Study of atomic properties for subshells of the systems have Z=12-16 by using Hartree-Fock approximation

The atomic properties are important to describe of the dynamics in atoms for the two-electron atomic systems studied in this work include the atom Mg, and like ions Al+1, Si+2, P+3 and S+4, Hartree-Fock approximation are used to determine the atomic properties like one electron radial density function D(r1)and its expectation value 〈r1n〉 , inter electron distribution function f(r 12) and its expectation value 〈r12n〉 , standard deviation for one and two electrons Δr 1 Δr 12, expectation values for all energies ⟨Ven ⟩, ⟨Vee ⟩, ⟨V⟩, ⟨T⟩, ⟨E⟩


1-Introduction
Classical physics is dominated by two fundamental concepts. The first is the concept of a particle, a discrete entity with definite position and momentum which moves in accordance with Newton's laws of motion. The second is the concept of an electromagnetic wave, an extended physical entity with a pres-ence at every point in space that is provided by electric and magnetic fields which change in accordance with Maxwell's laws of electromagnetism. The classical world picture is neat and tidy: the laws of particle motion account for the material world around us and the laws of electromagnetic fields account for the light waves which illuminate this world. Numerical computational methods for atomic Hartree-Fock equations have been developed by Froese Fischer. The one-particle Green's function approach and related manybody methods have been extensively used to calculate ionization and electron attachment spectra of atoms and molecules. Detailed accounts of the diverse techniques developed in this field and an overview of applications can be found in recent review articles [4,5].

2-theory
The Hartree-Fock (HF) approximation ignores the correlation between electrons but gives roughly 99% of the total electronic energy. We start from the standard N-electron Hamiltonian (in Hartree atomic units = ℏ = = 1 used throughout) Where: ̂ the electron kenitic energy ̂ attractive electron-nucleus potential operator ̂ is the two-electron Coulomb repulsion operator Where is ∇ 2 the Laplacian operator are the electron-nucleus distances for the electron(1)and electron(2) is the interelectronic distances The introduction of this effective potential means that each electron is subjected to a field that models the effect of the other electrons in the system. In The Hatree-Fock approximation, the many body wave function ( 1 , 2 , … … ) approximated by a single slater determinant . We can write wave function as The factor 1 √ ! ensures the normalization condition on the wavefunction . Here the variables xi include the coordinates of spin and space , ( ) terms are called spin orbitals and these spin orbitals are orthonormal functions, which are spatial orbitals times a spin functions. The wave function Ψ in equation (6) is clearly antisymmetric because interchanging any pair of particles is equivalent to interchanging two columns and hence changes the sign of the determinant. Moreover, if any pair of particles are in the same single-particle state, then two rows of the Slater determinant are identical and the determinant vanishes, in agreement with the Pauli exclusion principle.
The one-electron orbitals used to construct the each consist of a radial function ( ) a spherical harmonic ℓ ( , ) and a spin function The spatial part of one-electron spin orbital may be expressed as linear combination of Slater type orbital called basis functions ,Φ ( , , ) = ∑ Where the expansion coefficient determined by minimizing the energy using one of several procedures and this process is continued until ̂ and Ψ converge, at which point a self-consistent field (SCF) has been achieved. This usually yields the lowest-energy single determinant within the basis. is a Slater type orbital.
The radial part is: Here, > 0 is the orbital exponent. The quantity n occurring in eq. (9) is a positive principal quantum number of (STO). The determination of nonlinear parameters n and is very important for describing the atomic orbitals.

(2-2) Two-electron radial density distribution D (r1, r2)
The two-particle radial density distribution D (r1, r2) in each individual electronic shell is defined by: ( 1 , 2 ) = 1 Such that: This density function is a measure of the probability of finding the two-electrons such that, simultaneously, their radial coordinates are r1 and r2 respectively

(2-3) One -Particle Radial Density Function D(r 1 )
The radial density distribution function D (r1) is a measure of the probability of finding an electron in each shell and it is defined as [8]:

(2-4)One-particle expectation value〈 〉
The one-particle expectation value 〈r 1 n 〉 can be calculated from [9]: In the case (n=0) one can calculate the normalization condition as a result of 〈 1 〉

(2-5) Standard Deviation
The standard deviation of the distance of the test electron r1 from the nucleus, is defined as [10]:

(2-6) inter-particle distribution function f(r 12 )
The radial electron-electron distribution function, which describes the probability of locating two electrons separated by distance 12 from each other ( 12 ) representing a function of the distribution of the distance between electron 1 and electron 2 (2-7) The inter-particle expectation values〈 〉 The inter-particle expectation values 〈r 12 n 〉 can be calculated from [11] 〈r 12 n 〉 = ∫ ( 12 ) 12 12 (22)

(2-8) Standard Deviation ∆
The standard deviation of the inter electronic distance of the two electrons, is defined as [11] ∆ = √〈 〉 − 〈 〉 (23)       Fig. (1) observed when the distance equal to zero from the nucleus, the probability of finding an electron equal to Zero {when r =0 D(r 1 ) =0}. This means that the electron is not found in the nucleus and when the distance is far from nucleus, the probability of finding an electron equal to Zero also{ when = ∞ ( 1 ) = 0 }.That means the electron is not found out the atom. 2) when the positions of maximum values decrease when atomic number increases. these results agree with Coulomb law because increasing in attraction force leads to decreasing in the positions towards the nucleus 3) When r 12 =0 , f(r 12 )=0 ,and when 12 = ∞ , ( 12 ) = 0 , because the coulomb interaction neglect, when the distance between two electrons is very large   4) In the framework, we found the one-particle expectation value 〈 1 〉 increases when the atomic number Z increases, when n take negative values -2,-1 , where the 〈 1 −1 〉 related to the attraction energy expectation value 〈 〉 = − [ 〈 1 −1 〉] , N represents the number of electrons in the shell(the number of electrons in the Present work=2 for atom) 5) .The attraction energy expectation values〈 〉 are larger than the repulsion energy expectation value 〈 〉 because the distances between the electrons and the nucleus are smaller than the distances between the electrons

Conclusions
From these results, we Conclude 1) when the atomic number Z increases , for approximations the one-particle radial density distribution function D(r 1 ) and the inter-particle distribution function ( 12 ) are increased. 2) When Z increases the positions of these maximum values for approximations the oneparticle radial density distribution function D(r 1 ) ,and inter-particle distribution function ( 12 ) decreases. 3) For both one-particle expectation 〈 1 〉 , and two particle expectation 〈 12 〉 increase when Z increase and when m = -2,-1 and both decrease for m = +2 ,+1 4) The standard deviation of one-particle ∆ 1 and two-particle ∆ 12 decreases for all systems when the atomic number increases . 5) . All the expectation values of the energies〈 〉 , 〈 〉 , 〈 〉 , 〈 〉 〈 〉 , and increase when the atomic number increases.