review Letters, 24, 42. https://doi.org/10.1103/PhysRevLett.24.42Hansen, J.E., et al. (1991) Zeitschrift für Physik D, 21, 587. https://doi.org/10.1007/978-3-322-84218-3_3Madden, R.P. and Codling, K. (1965) Astrophysical Journal, 141, 364. https://doi.org/10.1086/148132Madden, R.P. and Codling, K. (1963) Physical Review Letters, 10, 516. https://doi.org/10.1103/PhysRevLett.10.516Drake, G.W.F. and Dalgarno, A. (1971) Proceedings of the Royal Society of London A, 320, 549-560. https://doi.org/10.1098/rspa.1971.0009Doscheek, G.A., Meekins, P., Kerptin, R.W., Chubb, T.A. and Friedman, H. (1971) The Astrophysical Journal, 164, 165. https://doi.org/10.1086/150827Kar, S. and Ho., Y.K. (2009) Journal of Physics B: Atomic, Molecular and Optical Physics, 42, Article ID: 044007. https://doi.org/10.1088/0953-4075/42/4/044007Sow, B., et al. (2016) American Journal of Modern Physics, 5, 146-153.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/10.1002/9781119476948Hylleraas, E.A. and Undheim, B. (1930) Zeitschrift für Physik, 65, 759-772. https://doi.org/10.1007/BF01397263Sakho, I. (2011) Journal of Atomic and Molecular Sciences, 2, 20-42. https://doi.org/10.4208/jams.062910.072810aSow, B., Diouf, A., Gning, Y., Diop, B., Sow, M. and Wague, A. (2018) International Journal of Physics, 6, 174-180.Dieng, M., Biaye, M., Gning, Y. and Wagu&eacute;, A. (2013) Chinese Journal of Physics, 51, 674-691. http://PSROC.phys.ntu.edu.tw/cjpGning, et al. (2015) Radiation Physics and Chemistry, 106, 1-6.Lindroth, E. (1994) Physical Review A, 49, 4473. https://doi.org/10.1103/PhysRevA.49.4473Ray, D. and Mukherjee, P.K. (1991) Journal of Physics B: Atomic, Molecular and Optical Physics, 24, 1241. https://doi.org/10.1088/0953-4075/24/6/013Sakho, I. (2011) European Physical Journal D, 61, 267-283. https://doi.org/10.1140/epjd/e2010-10294-8Sakho, I., Ndao, A.S., Biaye, M. and Wagu&eacute;, A. (2008) Physica Scripta, 77, Article ID: 055303. https://doi.org/10.1088/0031-8949/77/05/055303Sakho, I., Ndao, A.S., Biaye, M. and Wagu&eacute;, A. (2008) European Physical Journal D, 47, 37-44. https://doi.org/10.1140/epjd/e2008-00018-2Macias, A. and Riera, A. (1986) Physics Letters A, 119, 28-32. https://doi.org/10.1016/0375-9601(86)90639-0Conneely, M.J. and Lipsky, L. (1978) Journal of Physics B: Atomic and Molecular Physics, 24, 4135. https://doi.org/10.1088/0022-3700/11/24/008Ho, Y.K. (1986) Physical Review A, 34, 4402. https://doi.org/10.1103/PhysRevA.34.4402Ho, Y.K. (1981) Physical Review A, 23, 2137. https://doi.org/10.1103/PhysRevA.23.2137Ho, Y.K. (1980) Physics Letters. A, 79, 44. https://doi.org/10.1016/0375-9601(80)90313-8Bachau, 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-PRoy, K.A., Singh, R. and Deb, M.B. (1997) Journal of Physics B: Atomic, Molecular and Optical Physics, 30, 4763. https://doi.org/10.1088/0953-4075/30/21/014Hylleraas, E.A. (1929) Zeitschrift für Physik, 54, 347-366. https://doi.org/10.1007/BF01375457Hylleraas, E.A. (1928) Zeitschrift für Physik, 48, 469-494. https://doi.org/10.1007/BF01340013Ho, Y.K. and Bhatia, A.K. (1991) Physical Review A, 44, 2895. https://doi.org/10.1103/PhysRevA.44.2895Rodbro, M., Bruch, R. and Bisgaard, P. (1979) Journal of Physics B: Atomic and Molecular Physics, 12, 2413. https://doi.org/10.1088/0022-3700/12/15/009Gelebart, F., Tweed, R.J. and Peresse, J. (1976) Journal of Physics B: Atomic and Molecular Physics, 9, 1739. https://doi.org/10.1088/0022-3700/9/10/018Bordenave-Montesquieu, A., Gleizes, A., Rodiere, M. and Benoit-Cattin, P. (1973) Journal of Physics B: Atomic and Molecular Physics, 6, 1997. https://doi.org/10.1088/0022-3700/6/10/016Rudd, M.E. (1965) Physical Review Letters, 15, 580. https://doi.org/10.1103/PhysRevLett.15.580