Abstract

In this paper we present accurately calculated data on the resonance parameters (resonance energy and excitation energies) of the doubly excited singlet and triplet states ^1,3S^e, ^1,3P⁰, ^1,3D^e, ^1,3F⁰ and ^1,3G^e of helium and helium-like ions (Z = 2 – 10) located below the hydrogenic thresholds N = 3 – 8, using the variational method of the Screening Constant per Unit Nuclear Charge (SCUNC) formalism. These energies are calculated using special form wave functions of the Hylleraas type and a real Hamiltonian. The results obtained are in very good agreement with the experimental and theoretical values available in the literature. The results for the energies of the doubly excited ^1,3S^e, ^1,3P⁰, ^1,3D^e, ^1,3F⁰ and ^1,3G^e states associated with hydrogenic thresholds up to N = 8 listed for the first time in this paper may provide a useful guideline for future experimental and theoretical studies in the autoionization states of two-electron systems.

1. Introduction

The present paper is the third in a series of works devoted to the study of doubly excited states (DES) of two-electron atoms ^{1, 2}. Very recently, we presented precise values on the resonance parameters (resonance energies and excitation energies) of the symmetric doubly excited (nl²) ¹L^π states of helium-like ions (Z = 2 – 10) ¹ and the doubly excited singlet and triplet states ^1,3S^e, ^1,3P⁰, ^1,3D^e, ^1,3F⁰, and ^1,3G^e of the helium isoelectronic sequence located below the N = 2 hydrogenic threshold ².

The study of doubly excited states of two-electron atoms has become an important subject of theoretical study as it provides a fundamental testing ground for the accuracy of various theoretical calculations. Moreover, the precise determination of the resonance parameters of doubly excited states of different two-electron atoms is of great importance for the analysis of astrophysical data ³, for the diagnosis of lines observed in the solar corona ⁴, of high-temperature discharges ⁵ and for the diagnosis of plasmas ⁶. It should also be recalled that electronic correlation effects have played a primary role in the dynamics of doubly excited states of helium-like ions, and that studies of these states have contributed considerably to the development of various computational methods in the theory of two-electron atoms ^{7, 8, 9}. Since the first observations of these states by Madden and Codling ^{10, 11} in photoabsorption experiments on helium, the study of excited states in two-electron atomic systems remains an active area of research, both experimentally and theoretically. This is due to the multitude of theoretical methods that have been developed over the last two decades for a thorough understanding of these highly correlated states. We note, however, that most of the studies on doubly excited states of atomic systems in the literature concern the determination of the DES resonance parameters of helium-like ions associated with the N = 2, 3 and 4 hydrogenic thresholds.

There are very few results in the literature for higher values of N (N ≥ 5) and recent interests are now focused on resonances associated with higher hydrogen thresholds (N = 5, 6, 7, 8...). It is within this framework that we decided to present in this paper precise values on the resonance parameters (resonance energies and excitation energies) of the doubly excited states ^1,3S^e, ^1,3P⁰, ^1,3D^e, ^1,3F⁰, and ^1,3G^e associated with hydrogenic thresholds N = 3 – 8.

The results are obtained using the variational method of the Screening Constant per Unit Nuclear Charge (SCUNC) formalism. This approach has been successfully applied in the determination of the resonance parameters of the symmetric doubly excited nl² (l = s, p, d, f, h and g) states and the singlet and triplet states of the different valence electrons of type Nlnl' associated with the N = 2 hydrogenic threshold of helium and helium-like ions ^{1, 2}.

The goal of the present work is to extend our previous work ^{1, 2} to the study of doubly excited states of two-electron atomic systems associated with hydrogenic thresholds N = 3 – 8 in order to provide accurate values on the resonance parameters that can be used as a reference for future theoretical and experimental work on doubly excited states for high values of N.

Section 2 presents the theoretical procedure used in this work. In Section 3, a presentation and discussion of our calculations with other theoretical calculations and experimental data are also made.

2. Theory and Calculation

The description of the properties of matter at the atomic scale can be accurately described, taking into account correlations, by solving the time-independent Schrödinger equation

(1)

where Ĥ represents the Hamiltonian operator of the considered system (atom, molecule, solid), the trial wave function and E the associated energy.

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

(2)

Where r₁ and r₂ are the radial coordinates of the two electrons and r₁₂ is their relative distance.

However, it is not possible to solve such an equation rigorously, except for single-electron systems, because of the electronic correlation term One is therefore obliged to resort to approximate solutions. In order to be able to solve this equation in an approximate way several approximations and several methods of theoretical calculations using correlated wave functions had to be proposed.

The trial wave function we used in this work is obtained from the simplest helium wavefunction of Hylleraas ^{12, 13, 14} which we modified to fit the doubly excited states Nln'l' of different valence electrons of two-electron atoms (Z = 2 – 10). It is of the form ^{1, 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.

The interest of the Hylleraas type wave functions is explained by the fact that they contain an electronic correlation term which represents the angular part of the wave functions instead of the spherical harmonics as it is the case of the other wave functions used in the description of excited states. This electronic correlation term plays an important role in the test wave functions for the description of doubly excited states.

From the theoretical viewpoint, the Hylleraas variational method is based on the Hylleraas and Undheim theorem ¹⁵ according to which, a good approximation of the energy eigenvalue is obtained when the minima of the function 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 variational parameters α and C₀ can be determined by the following conditions:

(4)

and

(5)

In the framework of the Ritz’ variation principle, the energy is calculated from the relation:

(6)

In this equation, the correlated wave functions are given by (3) and the Hamiltonian H of the helium isoelectronic series in given by (2) in atomic units.

Furthermore, the closure relation represents the fact that are continuous bases in the space of the two – electron space, written as follow:

(7)

Using this relation, according to (6), we obtain:

(8)

By developping this expression (8), we find:

(9)

This means:

(10)

With the normalization constant

(11)

To make it easier to integrate equation (9), we operate the variable changes in elliptic coordonates by:

(12)

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

(13)

Using these elliptical coordinates, Eq. (10) is rewritten as follows

(14)

With respect to the correlated wave functions given by expression (3), it is expressed as follows :

(15)

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

(16)

In order to roughly resolve equation (1), we reduced the three-body interaction problem in a two-body interaction problem by introducing the concepts of internal and external interaction channels. These interaction channels are of three sorts: an internal interaction channel and two external interaction channels.

– The internal interaction channel reflects the attractive Coulomb nucleus–internal electron interaction, arbitrarily denoted “electron (1)”.

– The first external interaction channel takes account of the attractive Coulomb nucleus–external electron interaction, arbitrarily denoted “electron (2)”.

– The second external interaction channel takes account of the repulsive Coulomb internal electron (1)–external electron (2) interaction.

Using the two-body interaction model, we can then roughly resolve the Schrödinger equation (1) by combining the theory of stationary disturbances and the Ritz variation principle. This enables the screening constant by unit nuclear charge to be introduced.

Indeed, if designates the eigenwave function of the undisturbed operator, then the following is obtained according to the eigenvalue equation:

where E₀ is the eigenvalue of , given by the relation:

(17)

To determine the eigenvalue, W₀, of the Hamiltonian by the stationary disturbance theory, we write it in the form of a sum of eigenvalues obtained at various approximation orders, i.e.:

(18)

where is the first-order approximation of the disturbance energy, is the second-order approximation and so on, up to the approximation of any order, q. The eigenvalue, E (1s²; ¹S^e), of the Hamiltonian Ĥ is given by the equation:

This relation can be written using (17) and (18) in the form:

(19)

In the first-order approximation, the stationary disturbance theory gives, for the energy, the value ¹⁶:

(20)

Note that the first two terms on the right-hand side of equation (19) are proportional to Z²; therefore, intuitively, result (20) can be transformed as follows:

(21)

We can thereupon introduce a parameter denoted which we call the pthorder approximation of the disturbance coefficient. In the first-order approximation, this parameter has the following value: which then enables result (21) to be written in the form:

Equation (19) is then transformed as follows:

That is, in condensed form:

(22)

Using the concepts of interaction channels defined above, we consider expression (22) as the sum of the total energies of the hydrogen-like system {nucleus–electron (1)} and the {hydrogen-like–electron (2)} system. We can then write (22) in the form:

(23)

whereby:

– is the energy of the hydrogen-like system (in the internal interaction channel), which is given by the well-known relation:

(24)

– is the total energy in the external interaction channel, written in the ground state, by comparing (22), (23) and (24):

That is:

(25)

Introducing the effective charge, Z*, within the framework of the present formalism, we write:

(26)

Noting that the quantity:

(25) can be written as:

(27)

Written in this form, expression (27) reflects the energy of a hydrogen-like system of effective charge, Z*. Let us thus write:

(28)

In this expression, is a parameter that takes account of all of the electronic correlation effects. Its physical meaning is specified below. Considering relations (26) and (28), the following is obtained:

That is:

(29)

Result (29) gives the expression of the effective nuclear charge, Z*, at the ground state. This result can become generalized in the case of the doubly excited states designated by the label (Nlnl’; ^2S+1L^π) as follows:

(30)

By substituting result (30) into expression (27) of the total energy in the external interaction channel, we obtain:

(31)

This relation makes it possible to specify the physical meaning of the parameter by comparing it to the total energy of the {hydrogen-like ion-electron} system provided by Slater's atomic orbital theory ^{17, 18}:

(32)

The comparison of expressions (31) and (32) gives:

That is, thus:

(33)

Considering that designates the screening constant and Z denotes the nuclear charge (in elementary charge unit, e), the physical meaning of the parameter, , according to relation (33), is then clear: it is the screening constant by unit nuclear charge. From this definition stems the name of the new method to roughly calculate energies of multi-electron atomic systems: the Screening Constant by Unit Nuclear Charge (SCUNC) method ¹⁹

By substituting (24) and (27) into (23) and replacing the effective charge, Z*, by its expression (29), we establish the expression of the energy of the ground state of helium-like systems:

(34)

In this equation, the second term on the right-hand side corresponds to the firstionization energy. This makes it possible to generalize (34) to cases of autoionizing states of the type (Nl,nl’) ^2S+1L^π with n = N, N + 1, N + 2,..., i.e.:

(35)

In condensed form, we obtain, in Rydberg:

(36)

In this equation, N and n designate the principal quantum numbers of the inner and outer electrons, respectively, of helium and helium-like ions: on the basis of relation (33), the screening constant by unit nuclear charge is generally expressed in the form of a development in power of 1/Z ¹⁹, i.e.:

(37)

In this expression, the f_k parameters are screening constants determined either theoretically or empirically. The order of development, q, is linked to the accuracy of the calculations and the number of experimental values to be used to determine the f_k parameters empirically. In the general case, the value of q is set at 2 if the semi-empirical procedure is adopted.

In the framework of the SCUNC formalism, the resonance energies of the doubly excited (Nlnl') ^2S+1L^π states of two-electron atomic systems are expressed by the expression (36) above, which we recall.

where is the screening constant by unit nuclear charge expressed as a function of the variational parameter α evaluated variationaly using a wave function.

In this work, the screening constants per unit nuclear charge of the doubly excited Nsnl ^2S+1L^π (l = 0, 1, 2) and Npnl ^2S+1L^π (l = 1, 2, 3) states of helium-like ions (Z = 2 – 10) below the hydrogenic thresholds N = 3 – 8 are expressed as a function of the variational α-parameter as follows

• For Nsns ^1,3S^e, Nsnp ^1,3P⁰ and Nsnd ^1,3D^e doubly excited states

(38)

• For Npnp ^1,3D^e, Npnd ^1,3F⁰ and Ndnd ^1,3G^e doubly excited states

(39)

In these expressions, N and n denote the main quantum numbers of the inner and outer electrons respectively, L denotes the quantum state under consideration (S, P, D, F etc.), S is the total spin of the atomic system and α is the variational parameter.

Substituting expressions (38) and (39) into equation (36), the resonance energies of the doubly excited Nsns ^1,3S^e, Nsnp ^1,3P⁰, Nsnd ^1,3D^e, Npnp ^1,3D^e, Npnd ^1,3F⁰ and Ndnd ^1,3G^e (N = 3 – 8 and n = N, N + 1, N + 2...) states in the helium-like ions are then expressed as follows (in Ryd.)

• For Nsns ^1,3S^e, Nsnp ^1,3P⁰ and Nsnd ^1,3D^e doubly excited states

(40)

• For Npnp ^1,3D^e, Npnd ^1,3F⁰ and Ndnd ^1,3G^e doubly excited states

(41)

In these equations, only the parameter α is unknown. Considering the 3s4s ¹S^e level of heliumlike ions (Z = 2 – 10), we calculated the values of the variational parameters α and C₀, the results are presented in Table 1 below. The details of the calculation of these variational parameters are well explained in our previous work ^{1, 2}.

The equations (40) and (41) are used to calculate the resonance energies of the Nsns ^1,3S^e, Nsnp ^1,3P⁰, Nsnd ^1,3D^e, Npnp ^1,3D^e, Npnd ^1,3F⁰ and Ndnd ^1,3G^e (N = 3 – 8, n =N, N + 1, N + 2...) doubly excited states of heliumlike (Z = 2 – 10) ions without a complex calculation program.

3. Results and Discussions

The results of the currently calculations for the energy resonances (in a.u.) of ^1,3S^e, ^1,3P⁰, ^1,3D^e, ^1,3D^e, ^1,3F⁰ and ^1,3G^e states below the N = 3 – 8 hydrogenic thresholds are listed in Table 2 – Table 13.

Table 2 shows the comparison between current SCUNC calculations of the energy positions (– E, a.u) of doubly excited ¹S^e resonance states of helium-like ions (Z = 2 – 10) under N = 3 – 8 thresholds and other theoretical results. No experimental results are available for these states for direct comparison. Our present results are compared with the density function results of Roy et al. ²⁰, the complex coordinate results of Ho ²¹, and the theoretical data from the time variation perturbation method of Ray and Mukherjee ²². The comparison shows that the current SCUNC results are generally in very good agreement with those obtained by the above mentioned work. For 3s4s states up to Z = 5, the present results are in good agreement with the result of Roy et al. ²⁰ and Ray and Mukherjee ²². Moreover, the energy gaps almost never exceeded 0.02 a.u which proves the good accuracy of our present values.For the 3s5s levels, the comparison with the values of Ray and Mukherjee ²², the only ones available in the literature, shows a very good agreement up to Z = 5. Moreover the energy differences between our values and those of Ray and Mukherjee ²² never exceeded 0.01 a.u which allows us to consider our values as very accurate. For the 4s5s and 5s5s levels we compared our values with the results of Roy et al ²⁰ and Ray and Mukherjee ²², we also note a very satisfactory agreement with a very good accuracy. The very good agreement noted for the levels 3s4s, 3s5s, 4s5s and 5s5s, allows us to consider our results as accurate for higher values of N ≥ 5. Several new states are reported here, for example 5sns, 6sns, 7sns and 8sns with 5 ≤ n ≤ 11 for helium and its isoelectronic series.

Results for Nsns ³S^e states (4 ≤ n ≤ 12) below the N = 3 – 8 thresholds are shown in Table 3. For the 3s4s ³S^e levels our present results compared with the density-functional theory (DFT) results of Roy et al. ²⁰ and with the results of Bachau et al. ²³ who applied the Feshbach projection operator (FPO) method. The comparison shows a very good agreement. It should be mentioned that our results are closer to the accurate results of the Feshbach projection operator method of Bachau et al. ²³ and the largest difference between our values and those of these authors is 0.009 a.u which proves the good accuracy of our calculations. For the 3s5s ³S^e states, we compared our present results (SCUNC) with the multi-configuration calculations of Lipsky et al. ²⁴ and with the data of Pekeris ²⁵. Here again, we note a quite satisfactory agreement. Our values are closer to those of Lipsky et al. ²⁴ with a maximum energy difference of 0.005 a.u which proves the very good precision of our present values and allows us to consider the new values presented for the first time in the literature for the 3sns ³S^e levels with n ≥ 5 as very precise. Our results on the calculation of the resonance energies (– E, a.u) of the doubly excited 4s5s ³S^e states are with the data of Roy et al. ²⁰ and Pekeris ²⁵. For these states too, we note a very good agreement and our values are of very high accuracy. Except Z = 2, where the energy difference between our present value and that of Pekeris ²⁵ is 0.01 a.u, the large energy difference with respect to the values of Roy et al. ²⁰ and Pekeris ²⁵ is 0.007 a.u. This satisfactory agreement for the 4s5s ³S^e states allows us to consider our values for the 4sns ³S^e levels with n ≥ 5 as very accurate. Generally, we note a very good agreement between our present SCUNC results and other theoretical reference values in the literature. To the best of our knowledge, some of the states calculated here are new for example 5sns ³S^e, 6sns ³S^e, 7sns ³S^e and 8sns ³S^e with 5 ≤ n ≤ 12.

Table 4 reports the results for the Nsnp ¹P⁰ states (n = 4 – 13) for the isoelectronic series below the N =3 – 8 threshold. No experimental results are available for these states for direct comparison. For the 3s4p ¹P⁰ states, we compared our SCUNC results with the results of the semi - empirical procedure of the SCUNC method of Sakho et al. ²⁶ and with the theoretical results of the Feshbach projection operator method of Bachau et al. ²³. We note an excellent agreement up to Z = 10. It should be mentioned that the values are closer to the precise results of Bachau et al. ²². For the 3s5p ¹P⁰ levels, our results are also compared with the theoretical results of the semi-empirical procedure of the SCUNC method of Sakho et al. ²⁶, the only ones available to our knowledge. Here also we note a very satisfactory agreement. Although in some cases our energy values are overestimated and in some cases underestimated, the overall agreement with the theoretical reference values is good. To our knowledge, some of the states calculated here are presented for the first time in the literature for example Nsnp ¹P⁰ associated with the thresholds N ≥ 4. Thus we believe that these precise values listed in this table should be a good reference for future studies.

Table 5 shows the calculated resonance energies for the Nsnp ³P^o triplet states (n = 4 - 13) associated with the N = 3 – 8 hydrogenic thresholds. For the 3s4p ³P⁰ levels our present results are compared with the values of Sakho et al. ²⁶, Bachau et al. ²³ and Roy et al. ²⁰. The comparison shows an excellent agreement since the energy differences between our values and those of the other authors cited never exceeded 0.01 a.u. sometimes even less. For the 4s5p ³P⁰ states, we compared our results with the values of Roy et al. ²⁰, the only theoretical values available to our knowledge. For these states too we find a very satisfactory agreement overall and up to Z = 5, the maximum energy difference in absolute value is 0.009 a.u which sufficiently proves the accuracy of our calculations. For the Nsnp ³P⁰ states associated with the N ≥ 5 thresholds no theoretical or experimental results have been found in the literature to our knowledge. Hence the precise results listed in this table should serve as a good reference for future studies.

Results for Nsnd ¹D^e states (n = 4 – 13) below the N = 3 – 8 threshold are cited in Table 6. A comparison is made with the theoretical results of Sakho et al. ²⁶ and Bachau et al. ²³. For these states also we note a very satisfactory agreement. A multitude of new values for N ≥ 4 are presented for the first time in this table.

The results for the triplet states Nsnd ³D^e (n = 4 – 12) associated with thresholds N = 3 -8 are shown in Table 7. For the 3s4d ³D^e states, a comparison is made with the results of Bachau et al. ²³ and Roy et al. ²⁰. The agreement is globally satisfactory and the difference between the results never exceeded 0.02 a.u. Sometimes we even note a difference of less than 0.01 a.u. which justifies the high precision of our calculations. For the 3s5d ³D^e levels our SCUNC results are compared with the semi-empirical procedure values of Sakho et al. ²⁶. Here also the agreement is considered good. For the 4s5d ³D^e, 5s5d ³D^e and 5s6d ³D^e states, we compared our values with the only available values from Roy et al. ²⁰. For all these states also we note an excellent agreement with the reference values ¹⁹. Moreover, the differences between our values and those of the reference ²⁰ never reached 0.02 a.u or less. For this reason we consider our results to be very accurate. The values presented for the first time for N ≥ 5 will be, in my opinion, very useful for future studies on these states on both experimental and theoretical levels.

The doubly excited Npnp ¹D^e (n = 3 – 12) states below the N = 3 – 8 threshold are reported in Table 8. For the 3p3p ¹D^e states the comparison made with the values of the Density Functional formalism of Roy et al. ²⁰ and the values of Ho and Bhatia ²⁸ who used the complex rotation method with wave functions containing up to 1230 Hylleraas functions shows a very good agreement and moreover, the energy deviations with the quoted reference values never went beyond 0.03 a.u which testifies to the very good accuracy of our calculations. For the 3p4p ¹D^e levels, the comparison of our results with those of Ray et al. ³⁰ and Lipsky et al. ²³ indicates a satisfactory agreement up to Z = 5. However it should be mentioned that our values are closer to those of Lipsky et al. ²³ with a maximum energy difference of 0.007 a.u. Our results for the 4p4p ¹D^e states are compared with the only available values of Roy et al. ²⁰ and Ray and Mukherjee ²² here also, we note a very good agreement. For the 4p5p ¹D^e states, a comparison is made with the values of Roy et al. ²⁰ and Ray and Mukherjee ²² we also note for these states a very good agreement and the energy difference between our present calculations and that of the authors mentioned never exceeded 0.02 a.u which allows us to consider our values as very precise.For the 5p5p ¹D^e states, the comparison made with the only ones available from Roy et al. ²⁰ shows a very good agreement and the maximum deviation between our values and those of the reference which is 0.007 a.u proves sufficiently the precision of our calculations. Although we notice both an overestimation and an underestimation of the calculated positions, the overall agreement is again good, considering the simplicity of the present formalism. Moreover, the good agreement of our results with the cited reference values allows us to consider our presented values for the Npnp ¹D^e levels associated with the higher N thresholds (N = 5, 6, 7 and 8) as accurate.

Results for ³D^e states arising from Npnp configurations associated with N = 3 – 8 thresholds are shown in Table 9. A comparison is made with the results of Roy et al. ²⁰, Lipsky et al. ²³ and of Pekeris ²⁵. The overall agreement is quite satisfactory. For the Npnp ³D^e states with N ≥ 5, there are no values in the literature. These latter states are reported here for the first time.

Table 10 shows the results for the Npnd ¹F⁰ (n = 3 – 12) states of helium and its isoelectronic series below the N = 3 – 8 threshold. No experimental results are available for these states for direct comparison. Our present results are compared with the values of Roy et al. ²⁰, Bachau et al. ²³ and Sakho et al. ²⁶. For all the levels listed in this table our agreements are very satisfactory and our calculations are very accurate up to Z = 10. Note that for the Npnd ¹F⁰ states with N ≥ 5 (N = 5, 6, 7, 8...) we did not find any results in the literature hence the results presented for the first time in this table will be of great use for future studies.

In Table 11, a comparison is made with the theoretical results obtained by Roy et al. ²⁰, Bachau et al. ²³, Ivanov and Safronova ²⁷ and Sakho et al. ²⁶. For the 3p3d ³F⁰ state our results are in good agreement with those of Roy et al. ²⁰, Bachau et al. ²³ and Ivanov and Safronova ²⁷. For the 3p4d ³F⁰ state our results are in perfect agreement with the results of Roy et al. ²⁰ and Bachau et al. ²³. For the 3p5d ³F⁰ state, the comparison with the values of Roy et al. ²⁰ and Sakho et al. ²⁶. For the 4p4d ³F⁰, 4p5d ³F⁰ and 4p6d ³F⁰ states, the comparison made with the only available values of Roy et al. ²⁰ showed a very good agreement. Although we note slight discrepancies, the overall agreement with the theoretical reference values is good. To our knowledge, in this table also some of the calculated states are presented for the first time in the literature.

In Table 12 and Table 13, we present a comparison of the present SCUNC calculations of the energy positions (−E, a.u) of the doubly excited ^1,3G^e resonance states of helium-like ions (Z = 2 – 10) under the N = 3 – 8 thresholds with the results of Roy et al. ²⁰ who applied density functional theory (DFT), with data from the Feschbach projection formalism of Bachau et al. ²³, with values from the time-independent variational perturbation theory of Ray and Mukherjee ²², and with results from Pekeris ²⁵. Here again, the agreements between our calculations and those of these authors are generally good and with very good accuracy. It should be mentioned that for these states a multitude of new values never published in the literature are reported for the first time in this article.

In Table 14 – Table 19 are quoted our results for the excitation energies of the doubly excited Nsns ^1,3S^e, Nsnp ^1,3P⁰, Nsnd ^1,3D^e, Npnp ^1,3D^e, Npnd ^1,3F⁰, and Ndnd ^1,3G^e states of helium and heliumlike ions with nuclear charge Z ≤ 10 below the hydrogenic thresholds N = 3 – 8. Our excitation energies are calculated with respect to the precise ground state energies of Frankowski and Pekeris ³². Note that our excitation energies ΔE are deduced from the values of the resonance energies E (Nlnl') from which the precise values of the Frankowski and Pekeris ground state energy E (1s²) of the atomic system under consideration must be subtracted. Let us say by using the following relation:

(42)

The Frankowski and Pekeris ³² ground state energies are expressed in atomic units (a.u): – 2.90372 (He), – 7.27991 (Li⁺), – 13.65556 (Be²⁺) and – 22.03097 (B³⁺). The comparison of our present results with those of the authors quoted in these different tables shows a very good agreement and the energy differences between our present predictions and the results of other methods reported in these tables are very small as shown by the values given in the last three columns of Table 14 to Table 19. Analysis of these values clearly shows the very good accuracy of our calculations.

Overall, the very good agreements between the present SCUNC calculations of energy positions (– E, a.u) and excitation energies of the doubly excited ^1,3S^e, ^1,3P⁰, ^1,3D^e, ^1,3F⁰ and ^1,3G^e resonance states of helium-like ions (Z = 2 – 10) under the N = 3 – 8 thresholds and the various ab-initio results we have cited, sufficiently justify the validity of the present variational procedure of the screening constant per unit nuclear charge (SCUNC) formalism applied to the study of doubly excited states of two-electron atomic systems. It is worth mentioning that the calculations are directly obtained from simple analytical formulas, unlike all ab-initio methods cited in this paper.

4. Conclusion

In this paper, we have performed precise calculations on the resonance parameters (resonance energies and excitation energies) of the doubly excited states Nsns ^1,3S^e, Nsnp ^1,3P⁰, Nsnd ^1,3D^e, Npnp ^1,3D^e, Npnd ^1,3F⁰ and Ndnd ^1,3G^e two-electron atoms (Z = 2 – 10) below the N = 3 – 8 hydrogenic thresholds in the framework of the variational method of the Screening Constant per Unit Nuclear Charge (SCUNC) formalism. The results presented in this work are in very good agreement with the results of other authors in the literature. Moreover, in this paper, we have reported for the first time in the literature a multitude of new values on the resonance parameters (resonance energies and excitation energies) of the doubly excited ^1,3S^e, ^1,3P⁰, ^1,3D^e, ^1,3F⁰ and ^1,3G^e states of the two-electron systems located below the hydrogenic thresholds N = 3 – 8. It is believed that the results reported in this work are quite accurate and should serve as useful references for future theoretical and experimental work but also fill the gap in results on the resonance parameters of doubly excited states of two-electron systems for high N values.

References