Energy Positions of the 1,3De Rydberg Series of the He-Like Ions Up to the N = 9 Threshold

Matabara Dieng^{1}^{*},
Babou Diop^{2},
Malick Sow^{2},
Youssou Gning^{2},
Mamadi Biaye^{3},
Ababacar Sadikhe Ndao^{2},
Ahmadou Wagué^{2}

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1. Introduction

The double photoionization of helium has known a considerable importance because of the central role played by the electron-electron correlation in the ejection of two electrons by the absorption of a single photon. Electron-electron correlation is also an essential feature in the process of simultaneous excitation and ionization in which one electron is ejected and residual ion is left in an excited state. At present, we are witnessing a rapid and continuous development work on the study of autoionization states or resonances of atomic systems. Studies of these states resonances in atomic systems have become possible thanks to the development of new experimental methods and high resolving power through the use of synchrotron sources [1] [2] and lasers [3] in spectroscopic measurements. The study of the properties of atomic systems leads to the development of the theory of atomic processes and allows the development of approximate theoretical methods describing the excitation process. The study of these processes photoabsorption play a very important role in many branches of modern physics; it is necessary to carry out systematic calculations of the main spectroscopic characteristics to push further the development of theoretical methods for the study of these autoionizing states.

The energy positions calculated in the present paper are specified in terms of the supermultiplet classification scheme based on the
${}_{n}\left(K,T\right){}_{N}^{A}{}^{2S+1}L{}_{\pi}$ new notation of Herrick [4] as extended and reinterpreted by Lin [5] and replace the traditional Nlnl' notation used in the description of the electron independent model. The observed resonances have been also classified according to the scheme
${}_{n}{\left\{v\right\}}_{N}^{A}$ * *where
$v=1/2\left(N-K-T-1\right)$ is the vibrational quantum number [6].

In this paper, we present a novel approach to calculate the energy resonances of the ${}^{1,3}D{}_{e}$ helium-isoelectronic sequence below the N = 2 - 9 ionisation thresholds. We intend to investigate the ${}_{n}\left(K,T\right){}_{N}^{+,-}{}^{1,3}D{}_{e}$ doubly excited states energies of helium-like ions. We report total energies for several low-lying and high-lying inter-shell singlet and triplet doubly excited ${}_{n}\left(K,T\right){}_{N}^{+,-}{}^{1,3}D{}_{e}$ (N = 2 - 9, n = 3 - 10) states energies of the helium-like ions up to Z = 10. The new procedure used is to combine the variational method with the non-linear parameter of Hylleraas [7] and semi-empirical procedure of Screening Constant by Unit Nuclear Charge (SCUNC) [8].

Section 2, presents the theoretical procedure used in this work.

In Section 3 a presentation and discussion of our calculations along with other theoretical calculations and experimental observations are also made.

2. Theory

2.1. Hamiltonian and Hylleraas-Type Wave Functions

The Hamilton operator for the helium-like ions is written as:

$H=-\frac{{\hslash}^{2}}{2m}\left({\Delta}_{1}+{\Delta}_{2}\right)-\frac{Z{e}^{2}}{\left|{r}_{1}\right|}-\frac{Z{e}^{2}}{\left|{r}_{2}\right|}+\frac{{e}^{2}}{\left|{r}_{1}-{r}_{2}\right|}$ (1)

The vectors ${r}_{1}$ and ${r}_{2}$ denote the positions of the two electrons, m the mass of an electron, e the elementary charge and Z the nuclear charge number, ${\Delta}_{1}$ and ${\Delta}_{2}$ are the Laplace operators of the two electrons, $\left|{r}_{1}-{r}_{2}\right|$ represent the angular part of the wave functions instead of the spherical harmonic in the other Hylleraas type wave functions.

The Schrödinger equation can be written as:

$H{\Phi}_{Nln{l}^{\prime}}\left({r}_{1},{r}_{2}\right)=E{\Phi}_{Nln{l}^{\prime}}\left({r}_{1},{r}_{2}\right)$, (2)

${\Phi}_{Nln{l}^{\prime}}\left({r}_{1},{r}_{2}\right)$ represent, the trial non-orthogonal wave functions that we have considered for the description of the inter-shell singlet and triplet 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 [7]:

$\begin{array}{l}{\Phi}_{Nln{l}^{\prime}}\left({r}_{1},{r}_{2}\right)\\ =\langle {\left(2{r}_{1}2{r}_{2}\right)}^{l}\times {\displaystyle \underset{\upsilon =0}{\overset{\upsilon =N-l-1}{\sum}}{\left({N}^{2}{r}_{0}^{2}{\lambda}^{2}2{r}_{1}2{r}_{2}\right)}^{\upsilon}}+{\left(2{r}_{1}2{r}_{2}\right)}^{{l}^{\prime}}\times {\displaystyle \underset{{\upsilon}^{\prime}=0}{\overset{{\upsilon}^{\prime}=n-{l}^{\prime}-1}{\sum}}{\left({n}^{2}{r}_{0}^{2}{{\lambda}^{\prime}}^{2}2{r}_{1}2{r}_{2}\right)}^{{\upsilon}^{\prime}}}\rangle \\ \text{\hspace{0.17em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}\times {\left({r}_{1}+{r}_{2}\right)}^{j}{\left({r}_{1}-{r}_{2}\right)}^{k}{\left|{r}_{1}-{r}_{2}\right|}^{m}\mathrm{exp}\left(\lambda {r}_{1}+{\lambda}^{\prime}{r}_{2}\right)\end{array}$ (3)

where *j*, *k*, *m* are Hylleraas parameters with (*j*, *k*, *m *≥ 0), λ and λ' are variation parameters;

The wave functions ${\phi}_{Nln{l}^{\prime}}\left({r}_{1},{r}_{2}\right)$ 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, while the odd values define the antisymmetric wave functions for the triplet states;

N and n denote respectively the principal quantum numbers of the inner and of the outer electron, l and ${l}^{\prime}$ are orbital quantum numbers of the two electrons.

The form of the wave functions of the inter-shell singlet and triplet doubly excited state including the correlation effects to the mixing of configurations can be expressed as follows:

${\Psi}_{Nln{l}^{\prime}}\left({r}_{1},{r}_{2}\right)={\displaystyle \underset{jkm}{\sum}{a}_{jkm}{\Phi}_{Nln{l}^{\prime}}}$, (4)

where the set of Hylleraas parameters (j, k, m) defines the basis states (i.e. the configurations) and a_{jkm} are the eigenvectors which can be determined by solving the Schrödinger equation:

$H{\Psi}_{Nln{l}^{\prime}}\left({r}_{1},{r}_{2}\right)=E{\Psi}_{Nln{l}^{\prime}}\left({r}_{1},{r}_{2}\right)$, (5)

The representation of the Schrödinger equation on the non-orthogonal basis (3) leads to the general eigenvalue equation. In what follows, for the sake of brevity we shall denote the triad of Hylleraas parameter (j, k, m) by q.

$\underset{{q}^{\prime}}{\sum}\left({H}_{Nln{l}^{\prime}q{q}^{\prime}}-E{N}_{Nln{l}^{\prime}q{q}^{\prime}}\right){a}_{{q}^{\prime}}}=0$ (6)

with

${H}_{Nln{l}^{\prime}q{q}^{\prime}}=\langle {\Psi}_{Nln{l}^{\prime}q}\left({r}_{1},{r}_{2}\right)|{H}_{Nln{l}^{\prime}q{q}^{\prime}}|{\Psi}_{Nln{l}^{\prime}{q}^{\prime}}\left({r}_{1},{r}_{2}\right)\rangle $ (7)

and

${N}_{Nln{l}^{\prime}q{q}^{\prime}}=\langle {\Psi}_{Nln{l}^{\prime}q}\left({r}_{1},{r}_{2}\right)|{\Psi}_{Nln{l}^{\prime}{q}^{\prime}}\left({r}_{1},{r}_{2}\right)\rangle $ (8)

The inter-shell singlet and triplet doubly excited wave functions were found in the basis containing the configurations with the following condition for the Hylleraas parameters *j *+ *k *+ *m *≤ 3, corresponding to the basis dimension *D* = 13 or 7.

In order to obtain the minimum eigenvalue in which we are interested, the calculations are carried out for various values of the parameters λ and λ'.

The eigenvalues *E *obtained in the present calculations follow the Hylleraas-Undheim theorem [9] and do not include the Feshbach shifts because of the incomplete basis sets.

These calculations have been carried out in the framework of the variational method using interaction basis states with a real Hamiltonian.

According to the Hylleraas-Undheim theorem, a good approximation for the eigenvalues is obtained when the minima of the functions
$\left({\text{d}}^{2}\left(H\left(\lambda ,{\lambda}^{\prime}\right)\right)/\text{d}\lambda \text{d}{\lambda}^{\prime}=0\right)$ converge with increasing values of the dimension *D *and when the functions exhibit a plateau.
${\lambda}_{0}$ and
${{\lambda}^{\prime}}_{0}$ denotes the values of the λ and λ'-parameters corresponding to the minima of the function and the minimum eigenvalue.

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 energy resonances of the helium-isoelectronic sequence converging to the N = 2 - 9 hydrogenic thresholds. In the framework of the SCUNC-method, total energy of *Nlnl**'** *^{2S+1}*L** _{π}* excited states is expressed in the form (in Ry) [8] [10]:

$E\left(Nln{l}^{\prime};{}^{2S+1}L{}_{\pi};Z\right)=-{Z}^{2}\left\{\frac{1}{{N}^{2}}+\frac{1}{{n}^{2}}{\left[1-\beta \left(Nln{l}^{\prime};{}^{2S+1}L{}_{\pi};Z\right)\right]}^{2}\right\}\text{Ry}.$ (9)

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

$\beta \left(Nln{l}^{\prime};{}^{2S+1}L{}_{\pi};Z\right)={{\displaystyle \underset{k=1}{\overset{q}{\sum}}{f}_{k}\left(\frac{1}{Z}\right)}}^{k},$ (10)

where ${f}_{k}={f}_{k}\left(Nln{l}^{\prime};{}^{2S+1}L{}_{\pi}\right)$ are screening constants to be evaluated.

With the new classification scheme, Equation (9) takes the form (in Ry):

$E\left[{}_{n}\left(K,T\right){}_{N}^{A};{}^{2S+1}L{}_{\pi};Z\right]=-{Z}^{2}\left\{\frac{1}{{N}^{2}}+\frac{1}{{n}^{2}}{\left[1-\beta \left[{}_{n}\left(K,T\right){}_{N}^{A};{}^{2S+1}L{}_{\pi};Z\right]\right]}^{2}\right\},$ (11)

Furthermore, in the framework of the screening constant by unit nuclear charge formalism, the β-screening constant is expressed in terms of ${\lambda}_{0}$ and ${{\lambda}^{\prime}}_{0}$ who denotes the values of the λ and λ'-variational parameters of Hylleraas corresponding to the minima of the function and the minimum eigenvalue as follows.

$\begin{array}{l}\beta \left[{}_{n}\left(K,T\right){}_{N}^{A};{}^{2S+1}L{}_{\pi};Z,{\lambda}_{0},{{\lambda}^{\prime}}_{0}\right]\\ =\frac{1}{{Z}^{2}}\left(\frac{{N}^{2}{\lambda}_{0}+{n}^{2}{{\lambda}^{\prime}}_{0}}{{N}^{2}+{n}^{2}}\right)\left(1+\frac{L-1}{N+L+S\left(S+1\right)}+\frac{L-1}{n+L+S\left(S+1\right)}\right)\end{array}$ (12)

Using (11), total energy level of an inter-shell ${}_{n}\left(K,T\right){}_{N}^{A}{}^{2S+1}L{}_{\pi}$ doubly excited state in the helium-like ions is given by:

$E\left[{}_{n}\left(K,T\right){}_{N}^{A};{}^{2S+1}L{}_{\pi};Z\right]$

$\begin{array}{l}=-{\left(Z-\sigma \right)}^{2}\{\frac{1}{{N}^{2}}+\frac{1}{{n}^{2}}[1-\frac{1}{{Z}^{2}}\left(\frac{{N}^{2}{\lambda}_{0}+{n}^{2}{{\lambda}^{\prime}}_{0}}{{N}^{2}+{n}^{2}}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\times {\left(1+\frac{L-1}{N+L+S\left(S+1\right)}+\frac{L-1}{n+L+S\left(S+1\right)}\right)]}^{2}\},\end{array}$ (13)

where

$\sigma ={\left(48+\frac{14\left(N+K+1\right)\left(N+K-1\right)+14{T}^{2}-12L\left(L+1\right)+24}{{N}^{2}}\right)}^{-\frac{1}{2}}$ (14)

$\sigma $ is an effective screening factor or parameter. It is defined to adjust this formula, to accurately describe the resonances parameters for the inter-shell singlet doubly excited states of the helium-like ions.

We use the simple expression (13) to calculate the energy positions of the ${}^{1,3}D{}^{e}$ inter-shell autoionizing states (the two electrons occupy different shells) in helium with an easy calculation program.

3. Results and Discussion

The results of the currently calculations for inter-shell singlet and triplet doubly excited ${}_{n}\left(K,T\right){}_{N}^{+,-}{}^{1,3}D{}_{e}$ (N = 2 - 9, n = 3 - 10) states energies of the helium-like ions up to Z = 10 are listed in Tables 1(a)-(f) and Tables 2(a)-(f).

In Table 3, we have indicated the correspondence between various classification schemes for autoionizing states in two electron systems [4] [5] [6] [11].

Table 4 reports a comparison of the present doubly excited ${}_{3}\left(1,0\right){}_{2}^{+,-}{}^{1,3}D{}_{e}$ and ${}_{4}\left(1,0\right){}_{2}^{+,-}{}^{1,3}D{}_{e}$ states of helium-like ions with other theoretical results. The present results for ${}_{3}\left(1,0\right){}_{2}^{+}{}^{1}D{}_{e}$ match well with those reported by Sakho [8] [10] using a screening constant by unit nuclear charge method (SCUNC), by Ho and Bhatia [12] using a complex-coordinate rotation method (CCR) and by Ivanov and Safronova [13] using a computing double sum (CDS). For ${}_{4}\left(1,0\right){}_{2}^{+}{}^{1}D{}_{e}$ levels of the series, the present results agree well with the SCUNC results of Sakho [8] [10] and the CDS results of Ivanov and Safronova [13]. ${}_{3}\left(1,0\right){}_{2}^{-}{}^{3}D{}_{e}$ levels of up to Z = 10 are reported by Sakho [8] [10] using a SCUNC method, by Ho and Bhatia [12] using CCR and by Ivanov and Safronova [13] a CDS method and, in general, they are close to our results. For ${}_{4}\left(1,0\right){}_{2}^{-}{}^{3}D{}_{e}$ levels of the series, ours results agree well with the results of Sakho [8] [10] and those of Ivanov and Safronova [13].

In Table 5, we present a comparison of the present doubly excited ${}_{3}\left(1,0\right){}_{2}^{+}{}^{1}D{}_{e}$ and ${}_{4}\left(1,0\right){}_{2}^{+}{}^{1}D{}_{e}$ states of helium-like ions with other theoretical and experimental results. Eiglsperger et al. [14], use a spectral method (SM), Chen [15] using a saddle-point complex rotation method (SPCR), Oza [16] using a close-coupling method (CC) and Lipsky et al. [17] using a truncated diagonalization method (TDM) and above the present results. Our results is also quite close to the experimental observations of Hicks and Comer [18] using a Ejected Electron Spectroscopy (EES). The comparison of our calculated energies, with

Table 1. (a) Energy resonances of doubly excited ${}_{n}\left(K,T\right){}_{2}^{+}{}^{1}D{}_{e}$ (n = 3 - 10) states of helium-like ions (Z = 2 - 10). The results are expressed in Rydberg units: 1 Ry = 13.6056925 eV. (b) Energy resonances of doubly excited ${}_{n}\left(K,T\right){}_{3}^{+}{}^{1}D{}_{e}$ (n = 4 - 10) states of helium-like ions (Z = 2 - 10). The results are expressed in Rydberg units: 1 Ry = 13.6056925 eV. (c) Energy resonances of doubly excited ${}_{n}\left(K,T\right){}_{4}^{+}{}^{1}D{}_{e}$ (n = 3 - 10) states of helium-like ions (Z = 2 - 10). The results are expressed in Rydberg units: 1 Ry = 13.6056925 eV. (d) Energy resonances of doubly excited ${}_{n}\left(K,T\right){}_{5}^{+}{}^{1}D{}_{e}$ (n = 6 - 10) states of helium-like ions (Z = 2 - 10). The results are expressed in Rydberg units: 1 Ry = 13.6056925 eV. (e) Energy resonances of doubly excited ${}_{n}\left(K,T\right){}_{6}^{+}{}^{1}D{}_{e}$ (n = 7 - 10) states of helium-like ions (Z = 2 - 10). The results are expressed in Rydberg units: 1 Ry = 13.6056925 eV. (f) Energy resonances of doubly excited ${}_{n}\left(K,T\right){}_{7}^{+}{}^{1}D{}_{e}$ (n = 8 - 10), ${}_{n}\left(K,T\right){}_{8}^{+}{}^{1}D{}_{e}$ (n = 9, 10) and ${}_{10}\left(K,T\right){}_{9}^{+}{}^{1}D{}_{e}$ states of helium-like ions (Z = 2 - 10). The results are expressed in Rydberg units: 1 Ry = 13.6056925 eV.

Table 2. (a) Energy resonances of doubly excited ${}_{n}\left(K,T\right){}_{2}^{-}{}^{3}D{}_{e}$ (n = 3 - 10) states of helium-like ions (Z = 2 - 10). The results are expressed in Rydberg units: 1 Ry = 13.6056925 eV. (b) Energy resonances of doubly excited ${}_{n}\left(K,T\right){}_{3}^{-}{}^{3}D{}_{e}$ (n = 4 - 10) states of helium-like ions (Z = 2 - 10). The results are expressed in Rydberg units: 1 Ry = 13.6056925 eV. (c) Energy resonances of doubly excited ${}_{n}\left(K,T\right){}_{4}^{-}{}^{3}D{}_{e}$ (n = 5 - 10) states of helium-like ions (Z = 2 - 10). The results are expressed in Rydberg units: 1 Ry = 13.6056925 eV. (d) Energy resonances of doubly excited ${}_{n}\left(K,T\right){}_{5}^{-}{}^{3}D{}_{e}$ (n = 6 - 10) states of helium-like ions (Z = 2 - 10). The results are expressed in Rydberg units: 1 Ry = 13.6056925 eV. (e) Energy resonances of doubly excited ${}_{n}\left(K,T\right){}_{6}^{-}{}^{3}D{}_{e}$ (n = 7 - 10) states of helium-like ions (Z = 2 - 10). The results are expressed in Rydberg units: 1 Ry = 13.6056925 eV. (f) Energy resonances of doubly excited ${}_{n}\left(K,T\right){}_{7}^{-}{}^{3}D{}_{e}$ (n = 8 - 10), ${}_{n}\left(K,T\right){}_{8}^{-}{}^{1}D{}_{e}$ (n = 9, 10) and ${}_{10}\left(K,T\right){}_{9}^{-}{}^{1}D{}_{e}$ states of helium-like ions (Z = 2 - 10). The results are expressed in Rydberg units: 1 Ry = 13.6056925 eV.

Table 3. Correspondence between various classification schemes for autoionizing states in two-electron systems below the *N* = 2 - 9 hydrogenic threshold. (2*S *+ 1) is the multiplicity where *S *denotes the total spin angular momentum, *L *is the total orbital angular momentum, *π *represents the parity, *N *and *n *are respectively the principal quantum numbers of the inner and the outer electron. S, *L*, *π*, *N *and *n *are common to all the schemes. The correlation quantum numbers *K*, *T *and *A *used to label each state are introduced to complete the description.

Table 4. Comparison of the present doubly excited ${}_{3}\left(1,0\right){}_{2}^{+,-}{}^{1,3}D{}_{e}$ and ${}_{4}\left(1,0\right){}_{2}^{+,-}{}^{1,3}D{}_{e}$ states of helium-like ions with other theoretical results. SCUNC: screening constant by unit nuclear charge method, CCR: complex-coordinate rotation method, CDS: computing double sum. The energy resonances (−E) are expressed in Rydberg units.

Table 5. Comparison of the present doubly excited ${}_{n}\left(1,0\right){}_{2}^{+}{}^{1}D{}_{e}$ (n = 3 - 10) and ${}_{n}\left(1,1\right){}_{3}^{+}{}^{1}D{}_{e}$ (n = 4 - 10) states of helium-like ions with other theoretical results. SM: spectral method, SPCR: saddle-point complex-rotation method, CC: close-coupling method, TDM: truncated diagonalization method, EES: ejected electron spectroscopy, CIM: Configuration Interaction Method. The energy resonances (−E) are expressed in Rydberg units.

the very precise and recent values calculated by Restrepo [19] using the configuration interaction method (CIM), is very reasonable.

In Table 6, the results obtained for ${}_{n}\left(1,0\right){}_{2}^{-}{}^{3}D{}_{e}$ and ${}_{n}\left(1,1\right){}_{2}^{-}{}^{3}D{}_{e}$ resonances in the helium-like ions are displayed and compared with the data of Eiglsperger et al. [14], Argenti [20] using a B-spline K-matrix method (BKM), Chen [15], Lipsky et al. [17] and Cuartas and Vicario [19]. It can be seen that, the present calculations agree well with those of the cited authors.

4. Summary

In this work, we present an extensive analysis of the inter-shell singlet and triplet doubly excited
${}_{n}\left(K,T\right){}_{N}^{+,-}{}^{1,3}D{}_{e}$ (N = 2 - 9, n = 3 - 10) states energies of the helium-like ions up to Z = 10 using a simple approach. The calculations have been done by using the variational method with the no-linear parameters of Hylleraas and the β-parameters of screening constant by unit nuclear charge. We have used a simple equation (13) to calculate resonance parameters for inter-shell ^{1,3}D_{e} autoionizing states in helium (the two electrons occupy the different shells).

Comparisons with other theoretical calculations and experimental observations are in good agreement. For some states, we have noticed small differences

Table 6. Comparison of the present doubly excited ${}_{n}\left(1,0\right){}_{2}^{-}{}^{3}D{}_{e}$ (n = 3-10) and ${}_{n}\left(1,1\right){}_{3}^{-}{}^{3}D{}_{e}$ (n = 4 - 10) states of helium-like ions with other theoretical results. SM: Spectral Method, BKM: B-spline K-matrix method, SPCR: Saddle-Point Complex-Rotation method, TDM: Truncated Diagonalization Method, CIM: Configuration Interaction Method. The energy resonances (−E) are expressed in Rydberg units.

between ours results and other data, which are due by the approximations in the calculation of the variational parameters.

The prospect of this work is to improve our calculation method by modifying the parameters, to have a very good convergence of results. In this case, it will be possible to study the ^{1,3}F^{o}, ^{1,3}G^{e} and others states energies of Helium-Like Ions.

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