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

The ( ) 1,3 , N K T D doubly excited states of helium-like ions are investigated using a combination of the no-linear parameters of Hylleraas and the β-parameters of screening constant by unit nuclear charge. Calculations are performed for total energies of low-lying doubly excited states (N = 2 - 9) in He-like ions up to Z = 10. The results obtained from the novel method are in good agreement with the available theoretical calculations and experimental observations.


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 ( ) 2 1 , π + A S n N K T L 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 is the vibrational quantum number [6].
In this paper, we present a novel approach to calculate the energy resonances of the 1,3 e D helium-isoelectronic sequence below the N = 2 -9 ionisation thresholds. We intend to investigate the ( ) 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.

Hamiltonian and Hylleraas-Type Wave Functions
The Hamilton operator for the helium-like ions is written as: The vectors 1 r and 2 r denote the positions of the two electrons, m the mass of an electron, e the elementary charge and Z the nuclear charge number, 1 ∆ and 2 ∆ are the Laplace operators of the two electrons, 1 2 − r r 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: 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 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: 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: 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'. Journal of Applied Mathematics and Physics 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

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) [ (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: are screening constants to be evaluated.
With the new classification scheme, Equation (9) takes the form (in Ry):  ; , , Using (11) (14) σ 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 e D inter-shell autoionizing states (the two electrons occupy different shells) in helium with an easy calculation program.

Results and Discussion
The results of the currently calculations for inter-shell singlet and triplet doubly In Table 3, we have indicated the correspondence between various classification schemes for autoionizing states in two electron systems [4] [5] [6] [11].                         Table 3. Correspondence between various classification schemes for autoionizing states in two-electron systems below the N = 2 -9 hydrogenic threshold. (2S + 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.

Usual Classification
Nlnl' 2S+1 L π Conneely and Lipsky (N, n, α)   the very precise and recent values calculated by Restrepo [19] using the configuration interaction method (CIM), is very reasonable.

Summary
In this work, we present an extensive analysis of the inter-shell singlet and triplet doubly excited ( )