^{1}

^{*}

^{2}

^{2}

^{1}

^{1}

In this paper, we have declined the formalism of the method of the Modified Atomic Orbital Theory (MAOT) applied to the calculations of energies of doubly excited states 2
*snp*, 3
*snp*, and 4
*snp* Helium-like systems. Then we also applied the variational procedure of the Modified Atomic Orbital Theory to the computations of total energies, excitation energies of doubly-excited states 2
*snp*, 3
*snp*, 4
*snp* types of Helium-like systems. The results obtained in this work are in good agreement with the experimental and theoretical values available.

Much theoretical research has revealed that the helium atom exhibits a strong electron-electron correlation. Since the early experiment by Madden and Codling [

So, most atomic spectra can be treated in term of singly excitation of singly or mixed configurations [

Various methods have been performed to understand electron-electron correlation effects in doubly ( N l , n l ′ ) L 2 S + 1 π excited states of He-like systems. Although many accurate data have been tabulated for these doubly excited states, the methods used require in general, complexity in the Variationnal procedure along with the use of computational codes.

Many theoretical studies have been done on doubly-excited states ( N l , n l ′ ) L 2 S + 1 π . Among these methods, we have the theoretical and experimental methods [^{2}, 2p^{2} and 3d^{2} states of He, Li^{+}, Be^{2+}, and B^{3+} [^{1}P_{o} helium-like systems [

In all these ab initio methods, energies of ( N l , n l ′ ) L 2 S + 1 π doubly-excited states of He isoelectronic sequence can’t be expressed in an analytical formula. In addition, most of these preceding methods require large basis-set calculations involving a fair amount of mathematics complexity.

The Modified Atomic Orbital Theory is a purely theoretical method initiated by Sakho [^{2}2p^{4} (^{1}D_{2}) ns, nd, 2s^{2}2p^{4} (^{1}S_{0}) ns, nd, and 2s^{2}2p^{5} (^{3}P_{2}) states from of the metastable 2s^{2}2p^{5} (^{2}P_{1/2}) state and the ground 2s^{2}2p^{5} (^{2}P_{3/2}) state of the Ne^{+} ion. Thus the variational principle, which is a purely theoretical method, takes advantage of the principle of variation. This variational method is a computational technique to provide approximate solutions to solving the Schrödinger equation. In the following, after a brief review of Slater’s orbital theory, we apply for the first time the variational procedure of the Modified Atomic Orbitals Theory to the calculations of total energies, excitation energies of doubly excited 2snp, 3snp, 4snp states of types helium-like systems. This procedure consists of determining the variational parameter α and the screening constant σ , from the construction of a correlated wave function.

In the context of the Modified Atomic Orbitals Theory (MAOT), the total energy of a ( ν l ) —given orbital is expressed as Rydberg units [

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

For the ( N l , n l ′ ) L 2 S + 1 π doubly excited states, the total energy of an atomic system of many M electrons is expressed as follows

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

In the construction of the correlated wave function, a product of hydrogen-type wave functions is performed in which variational parameters are introduced. Thus, in the case of atomic systems, these criteria are generally determined by the screen effects exerted by the electrons on each other by the spin-orbit interaction, etc.

The hydrogen wave functions for | n , l , m l 〉 states are radial and have the same shape. They are non-normed and it’s obtained from the radial coordinates (r) and an exponential factor.

So for different states, we get:

For 4s (l = 0):

R 4 , 0 ( r ) = 24 96 × ( Z a 0 ) 3 2 × ( 1 − 3 × Z 4 × a 0 × r + Z 2 8 × a 0 2 × r 2 − Z 3 192 × a 0 3 × r 3 ) × e − Z × r 4 × a 0 (3)

For 4p (l = 1):

R 4 , 1 ( r ) = 5 16 2 × ( Z a 0 ) 3 2 × ( Z a 0 × r − 1 4 × Z 2 a 0 2 × r 2 + 1 80 × Z 3 a 0 3 × r 3 ) × e − Z × r 4 × a 0 (4)

For 3s (l = 0):

R 3 , 0 ( r ) = 1 3 3 π × ( z a 0 ) 3 2 × ( 1 − 2 3 × Z a 0 × r + 2 27 × Z 2 a 0 2 × r 2 ) × e − ( Z × r 3 × a 0 ) (5)

For 3p (l = 1):

R 3 , 1 ( r ) = 2 × 2 27 × π × ( Z a 0 ) 3 2 × ( Z a 0 × r − Z 2 6 × a 0 2 × r 2 ) × e − ( Z × r 3 × a 0 ) (6)

For 2s (l = 0):

R 2 , 0 ( r ) = 2 4 × 2 π × ( Z a 0 ) 3 2 × ( 1 − Z 2 × a 0 × r ) × e − ( Z × r 2 × a 0 ) (7)

For 2p (l = 1):

R 2 , 1 ( r ) = 1 4 × 2 π × ( Z a 0 ) 3 2 × ( Z a 0 × r ) × e − ( Z × r 2 × a 0 ) (8)

To build the wave functions of ( N l , n l ′ ) L 2 S + 1 π type, the product of the radial portions R_{n,l}(r) is produced while considering the electrons (1) and (2) heliumoid systems, whose radial coordinates are respectively r_{1} and r_{2}. As part of the independent particle model where electronic correlation phenomena are neglected, (Coulombian repulsion, spin-orbit interaction, etc.), the product of the functions is given as follows:

For the function 2s2p:

2 s = ( 1 − Z 2 × a 0 × r 1 ) × e − ( Z × r 1 2 × a 0 ) and 2 p = ( Z a 0 × r 2 ) × e − ( Z × r 2 2 × a 0 )

Ψ ( 2s2p ) = [ ( 1 − Z 2 × a 0 × r 1 ) × ( Z a 0 × r 2 ) ] × e − Z 2 × a 0 × r 1 × e − Z 2 × a 0 × r 2 (9)

For the function 3s3p:

3 s = ( 1 − 2 × Z 3 × a 0 × r 1 + 2 × Z 2 27 × a 0 2 r 1 2 ) × e − ( Z 3 × a 0 × r 1 ) and 3 p = ( Z a 0 × r 2 − Z 2 6 × a 0 2 × r 2 2 ) × e − ( Z 3 × a 0 × r 2 )

Ψ ( 3s3p ) = ( ( 1 − 2 × Z 3 × a 0 × r 1 + 2 × Z 2 27 × a 0 2 r 1 2 ) × ( Z a 0 × r 2 − Z 2 6 × a 0 2 × r 2 2 ) ) × e − ( Z 3 × a 0 × r 1 ) × e − ( Z 3 × a 0 × r 2 ) (10)

For the function 4s4p:

4 s = ( 1 − 3 × Z 4 × a 0 × r + Z 2 8 × a 0 2 × r 2 − Z 3 192 × a 0 3 × r 3 ) × e − ( Z × r 1 4 × a 0 )

4 p = ( Z a 0 × r − 1 4 × Z 2 a 0 2 × r 2 + 1 80 × Z 3 a 0 3 × r 3 ) × e − ( Z × r 2 4 × a 0 )

Ψ ( 4s4p ) = ( ( 1 − 3 × Z 4 × a 0 × r 1 + Z 2 8 × a 0 2 × r 1 2 − Z 3 192 × a 0 3 × r 1 3 ) × ( Z a 0 × r 2 − 1 4 × Z 2 a 0 2 × r 2 2 + 1 80 × Z 3 a 0 3 × r 2 3 ) ) × e − ( Z × r 1 4 × a 0 ) × e − ( Z × r 2 4 × a 0 ) (11)

Taking into account the phenomena of electron-electron correlation effects occurring in He-like systems, the nuclear charge of the exponential factor is substituted in favor of the effective charge Z^{*}, and in atomic unit, the Bohr radius a_{0} = 1.

So these functions become:

For the wave function 2snp:

Ψ ( 2snp ) = ( ( 1 − Z 2 × a 0 × r 1 ) × ( Z a 0 × r 2 ) ) × e − Z * n ( r 1 + r 2 ) (12)

For the wave function 3snp:

Ψ ( 3snp ) = ( ( 1 − 2 × Z 3 × a 0 × r 1 + 2 × Z 2 27 × a o 2 r 1 2 ) × ( Z a 0 × r 2 − Z 2 6 × a 0 2 × r 2 2 ) ) × e − Z * n ( r 1 + r 2 ) (13)

For the wave function 4snp:

Ψ ( 4snp ) = ( ( 1 − 3 × Z 4 × a 0 × r 1 + Z 2 8 × a 0 2 × r 1 2 − Z 3 192 × a 0 3 × r 1 3 ) × ( Z a 0 × r 2 − 1 4 × Z 2 a 0 2 × r 2 2 + 1 80 × Z 3 a 0 3 × r 2 3 ) ) × e − Z * n ( r 1 + r 2 ) (14)

where the effective charge number Z^{*} is given by:

Z * = Z ( 1 − σ ( N l , n l ′ ) Z ) (15)

With σ ( N l , n l ′ ) the screen constant relating to these states.

To determine the screen constant, we start from the relation:

E ( α ) = 〈 H 〉 ( α ) = 〈 Ψ ( α ) | H | Ψ ( α ) 〉 〈 Ψ ( α ) | Ψ ( α ) 〉 (16)

And Hamiltonian of the helium isoelectronic series in given by (in atomic units):

H = − 1 2 Δ 1 − 1 2 Δ 2 − Z r 1 − Z r 2 + 1 r 12 (17)

The average value of this expression (17), while using the closure relation reflecting the fact that the | r 1 , r 2 〉 kets are continuous bases in the state space of the two electrons:

∬ d r 1 3 d r 2 3 | r 1 , r 2 〉 〈 r 1 , r 2 | = 1 l (18)

From this relation we can from (21):

E ( α ) ∬ d 3 r 1 d 3 r 2 〈 Ψ ( α ) | | r 1 , r 2 〉 × 〈 r 1 , r 2 | | Ψ ( α ) 〉 = ∬ d 3 r 1 d 3 r 2 〈 Ψ ( α ) | | r 1 , r 2 〉 H ^ 〈 r 1 , r 2 | | Ψ ( α ) 〉 (19)

The development of (19) gives:

E ( α ) ∬ d 3 r 1 d 3 r 2 Ψ ( r 1 , r 2 , α ) × Ψ * ( r 1 , r 2 , α ) = ∬ d 3 r 1 d 3 r 2 Ψ ( r 1 , r 2 , α ) H ^ Ψ * ( r 1 , r 2 , α ) (20)

The normalization constant denoted N is given by:

N E ( α ) = ∬ d 3 r 1 d 3 r 2 Ψ ( r 1 , r 2 , α ) H ^ Ψ * ( r 1 , r 2 , α ) (21)

And from this relation (24), we obtain:

N = ∬ d 3 r 1 d 3 r 2 | ( r 1 , r 2 , α ) | 2 (22)

To facilitate the development of these expressions, we made a change of variable of some parameters of the Equation (20). It was later that we posed in elliptical coordinates:

s = ( r 1 + r 2 ) ; t = ( r 1 − r 2 ) ; u = r 12 (23)

And the element of elementary volume gives:

d τ = d r 1 3 d r 2 3 = 2 π 2 ( s 2 − t 2 ) u d s d t d u (24)

Applying these changes of variables in Equation (23) the preceding expression of the normalization constant denoted N is in elliptic coordinate:

N E ( α ) = ∫ 0 ∞ d s ∫ 0 s d u ∫ 0 u d t { u ( s 2 − t 2 ) × [ ( ∂ Ψ ∂ s ) 2 + ( ∂ Ψ ∂ t ) 2 + ( ∂ Ψ ∂ u ) 2 ] + 2 ( ∂ Ψ ∂ u ) × [ s ( u 2 − t 2 ) ∂ Ψ ∂ s + t ( s 2 − u 2 ) ∂ Ψ ∂ t ] − Ψ 2 [ 4 Z s u − s 2 + t 2 ] } (25)

Since we did not take into account the Coulomb repulsion, so: ∂ Ψ ∂ u = 0 .

The normalization constant becomes:

N E ( α ) = ∫ 0 ∞ d s ∫ 0 s d u ∫ 0 u d t { u ( s 2 − t 2 ) × [ ( ∂ Ψ ∂ s ) 2 + ( ∂ Ψ ∂ t ) 2 ] − Ψ 2 [ 4 Z s u − s 2 + t 2 ] } (26)

To determine the values of the screen constant σ and the variational parameter α , we start from this equation, which is the sum of three integral data as follows:

N E ( α ) = E 1 ( α ) + E 2 ( α ) + E 3 ( α ) (27)

The development of this expression (27) makes it possible to obtain the value of σ and α by the formula:

d E ( α i ) d α i = 0 (28)

The expressions corresponding to E 1 ( α ) , E 2 ( α ) , and E 3 ( α ) , are:

E 1 ( α ) = ∫ 0 ∞ d s ∫ 0 s d u ∫ 0 u d t u ( s 2 − t 2 ) × ( ∂ Ψ ∂ s ) 2 (29)

E 2 ( α ) = ∫ 0 ∞ d s ∫ 0 s d u ∫ 0 u d t u ( s 2 − t 2 ) × ( ∂ Ψ ∂ t ) 2 (30)

E 3 ( α ) = − ∫ 0 ∞ d s ∫ 0 s d u ∫ 0 u d t [ 4 Z s u − s 2 + t 2 ] Ψ 2 (31)

The normalization constant is as follows:

N = ∫ 0 ∞ d s ∫ 0 s d u ∫ 0 u d t u ( s 2 − t 2 ) × Ψ 2 (32)

With these changes of variables, the correlated wave functions of the states 2snp, 3snp, and 4snp become:

Ψ ( 2s2p ) = − 1 8 × ( ( s − t ) × ( s × z + t × z − 4 ) ) × e − α s (33)

Ψ ( 3s3p ) = 1 1296 × [ z × ( s − t ) × ( s × z − t × z − 12 ) × ( s 2 × z 2 + 2 × s × t × z 2 − 18 × s × z + t 2 × z 2 − 18 × t × z + 54 ) ] × e − α s (34)

Ψ ( 4s4p ) = − 1 983040 × ( z × ( s − t ) × ( s 2 × z 2 − 2 × s × t × z 2 − 40 × s × z + t 2 × z 2 + 40 × t × z + 320 ) × ( s 3 × z 3 + 3 × s 2 × t × z 3 − 48 × s 2 × z 2 + 3 × s × t 2 × z 3 − 96 × s × t × z 2 + 576 × s × z + t 3 × z 3 − 48 × t 2 × z 2 + 576 × t × z − 1536 ) ) × e − α s (35)

In this part, the procedure consists of determining the final expressions of energies, the value of the variational parameter α , and the screen constant σ . Since the calculations used are very complex, and require a lot of changes of variables, with matrices to be manipulated, we have found it necessary to make a first call to a computer program with the software matlab. In this program, we first defined the parameter s,t,u,α, andz of Equation (23), the expression of the derivative as a function of each parameter, and the square of its derivatives. In a second step, the expressions of (E_{1},E_{2},E_{3} and N) of the Equations (29)-(32), as well as their factorials were defined and detailed expression by expression. Then, to simplify some parameters, a matrix calculation was carried out in this program, and relations between these matrices were made to obtain a simple expression of the Equation (27) in order to apply the formula of the Equation (28) to have the approximate values of the screen constant σ and the variational parameter α .

In the case of the variational calculation of the Modified Atomic Orbital theory (MAOT, the expression of the total energy of the doubly-excited states (Nsnp) of an orbital is given by the formula (in Rydberg):

E ( N s n p ) = − ( ( Z − σ ( n s ) ) 2 N 2 ) − ( ( Z − σ ( n p ) ) 2 n 2 ) (36)

With N ≠ n and σ ( n s ) = σ ( n p ) .

In some cases, a corrective factor may be added to this expression to obtain results that are closer to those found in the literature consulted.

Thus the expressions of the states 2snp, 3snp, and 4snp are detailed as follows:

· For the state 2snp:

E ( 2 s 2 p ) = − ( ( Z − σ ( n s ) ) 2 N 2 ) − ( ( Z − σ ( n p ) ) 2 n 2 ) (37)

With σ ( n s ) = σ ( n p ) and N = n .

· For the state 3snp:

E ( 3s3p ) = − ( ( Z − σ ( n s ) ) 2 N 2 ) − ( ( Z − σ ( n p ) ) 2 n 2 ) (38)

With σ ( n s ) = σ ( n p ) et N = n .

· For the state 4snp:

E ( 4s4p ) = − ( ( Z − σ ( n s ) ) 2 N 2 ) − ( ( Z − σ ( n p ) ) 2 n 2 ) (39)

With σ ( n s ) = σ ( n p ) and N = n .

The determination of the variational parameter α comes from the expression (28) with:

E ( α i ) = ∑ i = 1 3 ( E i N ) (40)

Thus the calculation program is presented in the Appendix, and the variational parameter α of the states 2s2p, 3s3p and 4s4p is given as follows:

α = n + l + l ′ n Z ( 1 − σ ( N l , n l ′ ) Z ) (41)

For the state 2s2p:

α 2 ( 2s2p ) ≈ 3 2 Z ( 1 − 1 2 × 1 Z ) (42)

With σ ( 2s2p ) = 0.5 .

For the state 3s3p:

α 3 ( 3s3p ) ≈ 4 3 Z ( 1 − 3 12 × 1 Z ) (43)

With σ ( 3s3p ) = 0.25 .

For the state 4s4p:

α 4 ( 4s4p ) ≈ 5 4 Z ( 1 − 3 7 × 1 Z ) (44)

With σ ( 4s4p ) = 0.428 .

In this work, the results obtained are compared with those found in the theoretical and experimental literature. We have calculated the total energies of the states (3snp ^{1}P^{o}), (2snp ^{1}P^{o}), (4snp ^{1}P^{o}) as well as the excitation energies of the states (3snp ^{1}P^{o}), (2snp ^{1}P^{o}), (4snp ^{1}P^{o}). For the states (3snp ^{1}P^{o}) the total energies are given in Rydberg and in eV, shown in ^{1}P^{o}), (4snp ^{1}P^{o}) states, their results are given in eV and represented in ^{1}P^{o}), (2snp ^{1}P^{o}), (4snp ^{1}P^{o}) respectively. About excitations energies, we have taken the energies of the ground state given by Frankowski and Pekeris [^{+} (−7.27991), Be^{2+} (−13.65556), Be^{3+} (−22.03097).

In Tables 1-4, we used the variational computation of the modified atomic orbitals theory (MAOT) of the energies doubly-excited states (3snp ^{1}P^{o}), (2snp ^{1}P^{o}), (4snp ^{1}P^{o}). We compared the results obtained with theoretical results for all of these states, and experimental results existing only for the (3s3p ^{1}P^{o}), (2s2p ^{1}P^{o}), helium (He) states of Kossmann et al. [^{1}P^{o}), lithium (Li^{+}) from Diehl et al. [^{1}P^{o}) from Woodru et al. [

States | Z | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ||

3s3p ^{1}P^{o } | −E^{p} | 0.68056 | 1.68056 | 3.12500 | 5.01389 | 7.34722 | 10.12500 | 13.34722 | 17.01389 | 21.12500 |

−E^{s} | 0.66054 | 1.64784 | 3.07958 | 4.95577 | 7.27640 | 10.04147 | 13.25099 | 16.90496 | 21.00337 | |

−E^{a} | 0.67140 | 1.659 40 | 3.09000 | 4.96600 | 7.28600 | 10.04800 | 13.25600 | 16.91000 | 21.00000 | |

−E^{b} | 0.66268 | 1.67395 | 3.15417 | 5.10468 | 7.52607 | 10.41871 | 13.78253 | 17.61787 | 21.92494 | |

3s4p ^{1}P^{o} | −E^{p} | 0.53168 | 1.31293 | 2.44141 | 3.91710 | 5.74002 | 7.91016 | 10.42752 | 13.29210 | 16.50391 |

−E^{s} | 0.53206 | 1.31269 | 2.44053 | 3.91561 | 5.73790 | 7.90742 | 10.42415 | 13.28811 | 16.49930 | |

−E^{a} | 0.54240 | 1.31960 | 2.44400 | 3.91400 | 5.73000 | 7.89600 | 10.40800 | 13.26600 | 16.47200 | |

3s5p ^{1}P^{o} | −E^{p} | 0.46278 | 1.14278 | 2.12500 | 3.40944 | 4.99611 | 6.88500 | 9.07611 | 11.56944 | 14.36500 |

−E^{s} | 0.47259 | 1.15756 | 2.14475 | 3.43416 | 5.02579 | 6.91965 | 9.11573 | 11.61403 | 14.41455 |

P: Present results obtained from Equation (38); s: (Sakho et al., 2010) [

States | Z | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ||

3s3p ^{1}P^{o} | −E^{p} | 9.26 | 22.87 | 42.52 | 68.22 | 99.96 | 137.76 | 181.60 | 231.49 | 287.42 |

−E^{s} | 9.10 | 22.47 | 41.88 | 67.34 | 98.85 | 136.40 | 180.01 | 229.66 | 285.35 | |

−E^{h} | 8.28 | 21.14 | 40.05 | 65.01 | 96.01 | 133.06 | 176.16 | 225.30 | 280.49 | |

−E^{k} | 9.10 | 22.33 | 42.04 | 67.52 | 99.03 | |||||

−E^{l} | 9.11 | 22.54 | 42.00 | 67.51 | ||||||

−E^{i} | 9.10 | |||||||||

3s4p ^{1}P^{o} | −E^{p} | 7.23 | 17.86 | 33.22 | 53.29 | 78.10 | 107.62 | 141.87 | 180.85 | 224.55 |

−E^{s} | 7.66 | 18.42 | 33.90 | 54.10 | 79.03 | 108.68 | 143.06 | 182.16 | 225.98 | |

3s5p ^{1}P^{o} | −E^{p} | 6.30 | 15.55 | 28.91 | 46.39 | 67.98 | 93.68 | 123.49 | 157.41 | 195.45 |

−E^{s} | 7.02 | 16.58 | 30.25 | 48.03 | 69.93 | 95.94 | 126.05 | 160.05 | 198.63 |

P: Present results obtained from Equation (38); s: (Sakho et al., 2008) [

States | Z | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ||

2s2p ^{1}P^{o} | −E^{p } | 17.96 | 46.88 | 89.39 | 145.52 | 215.25 | 298.58 | 395.52 | 506.07 | 630.22 |

−E^{s} | 18.88 | 47.76 | 90.24 | 146.34 | 216.03 | 299.33 | 396.24 | 506.76 | 630.88 | |

E^{α} | 18.86 | 47.82 | 90.33 | 146.40 | 216.07 | 299.32 | 396.18 | 506.64 | 630.70 | |

−E^{h} | 19.42 | 48.23 | 90.63 | 146.66 | 216.28 | 299.51 | 396.34 | 506.78 | 630.84 | |

−E^{j} | 18.87 | 47.84 | 90.34 | 146.42 | 216.09 | 299.34 | 396.20 | 506.20 | 630.84 | |

−E^{f,i } | 18.88^{i } | 47.78f | ||||||||

2s3p ^{1}P^{o} | −E^{p } | 15.05 | 37.16 | 69.09 | 110.85 | 162.44 | 223.86 | 295.10 | 376.16 | 467.06 |

−E^{s} | 15.95 | 38.23 | 70.34 | 112.28 | 164.04 | 225.63 | 297.05 | 378.29 | 469.36 | |

−E^{h} | 15.95 | 37.99 | 69.86 | 111.55 | 163.07 | 224.41 | 295.59 | 376.58 | 467.41 |

P: Present results obtained from Equation (37); s: (Sakho et al., 2008) [

et al. [

Thus in ^{1}P^{o}), we have calculated the total energies of doubly-excited states types (3snp ^{1}P^{o}) ranging from (Z = 2 to 10) using Equation (38).

The results found are in perfect agreement with those found in the theoretical and experimental literature consulted and quoted above. For the (3s3p ^{1}P^{o}) helium (He) states, we compared our results with those obtained experimentally by Kossman et al. [

In ^{1}P^{o}), and (4s4p ^{1}P^{o}), we used the Equations ((37), (39)) respectively. Then we added to each of these equations a corrective factor to obtain results equivalent to those found in the theoretical and experimental. In ^{1}P^{o}), we calculated the total energies of doubly excited states of helium-like systems (Z = 2 to 10). Our results found are in good agreement with the theoretical results [

In ^{1}P^{o}), we also calculated the total energies of doubly-excited states of helium-like systems (Z = 2 to 10). The results found are in

State | Z | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ||

4s4p ^{1}P^{o } | −E^{p } | 5.24 | 12.91 | 23.98 | 38.46 | 56.33 | 77.61 | 102.29 | 130.37 | 161.85 |

−E^{s} | 5.35 | 13.03 | 24.10 | 38.58 | 56.46 | 77.75 | 102.43 | 132.03 | 162.00 | |

E^{α} | 5.29 | 12.95 | 24.01 | 38.46 | 56.31 | 77.56 | 102.43 | 130.27 | 161.72 | |

−E^{m} | 5.35 |

P: Present results obtained from Equation (39); m: Experimental data, Woodruff and Samson (1982) [

States^{ } | ^{ } | Z | |||
---|---|---|---|---|---|

^{ } | 2 | 3 | 4 | 5 | |

2s2p ^{1}P^{o} | E^{p} | 61.05 | 151.22 | 282.19 | 453.98 |

E^{s} | 60.13 | 150.34 | 281.35 | 453.15 | |

E^{j} | 60.13 | ||||

E^{f, i } | 60.13^{i} | 150.31^{f} | |||

2s3p ^{1}P^{o} | E^{p } | 63.97 | 160.94 | 302.50 | 488.64 |

E^{s} | 63.06 | 159.87 | 301.25 | 487.21 | |

3s3p ^{1}P^{o} | E^{p } | 69.75 | 175.23 | 329.07 | 531.28 |

E^{s} | 69.91 | 175.63 | 330.54 | 532.15 | |

E^{i} | 69.91 | ||||

3s4p ^{1}P^{o} | E^{p } | 71.78 | 180.23 | 338.37 | 546.20 |

E^{s} | 71.35 | 179.68 | 337.69 | 545.39 | |

3s5p ^{1}P^{o} | E^{p } | 72.72 | 182.55 | 342.67 | 553.11 |

E^{s} | 71.99 | 181.52 | 341.34 | 551.46 | |

4s4p ^{1}P^{o} | E^{p } | 73.78 | 185.19 | 347.60 | 561.04 |

E^{s} | 73.66 | 185.07 | 347.49 | 560.91 | |

E^{m} | 73.66 |

perfect agreement with those found in the literature consulted.

In ^{1}P^{o}) (N, n ≤ 5) types of helium and its assimilated ions. They are calculated from the energies of the ground state given by Frankowski and Pekeris [

In a global way, we applied the variational procedure of the modified atomic orbitals theory for the computation of total energies and excitation senergies doubly-excited states of the atomic system with several electrons. In order to achieve our results, we used a matlab program for the first time to reduce the complexity of the calculations. This program allowed us to determine the approximate expressions of the variational parameter, and of the screen constant.

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

Diallo, A., Sakho, I., Badiane, J.K., Ba, M.D. and Tine, M. (2021) Variational Calculation of the Doubly-Excited States Nsnp of He-Like Ions via the Modified Atomic Orbitals Theory. Journal of Modern Physics, 12, 105-121. https://doi.org/10.4236/jmp.2021.122011

The procedure for determining the radial wave function is given as follows:

R n , l ( r ) = { ( 2 Z n a 0 ) 3 ( n − l − 1 ) ! 2 n [ ( n + l ) ! ] 3 } 1 / 2 e − z r n a 0 ( 2 Z r n a 0 ) l L n + l 2 l + 1 ( 2 Z r n a 0 ) (A1)

The associated Laguerre polynomials are linked to the Laguerre polynomials L n + l ( r ) by the Rodrigue formula:

L n k ( r ) = ( − 1 ) d k d r k L n ( r ) (A2)

L n ( r ) = e r d n d r n ( r n e − r ) (A3)

For different values of n and l, the Laguerre polynomials are mutually orthogonal, which then determines the orthogonality of the radial wave functions.

Let’s give the example of the 4s wave function:

For the state 4s we have: n = 4, l = 0

L n + l 2 l + 1 ( 2 Z r n a o ) = L 4 1 ( 2 Z r n a o ) ⇒ d d r ( L 4 ( r ) ) ( 2 Z r n a o ) (A4)

And;

L 4 = e r d 4 d r 4 ( r n e − r ) (A5)

By developing this expression, we get:

L 4 ( r ) = e r d 3 d r 3 ( 4 r 3 e − r − r 4 e − r ) = e r d 2 d r 2 ( 12 r 2 e − r − 4 r 3 e − r − 4 r 3 e − r + r 4 e − r )

L 4 ( r ) = e r d d r ( 24 r e − r − 12 r 2 e − r − 12 r 2 e − r + 4 r 3 e − r − 12 r 2 e − r + 4 r 3 e − r + 4 r 3 e − r − r 4 e − r )

L 4 ( r ) = e r ( 24 e − r − 24 r e − r − 24 r e − r + 12 r 2 e − r − 24 r e − r + 12 r 2 e − r + 12 r 2 e − r − 4 r 3 e − r − 24 r e − r + 12 r 2 e − r + 12 r 2 e − r − 4 r 3 e − r + 12 r 2 e − r − 4 r 3 e − r − 4 r 3 e − r + r 4 e − r )

L 4 ( r ) = ( 24 − 96 r + 72 r 2 − 16 r 3 + r 4 ) (A6)

Then he comes:

L 4 1 = d d r L 4 ( r ) = 4 ( − 24 + 36 × r − 12 × r 2 + r 3 ) (A7)

Which give,

L n + l 2 l + 1 ( 2 Z × r n a 0 ) = L 4 1 ( 2 Z × r n a 0 ) = ( − 4 ) × ( − 24 × ( 2 Z 4 a 0 ) 0 + 36 × r ( 2 Z 4 a 0 ) 1 − 12 × r 2 ( 2 Z 4 a 0 ) 2 + r 3 ( 2 Z 4 a 0 ) 3 ) (A8)

So the determination of the first part of the expression (A1)

{ ( 2 Z n a 0 ) 3 × ( n − l − 1 ) ! 2 n × [ ( n + 1 ) ! ] 3 } 1 2

For n = 4and l = 0, we have:

{ ( 2 Z n a 0 ) 3 ( n − l − 1 ) ! 2 n × ( ( n + l ) ! ) 3 } 1 2 = { ( Z 2 × a 0 ) 3 × ( 6 8 × 24 3 ) } 1 2 = { ( Z 2 × a 0 ) 3 × ( 3 4 × 24 3 ) 1 2 } = { ( Z 2 × a 0 ) 3 2 ( 1 96 × 2 ) } = { 1 4 × 96 ( Z a 0 ) 3 2 } (A9)

{ ( 2 Z n a 0 ) 3 ( n − l − 1 ) ! 2 n × ( ( n + l ) ! ) 3 } 1 2 = { 1 4 × 96 ( Z a 0 ) 3 2 } (A10)

Thus, starting from (A8) and (A10);

{ 1 4 × 96 ( Z a 0 ) 3 2 } × ( − 4 ) × ( − 24 + 18 × Z r a 0 − 3 × Z 2 r 2 a 0 2 + Z 3 r 3 8 a 0 3 ) = [ 1 96 ( 24 − 18 Z a 0 r + 32 Z 2 a 0 2 r 2 − Z 3 8 a 0 3 r 3 ) ( Z a 0 ) 3 2 ] (A11)

Simplifying by 24 we finally obtain the expression of the radial wave function 4s as follows:

R 4 , 0 = 24 96 ( Z a 0 ) 3 2 ( 1 − 3 Z 4 a 0 r + Z 2 8 a 0 2 r 2 − Z 3 192 a 0 3 r 3 ) e − Z × r 4 a 0 (A12)

By analogy the wave function 4p is given as follows:

R 4 , 1 = 5 16 2 ( Z a 0 ) 3 2 ( Z a 0 r − 1 4 Z 2 a 0 2 r 2 + 1 80 Z 3 a 0 3 r 3 ) e − Z × r 4 a 0 (A13)

Appendix B: Principle of Determining the Screen ConstantTo determine the screen constant, we start from the relation:

E ( α ) = 〈 H 〉 ( α ) = 〈 Ψ ( α ) | H | Ψ ( α ) 〉 〈 Ψ ( α ) | Ψ ( α ) 〉 (B1)

And the Hamiltonian H (in atomic unit) is:

H = − 1 2 Δ 1 − 1 2 Δ 2 − z r 1 − z r 2 + 1 r 12 (B2)

The average value of this expression (B2), while using the closure relation reflecting the fact that the | r 1 , r 2 〉 kets are continuous bases in the state space of the two electrons:

∬ d r 1 3 d r 2 3 | r 1 , r 2 〉 〈 r 1 , r 2 | = 1 l (B3)

From this relation we can from (B3):

E ( α ) ∬ d 3 r 1 d 3 r 2 〈 Ψ ( α ) | | r 1 , r 2 〉 × 〈 r 1 , r 2 | | Ψ ( α ) 〉 = ∬ d 3 r 1 d 3 r 2 〈 Ψ ( α ) | | r 1 , r 2 〉 H ^ 〈 r 1 , r 2 | | Ψ ( α ) 〉 (B4)

The development of (B4) gives:

E ( α ) ∬ d 3 r 1 d 3 r 2 Ψ ( r 1 , r 2 , α ) × Ψ * ( r 1 , r 2 , α ) = ∬ d 3 r 1 d 3 r 2 Ψ ( r 1 , r 2 , α ) H ^ Ψ * ( r 1 , r 2 , α ) (B5)

The normalization constant denoted N is given by:

N E ( α ) = ∬ d 3 r 1 d 3 r 2 Ψ ( r 1 , r 2 , α ) H ^ Ψ * ( r 1 , r 2 , α ) (B6)

And from this relation (B6), we obtain:

N = ∬ d 3 r 1 d 3 r 2 | ( r 1 , r 2 , α ) | 2 (B7)

To facilitate the development of these expressions, we made a change of variable of some parameters of the Equation (B5). It was later that we posed in elliptical coordinates:

s = ( r 1 + r 2 ) ; t = ( r 1 − r 2 ) ; u = r 12 (B8)

And the element of elementary volume gives:

We know that, d τ = d r 1 3 d r 2 3

d τ = d r 1 3 d r 2 3 = 2 π 2 ( s 2 − t 2 ) u d s d t d u (B9)

Applying these changes of variables in Equation (B7) the preceding expression of the normalization constant denoted N is in elliptic coordinate:

N E ( α ) = ∫ 0 ∞ d s ∫ 0 s d u ∫ 0 u d t { u ( s 2 − t 2 ) × [ ( ∂ Ψ ∂ s ) 2 + ( ∂ Ψ ∂ t ) 2 + ( ∂ Ψ ∂ u ) 2 ] + 2 ( ∂ Ψ ∂ u ) × [ s ( s 2 − t 2 ) ∂ Ψ ∂ s + t ( s 2 − u 2 ) ∂ Ψ ∂ t ] − Ψ 2 [ 4 Z s u − s 2 + t 2 ] } (B10)

Since we did not take into account the Coulomb repulsion, so: ∂ Ψ ∂ u = 0

The normalization constant becomes:

N E ( α ) = ∫ 0 ∞ d s ∫ 0 s d u ∫ 0 u d t { u ( s 2 − t 2 ) × [ ( ∂ Ψ ∂ s ) 2 + ( ∂ Ψ ∂ t ) 2 ] − Ψ 2 [ 4 Z s u − s 2 + t 2 ] } (B11)

To determine the values of the screen constant σ and the variational parameter α , we start from this equation, which is the sum of three integral data as follows:

N E ( α ) = E 1 ( α ) + E 2 ( α ) + E 3 ( α ) (B12)

The development of this expression (B12) makes it possible to obtain the value of σ and α by the formula:

d E ( α i ) d α i = 0 (B13)

The expressions corresponding to E 1 ( α ) , E 2 ( α ) , and E 3 ( α ) , are:

E 1 ( α ) = ∫ 0 ∞ d s ∫ 0 s d u ∫ 0 u d t u ( s 2 − t 2 ) × ( ∂ Ψ ∂ s ) 2 (B14)

E 2 ( α ) = ∫ 0 ∞ d s ∫ 0 s d u ∫ 0 u d t u ( s 2 − t 2 ) × ( ∂ Ψ ∂ t ) 2 (B15)

E 3 ( α ) = − ∫ 0 ∞ d s ∫ 0 s d u ∫ 0 u d t [ 4 Z s u − s 2 + t 2 ] Ψ 2 (B16)

The normalization constant is as follows:

N = ∫ 0 ∞ d s ∫ 0 s d u ∫ 0 u d t u ( s 2 − t 2 ) × Ψ 2 (B17)

With these changes of variables, we obtain the equations presented above in section (2.3): Equation (33; 34; 35).