Modified Atomic Orbital Calculations of Energy of the (2 s 2 1 S ) Ground-State, the (2 p 2 1 D ); (3 d 2 1 G ) and (4 f 2 1 I ) Doubly Excited States of Helium Isoelectronic Sequence from H − to Ca 18+

We report in this paper the ground-state energy 2s 2 1 S and total energies of doubly excited states 2p 2 1 D, 3d 2 1 G, 4f 2 1 I of the Helium isoelectronic sequence from H − to Ca 18+ . Calculations are performed using the Modified Atomic Orbital Theory (MAOT) in the framework of a variational procedure. The purpose of this study required a mathematical development of the Hamiltonian applied to Slater-type wave function [1] combining with Hylle-raas-type wave function [2] . The study leads to analytical expressions which are carried out under special MAXIMA computational program. This first proposed MAOT variational procedure, leads to accurate results in good agreement as well as with available other theoretical results than experimental data. In the present work, a new correlated wave function is presented to ex-press analytically the total energies for the 2s 2 1 S ground state and each doubly 2p 2 1 D, 3d 2 1 G, 4f 2 1 I excited states in the He-like systems.The present accurate data may be a useful guideline for future experimental and theoretical studies in the (nl 2 ) systems. and Z ≤ 10, our results are compared with


Introduction
The resolution of the Schrödinger equation gives for the ground-state of the Helium atom the value E = 108.8 eV. The experimental result being equal to E 0 = −79.0 eV, one conceives that one takes into account the electronic correlation term, overestimating, numerous studies provided evidence of the importance of the electronic correlation in the ground-state and in the doubly excited state of He-like series [3] [4] [5].
For the ground states, energy calculations of Helium isoelectronic sequence are performed by using several correlated wavefunction and an analytical technique calculation. Thus, Moore [6], Radzig and Smirnov [7] and Arnaud [8] measured the ground state energies E (1s 2 1 S) of two-electron systems with atomic number Z (2 ≤ Z ≤ 10). Combining the perturbation theory [9] to the Ritz variational method [10], Sakho et al., [11] set in motion a technic of analytical calculation of the ground state energy E ( 1 S 0 ), the first ionization energy J ( 1 S 0 ) and the radial correlation expectation value 1 12 r − ( 1 S 0 ) for the Helium-like ions from Hydrogen ion H − (Z = 1) to silicon Si 12+ (Z = 14). Utpal and Talukdar [12] used an analytical approach to also calculate the ground-state energies of helium isoelectronic sequence from Hydrogen ion H − (Z = 1) up to the silicon ion Si 12+ (Z = 14). For some methods using variational wavefunction like Hylleraas's one, a good approximation of the eigenvalues is obtained when the minima of the  [2]. For other methods using a non variational wavefunction [13], the ground-state energy of He-like ions is determined by the use of proper core boundary conditions correct behavior for 12 0 r → and 12 r → ∞ and by taking recourse to a perturbative method. For some methods using an analytical technique calculation, some authors were interested in the setting in work of calculation techniques permitting one to succeed to an analytical expression of the ground-state energy of the He-like ions. Thus, an analytical calculation for the ground-state energy and radial expectation values of Helium isoelectronic sequence is managed by using a wavefunction of the type of Bhattacharyya [12] [14]. Besides, developing the orbital atomic theory, Slater in [15] introduced the notions of screen constant σ and effective quantum number n* for the calculation of the energy of an electronic configuration given containing N electrons.
On the basis of his theory, Slater expresses analytically the total energy of an atomic system of N electrons according to σ and n * determined from rules that he established. The analytical formula of Slater permits the simple calculation of the ground-state energy of He-like ions for which N = 2, σ = 0.30 and n * = 1.

Journal of Applied Mathematics and Physics
For the doubly excited states in He-like ions, since the early experiment [16] and theoretical explanation [17], Doubly Excited States (DES) of Helium isoelectronic sequence have been the target of a number of theoretical approaches.
Greatest attention have been concentrated on studying symmetric DES (nl 2 ) with excited electrons having equal values of principal quantum number n (intrashell states) where the electronic correlation effect may be predominant [18]. The investigations of the intrashell states of two-electrons systems are advanced and due to the group theoretical method [19] which allowed intrashell states to be approximatively classified and some of these properties studied [20]. Theoretical investigations of (nl 2 ) doubly excited states are performed by using various method. The projection operator method and group theoretical methods [20] have been used for energies calculations of the 2s 2 , 2p 2 states in Helium-like ions.
Time independent variational perturbation [21] was applied for total energies calculations of the 2s 2 , 2p 2 and 3d 2 states in He, Li + , Be 2+ and B 3+ . The correlation part of the energies for the 2s 2 , 2p 2 , 3s 2 , 3p 2 , 3d 2 states in He isoelectronic series have been investigated by using perturbation theory [4]. The Screening Constant by Unit Nuclear charge (SCUNC) method [22] used a semi-empirical procedure to calculate (ns 2 ) 1 S e , (np 2 ) 1 D e and (Nsnp) 1 P° excited state of He-like ions. Recently the Modified Atomic Orbital Theory (MAOT) has been applied successfully in the studies of high lying 1,3 P° of He-like ions [1]. In this paper, we apply the first MAOT variational procedure, to calculate the ground-state energy 2s 2 1 S and the total energies of the singlet DES 2p 2 1 D, 3d 2 1 G and 4f 2 1 I of He isoelectronic sequence from H − to Ca 18+ . In addition, for the first time in our knowledge, we have also calculated theoretical the screening constant theo

Brief Description of the MAOT Formalism
In the framework of Modified Atomic Orbital Theory (MAOT), total energy of (νℓ)-given orbital is expressed in the form [1] [23]: For an atomic system of several electrons M, the total energy is given by (in In the photoionization study, energy resonances are generally measured relatively to the E ∞ converging limit of a given ( 2S+1 L J )nl-Rydberg series. For these states, the general expression of the energy resonances is given by the formula [24] presented previously (in Rydberg units): In this equation m and q (m < q) denote the principal quantum numbers of the ( 2S+1 L J )nl-Rydberg series of the considered atomic system used in the empirical determination of the σ i ( 2S+1 L J )-screening constants, s represents the spin of the nl-electron (s = 1/2), E ∞ is the energy value of the series limit generally determined from the NIST atomic database, E n denotes the corresponding energy resonance, and Z represents the nuclear charge of the considered element. The only problem that one may face by using the MAOT formalism is linked to the determination of the In this Equation (3 bis), R is the Rydberg constant, Z core represents the electric charge of the core ion.
Z core is directly obtained by the photoionization process from an atomic X sys- are screening constants to be evaluated empirically with k taking the values from 1 to q. L J : denote the considered quantum state (S, P, D, F, ...).

Variational Procedure of Calculations
For the 2s 2 1 S ground state and each doubly excited states 2p 2 1 D, 3d 2 1 G, 4f 2 1 I, we constructed the basis wave functions below by combining Slater-type wave function [1] and Hylleraas-type wave functions [2]: We considered parameter  (5) and , , Z σ υ and 0 a are respectively the nucleus charge number, the screening constant, the principal quantum number and Bohr's radius The final form of the wave function of the singlet doubly excited state can be written as follow: (8) where the coefficients jkm β are determined by solving the Schrödinger equation: n n H E Ψ = Ψ r r r r (9) where the Hamiltonian operator H has the form: (11) where T is the kinetic energy, C is the Coulomb potential between the atomic nucleus and the two electrons, W is the Coulomb interaction between electrons.
In the Hamilton operator we neglected all magnetic and relativistic effects together with the motion of the atomic nucleus.
In this Equation (11) The wave function is written as follow: The matrix elements of the normalization factor is written as follow: The matrix elements of the Coulombian interaction Energy operator between the nucleus and the two electrons is written as follow: ( ) ( ) 21 (25) The matrix elements of the kinetic Energy operator of the two electrons is ex-Journal of Applied Mathematics and Physics ∇ + ∇ r r r r (26) Calculations for the other matrix elements of other states were obtained in the same procedure.
The mathematical development of the Hamiltonian applied to each wave function of each states leads to simplified analytical expressions which are carried out under MAXIMA computational program.
Concerning the screening constant σ and the variational parameter ξ, the procedure is as follow: From the Slater condition 0.3 1 σ ≤ ≤ , and taken into account Equation (5), the parameter ξ can be expressed as follow: In order to obtain the theoretical screening constant theo σ , the exponential parameter ξ and the minimum eigenvalue in which we are interested, the analytical expressions of each matrix elements of each state are carried out by our self MAXIMA computational program.
MAXIMA is a computer algebra system for the manipulation of symbolic and numerical expressions, including differentiation, integration, ordinary differential equations, and matrix elements. MAXIMA yields high precision numeric results by using exact fractions, arbitrary precision integers, and variable precision floating point numbers. Our MAXIMA source code is compiled on windows systems.
At the beginning, the variational parameter ξ is determined as shown (27). For each value of Z and n, we know a limited area of the variational parameter ξ. Thus for each value of Z, n and ξ, the program calculated directly the screening constant σ as shown the Equation (7) and then the eigenvalue E. To obtain the minimum eigenvalue and the theoretical screening constant in which we are interested and quoted in tables 1 -4, the variational parameter ξ and the Hylleraas parameters (J, K, M) are slightly varied that exhibit a plateau for the energy.

Results and Discussions
The main results of our calculations for the theoretical screening constant, the variational parameter ξ and energies concerning the 1s 2 1 S ground-state and the Doubly Excited State for 2p 2 1 D, 3d 2 1 G and 4f 2 1 I of helium isoelectronic sequence are quoted respectively in Tables 1-4. Our present results are compared with other theoretical calculations and experimental data. Then our results are converted into Rydberg for direct comparison by using the infinite Rydberg 1Ry = 0.5 a.u = 13.605698 eV. Table 1 shows a comparison of the present calculation for the 1s 2 1 S ground-state energy with the experimental data [6] [7] [8], and the theoretical results [9] [10] [11] [12] [15]. In addition the theoretical screening  (Sakho, 2006) [11]; E e (Pekeris, 1962); E c (Drake, 1988) [9]; E Slater (Minkine, 1982) [15]; E a,b Experimental data (Radzig, 1985 [7]; Arnaud, 1993)  Generally, this good agreement enables to expect our results with MAOT calculation for ground-state energy up to Z = 20 to be accurate.
In Table 2, we presented the theoretical screening constant σ theo = 0.3512 and reported the value of the variational parameter ξ and total energies for 2p 2 1 D Journal of Applied Mathematics and Physics

Summary and Conclusion
In this work, the variational procedure of the Modified Atomic Orbital Theory Journal of Applied Mathematics and Physics