jamp Journal of Applied Mathematics and Physics 2327-4379 2327-4352 Scientific Research Publishing 10.4236/jamp.2026.142040 jamp-149666 Article Physics Mathematics Calculation of Resonance Energies of 1,3P0, 1,3De, 1,3F0, 1,3Ge, 1,3H0 Doubly Excited States of Helium-Like Ions Systems Associated with n = 2, 3 and n = 4 Hydrogenic Thresholds Using the Hylleraas-Type Wave Functions Dieng Matabara 1 Department of Physics, UFR-SATIC, University Alioune Diop of Bambey, Bambey, Senegal

The author declares no conflicts of interest regarding the publication of this paper.

 02 02 2026 02 2026 14 02 770 784 31 12 2025 11 02 2026 14 02 2026 © 2026 by the authors and Scientific Research Publishing Inc. 2026 This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( https://creativecommons.org/licenses/by/4.0/ ). https://doi.org/10.4236/jamp.2026.142040

In this paper, the kinetic energy, the electrons-nucleus interaction energy, the electron-electron interaction energy and the total energy of doubly excited states of He-like ions are developed in the framework of the variational method using configuration interaction basis with a real Hamiltonian. Correlated Hylleraas-type wave functions are also used in this work in which several lower-lying doubly excited (nl1nl2;l1≠l2) 1,3P0, 1,3De, 1,3F0, 1,3Ge, 1,3H0 intrashell resonances associated with n = 2, 3 and 4 thresholds up to Z = 10 are reported. The results obtained are compared with some theoretical calculations of the available literature.

 Hylleraas Method Helium-Like Ions Systems Kinetic Energy Electrons-Nucleus Interaction Electron-Electron Interaction 1. Introduction

Doubly excited states (DES) of helium-like ions have been the subject of many studies by different physicists as the first seminal photoabsorption experiment realized by Madden and Codling [1][2] and the theoretical explanation by Cooper et al. [3]. The doubly excited states of two-electron atoms are highly correlated states and they cannot be described in general like a simple model based on independent-particle quantum numbers. This experiment pushed theoreticians and experimentalists to take an interest in the study of doubly excited states and particularly to the regime near the double ionization threshold, which represents a paradigm for electronic correlations in atomic physics.

Highly DES plays an important role in the ionization by low frequency intense laser pulses [4][5], the understanding of collisional and radiational processes which take place in hot astrophysical and laboratory plasma [6][7].

Over the years, a great effort has been made toward the investigation of electronic motions for the doubly excited resonances in helium atoms including model atoms, quantum dots systems, and natural two-electron systems, such as the helium atom. It’s for this purpose that many theorists and many experimenters are interested in these resonances concerning the classification of the intrashell doubly excited states for two electron systems. These DES resonances have been investigated by Herrick and Sinanoglu [8] and Kellman and Herrick [9] and the underlying symmetries of two-electron Hamiltonian [9][10].

As far as the excited states of helium-like ions are concerned, various methods of computations have been used. Among these methods, Oza [11] used the algebraic variational and a pseudostate close-coupling method to study resonances in He+ ions below the N = 2 threshold of the ion, Bhatia and Temkin used Feshbach projection formalism together with exchange scattering nonresonant continuum to calculate the width [12]. Seminario and Sanders [13] used a Feshbach projection Z-dependant perturbation method to investigate resonances in helium-like atoms sequence with Z = 2 to 10. Calculations involving complex Hamiltonians or complex wave functions have also been used to calculate the 2s2p 1P0 and 3P0 resonances in He. Use of the complex rotation method in Z-dependant perturbation theory simultaneously yields values of the resonance position and width for the 2s2p 1P0 autoionizing states of two-electron atoms to high order by Maning and Sanders [14]. A fully numerical multiconfiguration Hartree-Fock program has been modified for performance of calculations on atomic quasibound states using complex-coordinate technique by Bentley [15]. Using the complex-coordinate method, doubly excited states of helium sequence (Z = 1 - 10) are investigated by Ho [16] and complex-coordinate rotation by the present authors [17]. Xi Wang et al. [18] have made an investigation on the doubly excited 1P0 resonance states of helium atom in quantum plasmas using correlated exponential wave functions within the framework of the stabilization method.

The present work is an extension of the earlier calculations of two-photon excitation and ionization energies of the Rydberg helium [19].

In summary, our research is expanded toward DES intrashell states (nl1nl2) where both electrons occupy the same shell, and have high and equal values of the principal quantum number n. The electronic correlation effects may be as shown by Fano [20]. We employ special forms of the Hylleraas-type wave functions constructed without Slater type orbital and make use of the variational method combined to the configuration interaction states and a real Hamiltonian.

In Section 2, we present our wave functions, the analytical expression of kinetic energy, electrons-nucleus interaction energy, electron-electron interaction energy and total energy for (nl1nl2; l1≠l2) 1,3P0, 1,3De, 1,3F0, 1,3Ge, 1,3H0 of doubly excited states of He-like ions.

In Section 3, the presentation and the discussion of our results in the case of doubly excited states (2s2p) 1,3P0, (3s3p) 1,3P0, (3s3d) 1,3De, (3p3d) 1,3F0, (4s4p) 1,3P0, (4s4d) 1,3De, (4s4f) 1,3F0, (4p4d) 1,3F0 (4p4f) 1,3Ge and (4d4f) 1,3H0 of helium-like ions up to Z= 10 are made. Rydberg units are used throughout the present work. Some of our results are compared to available theoretical values. Here also, we have no experimental data and there is not much theoretical data available, for comparisons. Finally, we end with a conclusion.

 2. Theoretical Method 2.1. Hamiltonian and Wave Functions

In our present work, the method of variational is used to calculate doubly excited intrashell resonance parameters of He-like ions. The interest of using this method is that resonance parameters can be obtained by using bound-state-type wave functions and no product of Slater-type orbitals are necessarily used. The Schrödinger equation for the relative motion of the helium-like ion, which interacts with each other by a spherically symmetric potential, can be written as:

H Φ ( r 1 , r 2 ) = E Φ ( r 1 , r 2 )

When H is the non-relativistic Hamiltonian operator (in electron-volt) describing the three-body atomic system, with the nucleus being infinitely heavy, is given by:

H = − ℏ 2 2 m ( Δ 1 + Δ 2 ) − Z e 2 r 1 − Z e 2 r 2 + e 2 | r 1 − r 2 |

The vector r 1 and r 2 denote respectively the spatial coordinates of the electrons 1 and 2 from the nucleus, m the mass of an electron, e the elementary charge and Z the nuclear charge number, Δ 1 and Δ 2 are the Laplacian operators in position representation of the radius vectors r 1 and r 2 .

r 1 and r 2 are respectively used for | r 1 | and | r 2 | .

| r 1 − r 2 | = r 1 2 + r 2 2 − 2 r 1 r 2 cos θ 12

represente the relative distance between the two electrons.

θ 12 : is the mutual angle between the position vectors of the electrons.

The Hamilton operator can be of three parts:

H = T + C + W

where T , C and W are respectively the kinetic energy operator of the two electrons, the coulomb interaction operator between the atomic nucleus and the two electrons and the coulomb interaction operator between the two electrons:

T = − ℏ 2 2 m ( Δ 1 + Δ 2 ) C = − Z e 2 r 1 − Z e 2 r 2 W = e 2 | r 1 − r 2 |

In this case of the Hamilton operator, all magnetic and relativistic effects together with the motion of the atomic nucleus are neglected.

In this article, Φ ( r 1 , r 2 ) are the trials non-orthogonal of two-electron wave functions that we have considered for the description of the intrashell singlet doubly excited states of the helium-like ions. There are special constructions of the incomplete hydrogenic wave functions and Hylleraas type wave functions as follows:

Φ j , k , m ( r 1 , r 2 ) = χ j , k , m ( r 1 , r 2 ) φ j , k , m ( r 1 , r 2 )

with

φ j , k , m ( r 1 , r 2 ) = ( r 1 + r 2 ) j ( r 1 − r 2 ) k | r 1 − r 2 | m exp [ − λ ( r 1 + r 2 ) ]

which are Hylleraas ground state wave functions of the helium-like ions [21][22], including electron correlation effects.

j: takes into account the distance of the two electrons from the nucleus;

k: takes into account the approximation of the two electrons from the nucleus;

and m: takes into account the distance between the two electrons.

j, k, m are also called Hylleraas parameters with (j, k, m ≥ 0).

λ is the nonlinear variational parameter defined by:

λ = Z α n a 0

where Z , α , a 0 and n are respectively the nucleus charge number, variational parameters, Bohr’s radius and the principal quantum number.

These wave functions φ j , k , m ( r 1 , r 2 ) are not orthogonal.

The set of parameters ( j ,   k ,   m ) define the basis states (i.e. the configurations).

The even values of k define the symmetric wave functions describing the singlet states.

The wave functions Φ j , k , m ( r 1 , r 2 ) , are incomplete hydrogenic wave functions of the Hylleraas type for the doubly excited intrashell resonances and lower-lying states and can be expressed as follows:

Φ n , l 1 , l 2 ( r 1 , r 2 ) = { ( 2 r 1 2 r 2 ) l 1 ∑ υ 1 = 0 υ 1 = n − l 1 − 1 [ ( n 2 a 0 2 λ 2 ) ( 2 r 1 2 r 2 ) ] υ 1     + ( 2 r 1 2 r 2 ) l 2 ∑ υ 2 = 0 υ 2 = n − l 2 − 1 [ ( n 2 a 0 2 λ 2 ) ( 2 r 1 2 r 2 ) ] υ 2 }     × ( r 1 + r 2 ) j ( r 1 − r 2 ) k × | r 1 − r 2 | m exp [ − λ ( r 1 + r 2 ) ] ,

with

n = ( j , k , m ) is the principal quantum number of the two electrons;

l 1 and l 2 are orbital angular momentum for the two electrons.

The interesting feature in the wave functions Φ j , k , m ( r 1 , r 2 ) , is that they contain an electron correlation term: | r 1 − r 2 | , which represents the angular part of the wave functions instead of the spherical harmonic in the other Hylleraas type wave functions.

This electron correlation term plays an important role in our trial wave functions for the description of the intrashell singlet doubly excited states. The wave functions Φ j , k , m ( r 1 , r 2 ) have also the advantage that, in the eigenvalue calculations E, the exhibition of a plateau and the convergence of the minima of the functions ( d E / d α = 0 ) arise quickly for small basis set (13 terms).

2.2. Calculation Procedures

The final form of the wave functions of the intrashell singlet doubly excited state including the correlation effects due to the mixing of configurations can be expressed as follows:

ψ n , l 1 , l 2 ( r 1 , r 2 ) = ∑ n a n Φ n , l 1 , l 2

where a n are the eigenvectors which can be determined by solving the Schrödinger equation.

H ψ n , l 1 , l 2 ( r 1 , r 2 ) = E ψ n , l 1 , l 2 ( r 1 , r 2 )

The representation of the Schrodinger equation on the non-orthogonal basis leads to the general eigenvalue equation:

∑ n ( H n , n − E N n , n ) a n = 0

with:

N n , n = N J , K , M = 〈 ψ n , l 1 , l 2 | ψ n , l 1 , l 2 〉 H n , n = H J , K , M = 〈 ψ n , l 1 , l 2 | H | ψ n , l 1 , l 2 〉 H n , n = H J , K , M = 〈 ψ n , l 1 , l 2 | T + C + W | ψ n , l 1 , l 2 〉 H n , n = 〈 ψ n , l 1 , l 2 | T | ψ n , l 1 , l 2 〉 + 〈 ψ n , l 1 , l 2 | C | ψ n , l 1 , l 2 〉 + 〈 ψ n , l 1 , l 2 | W | ψ n , l 1 , l 2 〉 T n , n = T J , K , M = 〈 ψ n , l 1 , l 2 | T | ψ n , l 1 , l 2 〉 = T C n , n = C J , K , M = 〈 ψ n , l 1 , l 2 | C | ψ n , l 1 , l 2 〉 = C W n , n = W J , K , M = 〈 ψ n , l 1 , l 2 | W | ψ n , l 1 , l 2 〉 = W H J , K , M = T J , K , M + C J , K , M + W J , K , M = E = T + C + W

wherein N J , K , M are the matrix elements of normalisation factor, H J , K , M the matrix elements of Hamilton operator, T J , K , M the matrix elements of kinetic energy operator of the two electrons, C J , K , M the matrix elements of electrons-nucleus interaction energy operator and W J , K , M the matrix elements of electron-electron interaction energy operator.

For example, we present the result of the different parameters of 2s2p 1P0 state:

Matrix elements of normalization factor:

N J , K , M = 2 π 2 M + 2 ( 1 K + 1 − 1 K + 3 − 1 K + M + 3 + 1 K + M + 5 ) ( J + K + M + 5 ) ! ( 1 2 λ ) J + K + M + 6                           + 4 π 2 M + 2 ( 1 + n 2 a 0 2 λ 2 ) ( 1 K + 1 − 2 K + 3 + 1 K + 5 − 1 K + M + 3 + 2 K + M + 5                           − 1 K + M + 7 ) ( J + K + M + 7 ) ! ( 1 2 λ ) J + K + M + 8                           + 2 π 2 M + 2 ( 1 + n 2 a 0 2 λ 2 ) 2 ( 1 K + 1 − 3 K + 3 + 3 K + 5 − 1 K + 7 − 1 K + M + 3                           + 3 K + M + 5 − 3 K + M + 7 + 1 K + M + 9 ) ( J + K + M + 9 ) ! ( 1 2 λ ) J + K + M + 10                           for   K = 0 , 2 , 4 , 6 , ⋯ N J , K , M = 0     for   K = 1 , 3 , 5 , 7 , ⋯

Matrix elements of electrons-nucleus interaction energy:

C J , K , M = 2 π 2 M + 2 ( 1 K + 1 − 1 K + M + 3 ) ( J + K + M + 4 ) ! ( 1 2 λ ) J + K + M + 5                           + 16 π 2 M + 2 ( 1 + n 2 a 0 2 λ 2 ) ( 1 K + 1 − 1 K + 3 − 1 K + M + 3 + 1 K + M + 5 )                           × ( J + K + M + 6 ) ! ( 1 2 λ ) J + K + M + 7                           + 8 π 2 M + 2 ( 1 + n 2 a 0 2 λ 2 ) 2 ( 1 K + 1 − 2 K + 3 + 1 K + 5 − 1 K + M + 3                           + 2 K + M + 5 − 1 K + M + 7 ) ( J + K + M + 8 ) ! ( 1 2 λ ) J + K + M + 9                           for     K = 0 , 2 , 4 , 6 , ⋯ C J , K , M = 0     for     K = 1 , 3 , 5 , 7 , ⋯

Matrix elements of electron-electron interaction energy:

W J , K , M = e 2 N J , K , M − 1

Matrix elements of kinetic energy:

T J , K , M = 2 ( λ 2 N J , K , M − J λ N J − 1 , K , M + j j ′ N J − 2 , K , M + k k ′ N J , K − 2 , M + m m ′ N J , K , M − 2 )     + 1 2 [ − M λ ( C J , K , M − C J , K + 2 , M − 2 ) + ( m j ′ + j m ′ ) ( C J − 1 , K , M − C J − 1 , K + 2 , M − 2 )     + ( m k ′ + k m ′ ) ( C J + 1 , K , M − 2 − C J − 1 , K , M ) ]

All the other states are calculated with this same way.

The intrashell singlet doubly excited wave functions were found in the basics containing the configurations with the following condition for the Hylleraas parameters j + k + m ≤ 3, corresponding to the basis dimension D = 13.

In order to obtain the minimum eigenvalue in which we are interested in the calculations are carried out for various values of the parameter α.

The eigenvalues E obtained in the present calculations follow the Hylleraas-Undheim theorem [23] and do not include the Feshbach shifts because of the use of the incomplete basis sets of the wave functions.

According to the Hylleraas-Undheim theorem [23], a good approximation for the eigenvalues is obtained when the minima of the functions ( d E / d α = 0 ) converge with increasing values of the dimension D and when the functions exhibit a plateau.

In our approach, for example to calculate the resonance parameters of the 2s2p 1P0 state, we fix the variational parameter α and determine each time the value of the energy E. In Table 1, we notice that E varies slowly but we clearly see that il decreases until α = 1.1 and corresponds to E = −1.410176 Ry.

Table 1. Kinetic energies T, the electron-nucleus energies C, the electron-electron energies W and the total energies E of (2s2p) 1P0 state of helium-like ions (Z = 2) depending on the variational parameter α . The results are expressed in Rydberg units: 1 Ry = 13.6056925 eV.

α 0.300000 0.400000 0.500000 0.600000 0.700000 0.800000
T 5.755883 3.395141 2.342141 1.820539 1.564662 1.455149
C −6.425211 −4.996593 −4.195378 −3.728785 −3.472319 −3.353518
W 1.063901 0.796690 0.648398 0.562257 0.514444 0.491661
E 0.394573 −0.804761 −1.204838 −1.345988 −1.393212 −1.406707
α 0.900000 1.000000 1.100000 1.200000 1.300000 1.400000
T 1.419468 1.411248 1.409646 1.409165 1.408162 1.405059
C −3.312497 −3.302572 −3.300631 −3.300172 −3.299166 −3.295751
W 0.483339 0.481167 0.480808 0.480879 0.480956 0.480839
E −1.409689 −1.410156 −1.410176 −1.410128 −1.410047 −1.409851
α 1.500000 1.600000 1.700000 1.800000 1.900000 2.000000
T 1.398312 1.386401 1.367742 1.341044 1.305881 1.262946
C −3.288149 −3.274537 −3.252820 −3.221030 −3.178098 −3.124316
W 0.480461 0.479738 0.478416 0.476102 0.472428 0.467199
E −1.409374 −1.408397 −1.406661 −1.403882 −1.399788 −1.394169
3. Results and Discussions

In Tables 2-11, we show the variation, of the kinetic energies T, the electrons-nucleus interaction energies C, the electron-electron interaction energies W and the total energies E of the present work for (2s2p) 1,3P0, (3s3p) 1,3P0, (3s3d) 1,3De, (3p3d) 1,3F0, (4s4p) 1,3P0, (4s4d) 1,3De, (4s4f) 1,3F0, (4p4d) 1,3F0 (4p4f) 1,3Ge and (4d4f) 1,3H0 states of helium-like ions with Z = 2 - 10.

Table 2. Kinetic energies T, the electron-nucleus energies C, the electron-electron energies W and the total energies E of (2s2p) 1,3P0 state of helium-like ions (Z = 2 - 10). The results are expressed in Rydberg units: 1 Ry = 13.6056925 eV.

Z 2 3 4 5 6 7 8 9 10
2s2p 1 P 0
T 1.409646 3.556518 6.698735 10.842019 15.981076 22.119677 29.257977 37.396075 46.534037
C −3.300631 −7.931851 −14.564477 −23.201567 −33.836143 −46.471498 −61.107384 −77.743657 −96.380231
W 0.480808 0.818336 1.166202 1.519452 1.875488 2.233405 2.592559 2.952567 3.313191
E −1.410176 −3.556996 −6.699540 −10.840095 −15.979578 −22.118415 −29.256847 −37.395014 −46.533002
2s2p 3 P 0
T 1.551046 3.798303 7.045029 11.291638 16.538180 22.784692 30.031200 38.277715 47.524248
C −3.495628 −8.240897 −14.986652 −23.732740 −34.479017 −47.225433 −61.971965 −78.718599 −97.465329
W 0.393482 0.644273 0.896495 1.149354 1.402548 1.655939 1.909457 2.163060 2.416724
H −1.551099 −3.798320 −7.045127 −11.291747 −16.538289 −22.784801 −30.031307 −38.277823 −47.524356

Table 3. Kinetic energies T, the electron-nucleus energies C, the electron-electron energies Wand the total energies E of (3s3p) 1,3P0 state of helium-like ions (Z = 2 - 10). The results are expressed in Rydberg units: 1 Ry = 13.6056925 eV.

Z 2 3 4 5 6 7 8 9 10
3s3p 1 P 0
T 0.721806 1.743063 3.185168 5.168438 7.509442 10.285995 13.498918 17.148572 21.235117
C −1.629587 −3.791826 −6.816640 −10.828300 −15.640876 −21.333656 −27.907474 −35.362619 −43.699167
W 0.238319 0.391043 0.546509 0.706628 0.867911 1.031050 1.195586 1.361175 1.527570
H −0.669461 −1.657720 −3.084962 −4.953233 −7.263522 −10.016610 −13.212969 −16.852870 −20.936479
3s3p 3 P 0
T 0.768403 1.803668 3.344628 5.328445 7.625119 10.415003 13.638987 17.297331 21.390157
C −1.688254 −3.863953 −6.991190 −10.998171 −15.768679 −21.472579 −28.054519 −35.514806 −43.853579
W 0.208538 0.332778 0.461361 0.588854 0.713286 0.839675 0.966275 1.093033 1.219908
H −0.711312 −1.727506 −3.185201 −5.080871 −7.430273 −10.217900 −13.449255 −17.124441 −21.243513

Table 4. Kinetic energies T, the electron-nucleus energies C, the electron-electron energies W and the total energies E of (3s3d) 1,3De state of helium-like ions (Z = 2 - 10). The results are expressed in Rydberg units: 1 Ry = 13.6056925 eV.

Z 2 3 4 5 6 7 8 9 10
3s3d 1 D e
T 0.656972 1.676123 3.136619 5.107558 7.437827 10.203242 13.405236 17.044199 21.120237
C −1.509119 −3.683365 −6.739832 −10.738866 −15.536950 −21.214396 −27.773039 −35.213267 −43.535106
W 0.213194 0.377033 0.538811 0.699589 0.861030 1.024251 1.188866 1.354536 1.521008
H −0.638952 −1.630209 −3.064401 −4.931718 −7.238091 −9.986902 −13.178936 −16.814531 −20.893860
3s3d 3 D e
T 0.699180 1.785066 3.297155 5.217635 7.566095 10.347461 13.562951 17.212907 21.297440
C −1.568557 −3.814243 −6.920876 −10.867519 −15.683227 −21.375170 −27.945079 −35.393429 −43.720363
W 0.189600 0.326302 0.456298 0.582864 0.709178 0.835670 0.962348 1.089171 1.216102
H −0.679776 −1.702875 −3.167422 −5.067019 −7.407954 −10.192038 −13.419779 −17.091349 −21.206820

Table 5. Kinetic energies T, the electron-nucleus energies C, the electron-electron energies W and the total energies E of (3p3d) 1,3F0 state of helium-like ions (Z = 2 - 10). The results are expressed in Rydberg units: 1 Ry = 13.6056925 eV.

Z 2 3 4 5 6 7 8 9 10
3p3d 1 F 0
T 0.629939 1.583288 2.979055 4.812629 7.089337 9.809586 12.973643 16.581685 20.633831
C −1.469297 −3.516557 −6.456078 −10.280301 −14.994157 −20.597486 −27.090173 −34.472128 −42.743282
W 0.207525 0.349296 0.500465 0.656654 0.816564 0.979056 1.143405 1.309130 1.475898
H −0.631832 −1.583972 −2.976557 −4.811017 −7.088255 −9.808843 −12.973123 −16.581312 −20.633551
3p3d 3 F 0
T 0.671387 1.656249 3.083758 4.958420 7.274220 10.034445 13.239097 16.888183 20.981710
C −1.529178 −3.618737 −6.599089 −10.472152 −15.231182 −20.879443 −27.416816 −34.843238 −43.158672
W 0.184954 0.306024 0.430922 0.557411 0.684538 0.812194 0.940195 1.068431 1.196837
H −0.672837 −1.656464 −3.084407 −4.956320 −7.272424 −10.032803 −13.237523 −16.886622 −20.980124

Table 6. Kinetic energies T, the electron-nucleus energies C, the electron-electron energies W and the total energies E of (4s4p) 1,3P0 state of helium-like ions (Z = 2 - 10). The results are expressed in Rydberg units: 1 Ry = 13.6056925 eV.

Z 2 3 4 5 6 7 8 9 10
4s4p 1 P 0
T 0.382227 0.946782 1.750000 2.800205 4.098185 5.644330 7.438817 9.481716 11.773049
C −0.875729 −2.065196 −3.744224 −5.921253 −8.597080 −11.771979 −15.445986 −19.619041 −24.291050
W 0.123072 0.200570 0.282260 0.367234 0.454818 0.544460 0.635731 0.728300 0.821914
H −0.370430 −0.917843 −1.711962 −2.753813 −4.044076 −5.583188 −7.371438 −9.409024 −11.696086
4s4p 3 P 0
T 0.404489 0.967613 1.773480 2.866925 4.181971 5.745029 7.556065 9.615036 11.921905
C −0.903688 −2.096307 −3.782376 −6.007075 −8.703991 −11.899234 −15.592654 −19.784134 −24.473587
W 0.112622 0.183071 0.255860 0.330398 0.405921 0.482096 0.558718 0.635659 0.712831
H −0.386576 −0.945622 −1.753035 −2.809751 −4.116099 −5.672108 −7.477870 −9.533439 −11.838849

Table 7. Kinetic energies T, the electron-nucleus energies C, the electron-electron energies W and the total energies E of (4s4d) 1,3De state of helium-like ions (Z = 2 - 10). The results are expressed in Rydberg units: 1 Ry = 13.6056925 eV.

Z 2 3 4 5 6 7 8 9 10
4s4d 1 D e
T 0.378690 0.935032 1.740104 2.790332 4.087033 5.631502 7.424205 9.465291 11.754807
C −0.866906 −2.048387 −3.730073 −5.907102 −8.581152 −11.753745 −15.425303 −19.595870 −24.265385
W 0.120289 0.198552 0.280931 0.366140 0.453767 0.543411 0.634677 0.727241 0.820851
H −0.367926 −0.914802 −1.709037 −2.750629 −4.040351 −5.578830 −7.366420 −9.403337 −11.689727
4s4d 3 D e
T 0.393815 0.956434 1.790212 2.857389 4.171413 5.733097 7.542654 9.600113 11.905458
C −0.887971 −2.080239 −3.795150 −5.993688 −8.689163 −11.882497 −15.573870 −19.763257 −24.450599
W 0.110205 0.181337 0.254793 0.329543 0.405129 0.481329 0.557965 0.634915 0.712095
H −0.383950 −0.942467 −1.750144 −2.806755 −4.112620 −5.668070 −7.473249 −9.528228 −11.833045

Table 8. Kinetic energies T, the electron-nucleus energies C, the electron-electron energies W and the total energies E of (4s4f) 1,3F0 state of helium-like ions (Z = 2 - 10). The results are expressed in Rydberg units: 1 Ry = 13.6056925 eV.

Z 2 3 4 5 6 7 8 9 10
4s4f 1 F 0
T 0.360118 0.923018 1.739458 2.780225 4.076135 5.618920 7.409806 9.449062 11.736750
C −0.833461 −2.020785 −3.721165 −5.892338 −8.565492 −11.735817 −15.404895 −19.572955 −24.239965
W 0.114007 0.193102 0.277955 0.365006 0.452738 0.542382 0.633638 0.726194 0.819800
H −0.359335 −0.904664 −1.703750 −2.747106 −4.036619 −5.574514 −7.361450 −9.397698 −11.683414
4s4f 3 F 0
T 0.375815 0.949770 1.776567 2.849558 4.162967 5.723109 7.530997 9.586784 11.890474
C −0.855111 −2.058308 −3.774791 −5.982080 −8.676859 −11.868116 −15.557213 −19.744312 −24.429384
W 0.104342 0.176538 0.253193 0.328784 0.404453 0.480653 0.557283 0.634228 0.711405
H −0.374953 −0.931999 −1.745030 −2.803737 −4.109437 −5.664353 −7.468933 −9.523299 −11.827503

Table 9. Kinetic energies T, the electron-nucleus energies C, the electron-electron energies W and the total energies E of (4p4d) 1,3F0 state of helium-like ions (Z = 2 - 10). The results are expressed in Rydberg units: 1 Ry = 13.6056925 eV.

Z 2 3 4 5 6 7 8 9 10
4p4d 1 F 0
T 0.377564 0.931412 1.731863 2.779309 4.074296 5.617245 7.408402 9.447886 11.735752
C −0.864954 −2.042848 −3.718223 −5.891501 −8.563214 −11.733712 −15.403133 −19.571486 −24.238724
W 0.119979 0.197904 0.279823 0.364929 0.452580 0.542256 0.633545 0.726125 0.819746
H −0.367410 −0.913531 −1.706535 −2.747261 −4.036337 −5.574209 −7.361185 −9.397474 −11.683225
4p4d 3 F 0
T 0.392623 0.953248 1.782548 2.847358 4.159928 5.720312 7.528542 9.584624 11.888552
C −0.885930 −2.075279 −3.784310 −5.979674 −8.673157 −11.864684 −15.554207 −19.741677 −24.427046
W 0.109914 0.180821 0.253942 0.328651 0.404275 0.480512 0.557176 0.634146 0.711341
H −0.383392 −0.941210 −1.747819 −2.803664 −4.108953 −5.663859 −7.468488 −9.522906 −11.827153

Table 10. Kinetic energiesT, the electron-nucleus energies C, the electron-electron energies W and the total energies E of (4p4f) 1,3Ge state of helium-like ions (Z = 2 - 10). The results are expressed in Rydberg units: 1 Ry = 13.6056925 eV.

Z 2 3 4 5 6 7 8 9 10
4p4f 1 G e
T 0.361420 0.917805 1.728398 2.767067 4.062856 5.604830 7.394425 9.432152 11.718194
C −0.835491 −2.014723 −3.705335 −5.873392 −8.546556 −11.715846 −15.383184 −19.549156 −24.213909
W 0.114311 0.192470 0.276560 0.363548 0.451491 0.541235 0.632533 0.725107 0.818721
E −0.359759 −0.904447 −1.700376 −2.742776 −4.032208 −5.569780 −7.356224 −9.391896 −11.676993
4p4f 3 G e
T 0.377083 0.944754 1.765231 2.836166 4.149753 5.709380 7.516280 9.570844 11.873193
C −0.857059 −2.052467 −3.758744 −5.963291 −8.658429 −11.849018 −15.536761 −19.722167 −24.405376
W 0.104606 0.175948 0.251961 0.327613 0.403489 0.479792 0.556474 0.633448 0.710644
E −0.375369 −0.931764 −1.741551 −2.799512 −4.105187 −5.659845 −7.464006 −9.517873 −11.821538

Table 11. Kinetic energies T, the electron-nucleus energies C, the electron-electron energies W and the total energies E of (4d4f) 1,3H0 state of helium-like ions (Z= 2 - 10). The results are expressed in Rydberg units: 1 Ry = 13.6056925 eV.

Z 2 3 4 5 6 7 8 9 10
4d4f 1 H 0
T 0.363752 0.913058 1.711677 2.763292 4.063333 5.611884 7.409275 9.455792 11.751644
C −0.839208 −2.008834 −3.681602 −5.858547 −8.534613 −11.709765 −15.384300 −19.558471 −24.232439
W 0.114866 0.192091 0.274470 0.359654 0.447211 0.536694 0.627752 0.720101 0.813511
E −0.360588 −0.903683 −1.695454 −2.735600 −4.024068 −5.561185 −7.347272 −9.382578 −11.667283
4d4f 3 H 0
T 0.378224 0.939796 1.749296 2.813019 4.124580 5.683574 7.490021 9.544009 11.845612
C −0.859352 −2.046876 −3.736228 −5.930864 −8.623454 −11.813381 −15.500651 −19.685368 −24.367622
W 0.104930 0.175942 0.250210 0.325565 0.401632 0.478163 0.555027 0.632136 0.709432
E −0.376197 −0.931137 −1.736720 −2.792278 −4.097241 −5.651643 −7.455602 −9.509222 −11.812576

Table 12 and Table 13 contain comparison electron-electron interaction energies W of the present work with results of Sakho et al. have used the Screening Constant by Unit Nuclear Charge (SCUNC) method [24][25], and Ivanov and Safronova, have used the method of computing double sums over the complete hydrogen spectrum [26] for (2s2p) 1P0 and (3s3p) 1P0 states of helium-like ions (Z = 2 - 10). Electron-electron correlation is essential for the process of excitation and ionization and is great when the nucleus charge number Z increase. In these cases, we did not find any experimental results for comparison.

Table 12. Comparison of the calculations of electron-electron energies W of the (2s2p) 1P0 state of helium-like ions (Z = 2 - 10) with results from other authors.

Z 2 3 4 5 6 7 8 9 10
2s2p 1 P 0
W a 0.480808 0.818336 1.166202 1.519452 1.875488 2.233405 2.592559 2.952567 3.313191
W b 0.612243 0.989291 1.367074 1.744122 2.121905 2.498953 2.876735 3.253783 3.631566
W c 0.572554 0.955482 1.338409 1.720602 2.103530 2.486458 2.869386 3.252313 3.635241

aPresent work, bI. Sakho et al. [24][25], cA. I. Ivanov, I. U. Safronova [26].

Table 13. Comparison of the calculations of electron-electron energies W of the (3s3p) 1P0 state of helium-like ions (Z = 2 - 10) with results from other authors.

Z 2 3 4 5 6 7 8 9 10
3s3p 1 P 0
W a 0.238319 0.391043 0.546509 0.706628 0.867911 1.031050 1.195586 1.361175 1.527570
W b 0.219760 0.348383 0.477006 0.605628 0.734251 0.862873 0.991496 1.120118 1.249476
W c 0.280029 0.445401 0.611508 0.777615 0.942987 1.109094 1.274466 1.440572 1.606679

aPresent work, bI. Sakho et al. [24][25], cA. I. Ivanov, I. U. Safronova [26].

In Table 12, we note a difference between our results and those of Sakho et al. and Ivanov and al. In Table 13, our results are much closer to those of Ivanov et al. than those of Sakho et al. In these two tables, the difference in behavior can be explained by the interactions created by the significant presence of the nuclear charge.

We compared the Table 14 and Table 15, to our results of the total energies E for (2s2p) 1,3P0 and (3s3p) 1,3P0 the states of helium-like ions (Z = 2 - 10) with the theoretical values of Ho. [16], Drake and Delgarno [27], Seminario and Sanders [13], Sakho et al. [24][25], and Bachau et al. [28] which have respectively used the complex-coordinate method, the energy maximization method, the Feshbach projection method, the Screening Constant by Unit Nuclear Charge (SCUNC) method, the pseudo-potential-Feshbach method (PPF) with the character of the closed-channel wave function. Comparison shows a good agreement between the present calculations and theoretical results of these authors. The disagreements noted between our results and those of the other calculations can be explained by the fact that we neglected in the present calculations the Feshbach shifts. These disagreements can also be explained by the choice of the angular part of the wave functions used for the description of the doubly excited states of the helium like-ions.

Table 14. Comparison of the calculations of energies E of the (2s2p) 1,3P0 state of helium-like ions (Z = 2 - 10) with results from other authors.

Z 2 3 4 5 6 7 8 9 10
2s2p 1 P 0
− E a 1.410176 3.556996 6.699540 10.840095 15.979578 22.118415 29.256847 37.395014 46.533002
− E b 1.38627 3.51512 6.63896 10.76042 15.88056 21.99993 29.11878 37.23735 46.35555
− E c 1.38708 3.51526 6.64030 10.76186 15.88206 22.00146 29.12034 37.23886 46.35712
− E d 1.38508 3.51402 6.63778 10.75916 15.87924 21.99854 29.11734 37.23574 46.35394
− E e 1.38875 3.51124 6.63574 10.75592 15.88274 22.00624 29.12974 37.25324 46.37670
2s2p 3 P 0
− E a 1.551099 3.798320 7.045127 11.291747 16.538289 22.784801 30.031307 38.277823 47.524356
− E a 1.52099 3.75637 6.99127 11.22598 16.46057 22.69511 29.92561 38.16408 47.39853
− E c 1.52114 3.75650 6.99140 11.22610 16.46069 22.69522 29.92972 38.16419 47.39862
− E d 1.52294 3.75877 6.99391 11.22875 16.46345 22.69805 29.93260 38.16705 47.40155
− E e 1.52765 3.75406 6.68046 11.20686 16.43326 22.36966 29.88606 38.11246 47.33886

aPresent work, bHo [16], cDrake and dalgarno [27], dSeminario and sanders [13], eSakho et al. [24][25].

Table 15. Comparison of the calculations of energies E of the (3s3p) 1,3P0 state of helium-like ions (Z = 2 - 10) with results from other authors.

Z 2 3 4 5 6 7 8 9 10
3s3p 1 P 0
− E a 0.669461 1.65772 3.084962 4.953233 7.263522 10.01661 13.212969 16.85287 20.936479
− E b 0.67125 1.65760 3.08770 4.96180 7.28050 10.04350 13.25100 16.90280 20.99900
− E c 0.67140 1.65940 3.09000 4.96600 7.28600 10.04800 13.25600 16.91000 21.00000
− E d 0.66884 1.65151 3.07812 4.94939 7.26533 10.02521 13.23048 16.87969 20.97283
3s3p 3 P 0
− E a 0.711312 1.727506 3.185201 5.080871 7.430273 10.217900 13.449255 17.124441 21.243513
− E a 0.70080 1.71010 3.16390 5.06210 7.40480 10.19190 13.42350 17.09950 21.22000
− E c 0.70360 1.71380 3.16390 5.06210 7.40480 10.19190 13.42350 17.09950 21.22000

aPresent work, bHo [16], cBachau et al. [28], dSakho et al. [24][25].

Kinetic energy is the first term of the Hamiltonian, which defines the total energy of system. The interest in calculating the kinetic energy of electrons lies the equilibrium and stability of matter, as stipulated by the virial theorem. It balances the attractive potential energy of the nucleus and prevents the electron from colliding with it. This calculation is central to the design of new materials and advanced quantum technologies such as superconductivity, semiconductors and nanotechnology.

The coulomb interaction is an essential attractive force between the positive charge of the nucleus (proton) and the negative charge of electron, representing the cohesive force of the atom. It is defined in a potential «well» where the electron is trapped and partially counteracts the attraction of the nucleus. This energy becomes negative and significant as the electron moves closer to the nucleus, which explains its strong bound. To identify chemical elements using spectroscopy, it is necessary to study the transitions of electrons between the different levels of coulomb interaction. Understanding these interactions allows us to design powerful lasers, explore imaging or radiotherapy techniques and plasma phenomena.

The coulomb interaction energy between electrons represents the repulsive electrostatic force that electrons exert on each other within an atom. This positive energy destabilizes the atom pushing electrons apart. This study is part of one of the most complex and important challenges in modern physics, especially in the design of materials at the atomic scale. The stable state of atom is the precise point where the total energy is minimal. This work provides theorists and experimentalists in the field of atomic and nuclear physics with a database for their different branches of research.

 4. Conclusion

We have presented this paper, independently, using special forms of Hylleraas-type wave functions, the kinetic energies, the electrons-nucleus interaction energies, the electron-electron interaction energies and the total energies for (2s2p) 1,3P0, (3s3p) 1,3P0, (3s3d) 1,3De, (3p3d) 1,3F0, (4s4p) 1,3P0, (4s4d) 1,3De, (4s4f) 1,3F0, (4p4d) 1,3F0 (4p4f) 1,3Ge and (4d4f) 1,3H0 resonance states for He-like ions below the n = 2, 3 and 4 hydrogenic thresholds up to Z = 10. The calculations have been done in the framework of the variation method using configuration interaction basis states with a real Hamiltonian. Our results for total energies are in good agreement with cited theoretical literatures values and other methods. For the electron-electron interaction energies, we have noted a slight disagreement between our results and those of the other calculations. In a general way, we have presented in this paper satisfactory results of some singlet doubly excited states of two-electron atoms for nl1nl2 (l1 ≠ l2). The calculation of these different parameters will allow the community working in the field of atomic physics to understand the resonance phenomena linked to three-body atomic systems. Our future project is to predict the theoretical calculations for non-relativistic theories of singlet and triplet doubly excited resonances states n1l1n2l2 (n1 ≠ n2, l1 ≠ l2).

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