^{1}

^{*}

^{2}

^{1}

^{1}

^{1}

^{1}

^{3}

^{4}

^{4}

^{4}

^{5}

^{1}

We report in this paper the ground-state energy 2
s
^{2}
^{1}
S and total energies of doubly excited states 2
p
^{2}
^{1}
D, 3
d
^{2}
^{1}
D, 4
f
^{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 Hylleraas-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 express analytically the total energies for the 2
s
^{2}
^{1}
S ground state and each doubly 2
p
^{2}
^{1}
D, 3
d
^{2}
^{1}
D, 4
f
^{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 (
n
l
^{2}) systems.

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 [

For the ground states, energy calculations of Helium isoelectronic sequence are performed by using several correlated wavefunction and an analytical technique calculation. Thus, Moore [^{2} ^{1}S) of two-electron systems with atomic number Z (2 ≤ Z ≤ 10). Combining the perturbation theory [^{1}S_{0}), the first ionization energy J (^{1}S_{0}) and the radial correlation expectation value 〈 r 12 − 1 〉 (^{1}S_{0}) for the Helium-like ions from Hydrogen ion H^{−} (Z = 1) to silicon Si^{12+} (Z = 14). Utpal and Talukdar [^{−} (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

function ( d E d α , with α a variational parameter, E is the energy) converges with

the increasing values of the dimension D and when the function E = f ( α , D ) exhibits a plateau [

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.

For the doubly excited states in He-like ions, since the early experiment [^{2}) with excited electrons having equal values of principal quantum number n (intrashell states) where the electronic correlation effect may be predominant [^{2}) doubly excited states are performed by using various method. The projection operator method and group theoretical methods [^{2}, 2p^{2} states in Helium-like ions. Time independent variational perturbation [^{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 [^{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 [^{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 σ t h e o which is compared with experimental Slater screening constant ( σ e x p ) determined from his rule. Our present procedure leads to analytical expression which are carried out under MAXIMA computational program. Our energies positions are compared to other available theoretical and experimental data.

In the framework of Modified Atomic Orbital Theory (MAOT), total energy of (νℓ)-given orbital is expressed in the form [

E ( υ l ) = − [ Z − σ ( l ) ] 2 υ 2 (1)

For an atomic system of several electrons M, the total energy is given by (in Rydberg):

E = − ∑ i = 1 M [ Z − σ i ( l ) ] 2 υ i 2

With respect to the usual spectroscopic notation ( N l , N l ′ ) L 2 S + 1 π , this equation becomes

E = − ∑ i = 1 M [ Z − σ i ( L 2 S + 1 π ) ] 2 υ i 2 (2)

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 [

E n = E ∞ − 1 n 2 { Z − σ 1 ( 2 S + 1 L J ) − σ 2 ( 2 S + 1 L J ) × 1 n − σ 2 α ( 2 S + 1 L J ) × ( n − m ) × ( n − q ) ∑ k 1 f k ( n , m , q , s ) } 2 (3)

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 ∑ k 1 f k ( n , m , q , s ) term. The correct expression of this term

is determined iteratively by imposing general Equation (3) to give accurate data with a constant quantum defect values along all the considered series. The value of α in the σ 2 of the last term is fixed to 1 and 2 during the iteration. The quantum defect δ is calculated from the standard formula

E n = E ∞ − R Z c o r e 2 ( n − δ ) 2 ⇒ δ = n − Z c o r e R ( E ∞ − E n ) (3 bis)

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 system X + h ν → X p + + p e −

f k = f k ( L 2 S + 1 J , n , s , m , q ) 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, ...).

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 [

ϕ j k m n ( r 1 , r 2 ) = ( r 1 r 2 ) ( υ − 1 ) × exp ( − ( Z − σ υ × a 0 ) ( r 1 + r 2 ) ) ( r 1 + r 2 ) j ( r 1 − r 2 ) k | r 1 − r 2 | m (4)

We considered parameter

ξ = Z − σ υ × a 0 (5)

and Z , σ , υ and a 0 are respectively the nucleus charge number, the screening constant, the principal quantum number and Bohr’s radius

ϕ j k m n ( r 1 , r 2 ) = ( r 1 r 2 ) ( υ − 1 ) × exp ( − ξ ( r 1 + r 2 ) ) ( r 1 + r 2 ) J ( r 1 − r 2 ) K | r 1 − r 2 | M (6)

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

where r 1 and r 2 denote the positions of the two electrons;

r 1 and r 2 are respectively used for | r 1 | and | r 2 | , J , K , M are Hylleraas parameters with ( J , K , M ≥ 0 ) .

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.

M takes into account the distance between the two electrons.

From Equation (5) the screening constant can be expressed

σ = Z − a 0 υ ξ (7)

The final form of the wave function of the singlet doubly excited state can be written as follow:

Ψ n ( r 1 , r 2 ) = ∑ j k m β j k m ϕ j k m n ( r 1 , r 2 ) (8)

where the coefficients β j k m are determined by solving the Schrödinger equation:

H Ψ n ( r 1 , r 2 ) = E Ψ n ( r 1 , r 2 ) (9)

where the Hamiltonian operator H has the form:

H = T + C + W with (10)

T = − ℏ 2 2 m ( ∇ 1 + ∇ 2 ) ; C = − ( Z e 2 r 1 + Z e 2 r 2 ) ; W = e 2 | r 1 − r 2 | (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), Z is the nuclear charge ∇_{1} is the Laplacian with reference to the coordinates of the vector radius r 1 which detect the position of the electron 1. ∇_{2} Laplacian defines the coordinates of the vector radius r 2 which detect the position of the electron 2 and | r 1 − r 2 | inter-electronic distance.

The representation of the Schrödinger equation on the non-orthogonal basis leads to the general eigenvalue equation [

∑ ( j , k , m ) , ( j ′ k ′ m ′ ) ( H J K M n l − E N J K M n l ) = 0 (12)

With J = j + j ′ ; K = k + k ′ ; M = m + m ′

N J K M n l = 〈 ϕ j k m n l | ϕ j ′ k ′ m ′ n l 〉 : is the normalization factor (13)

H J K M n l = 〈 ϕ j k m n l | H | ϕ j ′ k ′ m ′ n l 〉 : is the matrix elements of Hamilton operator (14)

E: is the eigenvalue of the energy

H J K M n l = T j k m , j ′ k ′ m ′ n l + C J K M n l + W J K M n l (15)

T j k m , j ′ k ′ m ′ n l : is the matrix elements of the kinetic Energy operator of the two electrons

C J K M n l : is the matrix elements of the Coulombian interaction Energy operator between the nucleus and the two electrons

W J K M n l : is the matrix elements of the Coulombian interaction Energy operator between the two electrons

Thus following the form of the basis wave function above (4), we have constructed for each state a special wave function and then calculated the matrix elements N J K M n l , C J K M n l , W J K M n l , T j k m , j ′ k ′ m ′ n l

For example on 2p^{2} states: n = υ = 2 ; l = 1

The wave function is written as follow:

ϕ j k m 21 ( r 1 , r 2 ) = ( r 1 r 2 ) × exp ( − ξ ( r 1 + r 2 ) ) ( r 1 + r 2 ) J ( r 1 − r 2 ) K | r 1 − r 2 | M (16)

The matrix elements of the normalization factor is written as follow:

N J K M 21 = 〈 ϕ j k m 21 | ϕ j ′ k ′ m ′ 21 〉 (17)

N J K M 21 = ∭ d 3 r 1 d 3 r 2 ϕ j k m 21 ( r 1 , r 2 ) × ϕ j ′ k ′ m ′ 21 ( r 1 , r 2 ) (18)

N J K M 21 = ∭ d 3 r 1 d 3 r 2 ( r 1 r 2 ) 2 ( r 1 + r 2 ) J ( r 1 − r 2 ) K | r 1 − r 2 | M × exp ( − 2 ξ ( r 1 + r 2 ) ) (19)

With ∭ d 3 r i = ∫ r i 2 d r i ∫ 0 π sin θ i d θ i ∫ 0 2 π d φ i = 4 π ∫ r i 2 d r i ( i = 1 , 2 ) (20)

The matrix elements of the Coulombian interaction Energy operator between the nucleus and the two electrons is written as follow:

C J K M n l = 〈 ϕ j k m 21 ( r 1 , r 2 ) | C | ϕ j ′ k ′ m ′ 21 ( r 1 , r 2 ) 〉 (21)

C J K M 21 = 〈 ϕ j k m 21 ( r 1 , r 2 ) | − Z e 2 ( 1 r 1 + 1 r 2 ) | ϕ j ′ k ′ m ′ 21 ( r 1 , r 2 ) 〉 (22)

C J K M 21 = − Z e 2 ∭ d 3 r 1 d 3 r 2 ϕ j k m 21 ( r 1 , r 2 ) ( 1 r 1 + 1 r 2 ) ϕ j ′ k ′ m ′ 21 ( r 1 , r 2 ) (23)

The matrix elements of the Coulombian interaction Energy operator between the two electrons is expressed as follow:

W J K M 21 = 〈 ϕ j k m 21 ( r 1 , r 2 ) | 1 | r 1 − r 2 | | ϕ j ′ k ′ m ′ 21 ( r 1 , r 2 ) 〉 (24)

W J K M 21 = ∭ d 3 r 1 d 3 r 2 ϕ j k m 21 ( r 1 , r 2 ) ( 1 | r 1 − r 2 | ) ϕ j ′ k ′ m ′ 21 ( r 1 , r 2 ) (25)

The matrix elements of the kinetic Energy operator of the two electrons is expressed as follow:

T j k m , j ′ k ′ m ′ 21 = 〈 ϕ j k m 21 ( r 1 , r 2 ) | − ℏ 2 2 m ( ∇ 1 + ∇ 2 ) | ϕ j ′ k ′ m ′ 21 ( r 1 , r 2 ) 〉 (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 ≤ σ ≤ 1 , and taken into account Equation (5), the parameter ξ can be expressed as follow:

Z − 1 n ≤ ξ ≤ Z n (27)

In order to obtain the theoretical screening constant σ t h e o , 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.

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. ^{2} ^{1}S ground-state energy with the experimental data [

Theory | Experiment | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

ξ | −E^{MAOT} | −E^{Sakho} | −E^{e} | −E^{c} | −E^{d} | −E^{Slater} | −E^{f} | −E^{exp} | −E^{a}^{,b} | |

H^{− } | 0.6899 | 1.0098 | 1.1406 | 1.0164 | 0.9745 | |||||

He | 1.6899 | 5.7951 | 5.8872 | 5.8071 | 5.8078 | 5.7784 | 5.7799 | 5.7233 | 5.8063 | 5.8071^{a} |

Li^{+} | 2.6899 | 14.5063 | 14.6401 | 14.5593 | 14.5615 | 14.5321 | 14.6049 | 14.4726 | 14.4645 | 14.5593^{b} |

Be^{2+} | 3.6899 | 27.2554 | 27.3907 | 27.3113 | 27.3150 | 27.2844 | 27.4260 | 27.2224 | 27.2238 | 27.3120^{b} |

B^{3+} | 4.6899 | 43.9813 | 44.1403 | 44.0616 | 44.0690 | 44.0352 | 44.2542 | 43.9720 | 43.9815 | 44.0683^{b} |

C^{4+} | 5.6899 | 64.7582 | 64.8904 | 64.8125 | 64.8228 | 64.7860 | 65.0896 | 64.7221 | 64.7449 | 64.8294^{b} |

N^{5+} | 6.6899 | 89.5054 | 89.6403 | 89.5624 | 89.5779 | 89.5367 | 89.9314 | 89.4720 | 89.5999^{b} | |

O^{6+} | 7.6899 | 118.3513 | 118.3908 | 118.2886 | 118.3342 | 118.2872 | 118.7803 | 118.2952 | 118.3834^{b} | |

F^{7+} | 8.6899 | 151.0853 | 151.1403 | 151.0631 | 151.0903 | 151.0374 | 151.6357 | 150.9721 | 151.2366^{b} | |

Ne^{8+} | 9.6899 | 187.9563 | 187.8903 | 187.8840 | 187.8477 | 187.7874 | 188.4974 | 187.7221 | 188.0050^{b} | |

Na^{9+} | 10.6899 | 228.6006 | 228.6395 | 228.6056 | 228.5373 | 228.9797 | 228.4719 | |||

Mg^{10+} | 11.6899 | 273.4513 | 273.3906 | 273.3648 | 273.2877 | 273.7801 | 273.2222 | |||

Al^{11+} | 12.6899 | 322.0992 | 322.1407 | 322.1245 | 322.0378 | 322.5802 | 321.9724 | |||

Si^{12+} | 13.6899 | 374.9353 | 374.8907 | 374.8855 | 374.7878 | 375.3802 | 374.7224 | |||

P^{13+} | 14.6899 | 431.4453 | ||||||||

S^{14+} | 15.6899 | 492.1952 | ||||||||

Cl^{15+} | 16.6899 | 556.9453 | ||||||||

Ar^{16+} | 17.6899 | 625.6953 | ||||||||

K^{17+} | 18.6899 | 698.4452 | ||||||||

Ca^{18+} | 19.6899 | 775.1954 |

E^{MAOT}: Energy E of the Modified Atomic Orbital Theory (MAOT), present work. E^{sakho} (Sakho, 2006) [^{e} (Pekeris, 1962); E^{c} (Drake, 1988) [^{Slate}^{r} (Minkine, 1982) [^{a}^{,b} Experimental data (Radzig, 1985 [^{ex}^{p} (Moore, 1971) [^{d} (Utpal and Talukdar, 1999) [^{f} (Roothaan et al., 1960) [

constant σ_{the}_{o} = 0.3101 is also presented and agree very good with the experimental screening constant of Slater σ_{Slater} = 0.30 determined from his rule (σ_{Slate}_{r} = 0.30 for 1s^{2} state, n = 1). For H^{−} (Z = 1), our results at −1.0098 Ry are compared with those of [

In _{theo} = 0.3512 and reported the value of the variational parameter ξ and total energies for 2p^{2} ^{1}D

Theory | Experiment | ||||||||
---|---|---|---|---|---|---|---|---|---|

ξ | −E^{MAOT} | −SCUNC | −E^{BIA } | −E^{c} | −E^{g} | −E^{h} | −E^{e}^{,f} | −E^{Ray} | |

H^{−} | 0.3244 | 0.2335 | |||||||

He | 0.8244 | 1.3841 | 1.4052 | 1.4097 | 1.4266 | 1.4052^{e} | 1.3728 | ||

Li^{+} | 1.3244 | 3.5398 | 3.5418 | 3.5565 | 3.5110 | 3.5411 | 3.5565 | 3.5396^{f} | 3.3906 |

Be^{2+} | 1.8244 | 6.6838 | 6.6788 | 6.6993 | 6.6545 | 6.6751 | 6.6861 | 6.5097 | |

B^{3+} | 2.3244 | 10.8132 | 10.8153 | 10.8403 | 10.7976 | 10.8072 | 10.8160 | 10.6266 | |

C^{4+} | 2.8244 | 15.9892 | 15.9514 | 15.9793 | 15.9455 | ||||

N^{5+} | 3.3244 | 22.0819 | 22.0885 | 22.1186 | 22.0753 | ||||

O^{6+} | 3.8244 | 29.1999 | 29.2252 | 29.2568 | 29.2046 | ||||

F^{7+} | 4.3244 | 37.3396 | 37.3615 | 37.3946 | 37.3350 | ||||

Ne^{8+} | 4.8244 | 46.5156 | 46.4981 | 46.5327 | 46.4643 | ||||

Na^{9+} | 5.3244 | 56.6209 | |||||||

Mg^{10+} | 5.8244 | 67.7463 | |||||||

Al^{11+} | 6.3244 | 79.9033 | |||||||

Si^{12+} | 6.8244 | 93.0931 | |||||||

P^{13+} | 7.3244 | 107.2076 | |||||||

S^{14+} | 7.8244 | 122.3587 | |||||||

Cl^{15+} | 8.3244 | 138.4988 | |||||||

Ar^{16+} | 8.8244 | 155.6603 | |||||||

K^{17+} | 9.3244 | 173.8364 | |||||||

Ca^{18+} | 9.8244 | 193.0401 |

E^{MAOT}: Energy E of the Modified Atomic Orbital Theory (MAOT), present work. SCUNC (Sakho, 2008) [^{BIA} (Biaye et al., 2005) [^{Ray} Experimental data (Ray et al., 1991) [^{c} (Roy et al., 1997) [^{g} (Ho and Bhatia, 1991) [^{h} (Ivanov and Safronova, 1993) [^{e} Experimental data (Hicks and Comer, 1975) [^{f} Experimental data (Diehl et al., 1999) [

doubly excited states of He-like ions up to Z = 20. For this state, comparison shows that MAOT results agree well with the theoretical results of [^{−} where there is no available data for comparison. For Z = 10, our results at −46.5156 Ry are compared with those of [

^{2} ^{1}G DES that are compared with those of [^{−} (Z = 1) where there is no available results, the agreements between the calculation are generally good up to Z = 5. Thus for 5 < Z ≤ 10, our results are compared with

Theory | ||||||||
---|---|---|---|---|---|---|---|---|

ξ | −E^{MAOT} | −E^{Sakho} | −E^{a} | −E^{b } | −E^{c} | −E^{d} | −E^{Ray} | |

H^{−} | 0.2162 | 0.0152 | - | - | - | - | - | - |

He | 0.5495 | 0.6260 | 0.6104 | 0.6166 | 0.6308 | 0.6303 | 0.5849 | 0.6232 |

Li^{+} | 0.8829 | 1.6004 | 1.5486 | 1.5538 | 1.5837 | 1.5618 | 1.5247 | 1.4601 |

Be^{2+} | 1.2162 | 2.9529 | 2.9313 | 2.9320 | 2.9762 | 2.9377 | 2.9085 | 2.8421 |

B^{3+} | 1.5495 | 4.7596 | 4.7585 | 4.7540 | 4.8105 | 4.7581 | 4.7360 | 4.6669 |

C^{4+} | 1.8829 | 7.1927 | 7.0301 | 7.0180 | 7.0875 | 7.0229 | ||

N^{5+} | 2.2162 | 9.8973 | 9.7461 | 9.7280 | 9.8079 | 9.7321 | ||

O^{6+} | 2.5495 | 13.0334 | 12.9066 | 12.882 | 12.9718 | 12.8858 | ||

F^{7+} | 2.8829 | 16.5999 | 16.5116 | 16.4800 | 16.5797 | 16.4840 | ||

Ne^{8+} | 3.2162 | 20.5969 | 20.5609 | 20.5200 | 20.6316 | 20.5265 | ||

Na^{9+} | 3.5495 | 25.0246 | ||||||

Mg^{10+} | 3.8829 | 29.8814 | ||||||

Al^{11+} | 4.2162 | 35.1721 | ||||||

Si^{12+} | 4.54959 | 40.8918 | ||||||

P^{13+} | 4.88293 | 47.0424 | ||||||

S^{14+} | 5.21626 | 53.6234 | ||||||

Cl^{15+} | 5.54959 | 60.6351 | ||||||

Ar^{16+} | 5.88293 | 68.0777 | ||||||

K^{17+} | 6.21626 | 75.9507 | ||||||

Ca^{18+} | 6.54959 | 84.2544 |

E^{MAOT}: Energy E of the Modified Atomic Orbital Theory (MAOT), present work. E^{sakho} (Sakho,2010) [^{a} (Bachau et al., 1991) [^{b} (Biaye,2005) [^{c} (Ivanov and Safronova, 1993) [^{d} (Ho, 1989) as quoted in (Biaye, 2005) [^{Ray} (Ray et al., 1991) [

those of [_{theo} = 0.3512). This good agreement allows us to expect our results with MAOT calculation for 3d^{2} ^{1}G doubly excited state up to Z = 20 to be accurate.

In ^{2} ^{1}I doubly excited state and compare our results with theoretical results of [^{−} (Z = 1). Here the agreements between the calculations are seen to be very satisfactory. As far as comparisons with the SCUNC results of [

Theory | |||||
---|---|---|---|---|---|

ξ | −E^{MAOT} | −E^{Sakho} | −E^{Biaye } | −E^{Ho} | |

H^{−} | 0.1623 | 0.04848 | - | - | - |

He | 0.4123 | 0.32906 | 0.32380 | 0.3591 | 0.34080 |

Li^{+} | 0.6623 | 0.85937 | 0.83464 | 0.89778 | |

Be^{2+} | 0.9123 | 1.63938 | 1.59547 | 1.68285 | |

B^{3+} | 1.1623 | 2.66909 | 2.60631 | 2.71550 | |

C^{4+} | 1.4123 | 3.95001 | 3.86714 | 3.99645 | |

N^{5+} | 1.6623 | 5.47766 | 5.37798 | 5.52616 | |

O^{6+} | 1.9123 | 7.25650 | 7.13881 | 7.30493 | |

F^{7+} | 2.1623 | 9.28507 | 9.14965 | 9.33300 | |

Ne^{8+} | 2.4123 | 11.56332 | 11.41048 | 11.61005 | |

Na^{9+} | 2.6623 | 14.09130 | |||

Mg^{10+} | 2.9123 | 16.86898 | |||

Al^{11+} | 3.1623 | 19.89637 | |||

Si^{12+} | 3.4123 | 23.17405 | |||

P^{13+} | 3.6623 | 26.70029 | |||

S^{14+} | 3.9123 | 30.47681 | |||

Cl^{15+} | 4.1623 | 34.50304 | |||

Ar^{16+} | 4.4123 | 38.77898 | |||

K^{17+} | 4.6623 | 43.30464 | |||

Ca^{18+} | 4.9123 | 48.07999 |

E^{MAOT}: Energy E of the Modified Atomic Orbital Theory (MAOT), present work. E^{Sakho} (Sakho, 2010) [^{Biaye} (Biaye et al., 2005) [^{Ho} (Ho, 1989) as quoted in (Biaye et al., 2005) [

concerned, our results for Z = 2, at −0.32906 Ry, agree well with them respectively at −0.32380 Ry; −0.35913 Ry and −0.34080 Ry. For 2< Z ≤ 10, it can be seen the present MAOT results agrees with each other, and the agreement can be seen up to Z = 10 between our results at −11.56322 Ry and that from theoretical calculations [_{theo} = 0.3508. The good agreement between the results quoted in Tables 1-4 indicate the validity of this MAOT variational procedure and his merit to calculate the (nl^{2}) ground-state and doubly excite state energies.

In this work, the variational procedure of the Modified Atomic Orbital Theory (MAOT) has been applied for the first time to the calculations of the ground-state energy 2s^{2} ^{1}S and excitation energies of the doubly 2p^{2} ^{1}D, 3d^{2} ^{1}G, 4f^{2} ^{1}I excited states of the Helium isoelectronic sequence from Hydrogen ion H^{−} to Calcium ion Ca^{18+}. It has demonstrated the possibilities to construct a new correlated wave function adapted to the correct description of the electron-electron correlations phenomena in the ground and in the doubly excited nl^{2} states of the He-like systems. These very important results obtained in this work indicate the possibility to apply the MAOT variational procedure to the treatment of atomic spectra in two electron systems and probably in more complex atomic systems. The good results give also the possibility to analyze resonance energies via a very MAOT flexible procedure, in contrast to the complex procedures of experimental and theoretical methods based on the determination of the photoionization cross-section.

The authors are grateful to the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, the Orsay Institute of Molecular Sciences (OIMS), Paris, France and the Swedish International Development Agency (SIDA) for support.

The authors declare no conflicts of interest regarding the publication of this paper.

Sow, M., Sakho, I., Sow, B., Diouf, A., Gning, Y., Diop, B., Dieng, M., Diallo, A., Ba, M.D., Badiane, J.K., Biaye, M. and Wagué, A. (2020) Modified Atomic Orbital Calculations of Energy of the (2s^{2} ^{1}S) Ground-State, the (2p^{2} ^{1}D); (3d^{2} ^{1}G) and (4f^{2} ^{1}I) Doubly Excited States of Helium Isoelectronic Sequence from H^{−} to Ca^{18+}. Journal of Applied Mathematics and Physics, 8, 85-99. https://doi.org/10.4236/jamp.2020.81007

σ_{theo}: (theo = theory): screening constant σ determined theoretically

E: Energy

H: Hamilton operator

N: Normalization factor

T: Kinetic Energy

W: Coulombian interaction potential operator between the two electrons

C: Coulombian interaction potential between the nucleus and the two electrons

∇_{i}: denotes the Laplacian operator ∇ with reference to the coordinates of the vector radius r_{i} which detect the position of the electron i.

∇_{1}: is the Laplacian with reference to the coordinates of the vector radius r_{1} which detect the position of the electron 1.

∇_{2}: Laplacian defines the coordinates of the vector radius r_{2} which detect the position of the electron 2.

r_{i} = denotes the position r of the electron i relative to the nucleus

r_{1} and r_{2} = denote respectively the position r of the electron 1 and the electron 2 relative to the nucleus.

| r 1 − r 2 | : is an inter-electronic distance.

θ i ; φ i = angular variables of the electron i

S: total spin

ℏ = reduced Planck constant such as ℏ = h / 2 π with h the Planck constant

N = principal quantum number

l = orbitalar quantum number

Θ_{12}: angle between electron 1 and electron 2.

ξ : variational parameter zeta, a Greek alphabet

H^{−}: Hydrogen ion

He: Helium atom

Li^{+}: Lithium ion

Be^{2+}: Beryllium ion

B^{3+}: Boron ion

C^{4+}: Carbon ion

N^{5+}: Nitrogen ion

O^{6+:} Oxygen ion

F^{7+}: Fluorine ion

Ne^{8+}: Neon ion

Na^{9+}: Sodium ion

Mg^{10+}: Magnesium ion

Al^{11+}: Aluminium ion

Si^{12+:} Silicon ion

P^{13+}: Phosphorus ion

S^{14+}: Sulfur ion

Cl^{15+}: Chlorine ion

Ar^{16+}: Argon ion

K^{17+}: Potassium ion

Ca^{18+}: Calcium ion