Calculation of the atomic properties of excited states for B-atom using Hartree – Fock approximation 1

In this paper were studied some atomic properties for Boron atom and like ions such as ( C +1 and N +2 ) for different excited states(1S2S nS)where nequal (3,4,5). This system consists of five –electron : two electron each in(1S,2S-Level) with same quantum number except spin component ,where the first electron with spin up (α)and the other –down(β) and the Fifth electron puts in(n S-Level)with spin up(α).using Hartree –Fock approximation in position spaceand electron density at the nucleus ρ(0) have been calculated for these states of the same atom .


Introduction
The Schrödinger equation for atoms (ions) with more than one electron has not been solved analytically.Approximate methods must be applied in order to obtain the wave functions or another physical attributes from quantum mechanical calculation [1].
The Hartree-Fock(HF) equations were first proposed by Fock in 1930.sincethen, the Hartree-Fock method has taken a central role in studies of atomic and molecular electronic structure the development of effective computational methods for Hartree-Fock equations there are two computation methods for Hartree-Fock equations numerical and algebraic [2].Numerical computational methods for atomic HF equations have been developed by Froese Fischer [3,4].
The Hartree-Fock method(HF) is known to be successful in calculating properties of electron systems, in particular , the ground state properties of atoms.Based on a variational principle ,the HF method estimate the ground state energy E of the electron system ,from above,i.e ≥ ,where   is the ground state energy calculated within the HF method if the ground state wave function of N-electrons is approximated by a single N-electron Slater determinant, the HF solution delivers minimum value   on the set of all such determinants.Agreement, or otherwise, with the HF result is often used to estimate the success of other approximate computational schemes [5] 1.Two-particle density distribution function Г  (  ,   ) The Two-particle density   (, ) for N electron system is given by [6]     Where N is the Binomial factor defined as [6].
Where N is the number of electrons within system and   …..  indicates that the integration is over N electrons except i and j .Since the partitioning technique enable correlation to be examined in depth for various intra and inter shells electron pairs thus we had employed this technique to pair-wise components (p,q) [7].
Where Ψ is the occupied normalized Hartree -Fock spin orbital, the symbol (p,q) represent the spin orbital labels and (  ,   ) indicates the electron labels [7].The standard deviation of distance of the test electron from the nucleus is defined as [10][11].
While the standard deviation of the inter electronic distance of the two electrons, is defined as .Coulson and Neilson proposed a distribution function for inter-electronic separation of    12 of s-states associated with spin -orbital pair (p,q) 11 .
Where the function    12 is the probability distribution distance between electron 1 and electron 2; this function of great importance in both Fermi and Coulomb holes and  12 is the distance between the two electrons, this function satisfies the normalization condition [12][13].

5.Electron density at the nucleus ρ(0)
The electron density at the nucleus can be evaluated using the following form [14] .

Results and discussion
The results obtained in the present of one-particle expectation values for different powers (k =-2 to 2)in addition to calculate the standard deviation for different excited states (1 2 2 2 3 1 ), (1 2 2 2 4 1 )  (1 2 2 2 5 1 )of B-like ions up to Z=7 are listed in tables (1,2,3).From this tables we noted effect the increase in atomic number with fixed the number of electrons, where observed when (k ) takes negative values the expectation values of  1  increase with increasing the atomic number while when(k ) takes positive values the expectation values of  1  are decreases because the distance between the electron and the nucleus become smallest as nuclear charge increases .Normalization condition has been applied for all wave functions.The values of standard deviation are decreases as atomic number increase because it depend on the values of  1 1 and  1 2 .
Tables: (4,5,6) contain The results of inter-particle expectation value  12  ,we noted when(k ) takes negative values the expectation values  12  increase as atomic number increases for all studied system ,while when n takes positive values the  12  decreases.also standard deviation decreases for all intra and inter shells.
Table (6) represents the electron density at the nucleus calculated from equation ().we could see the for (1s)-shell is larger than that for other shells because (1s) shell closer to the nucleus than other shells.We could see from those tables The effect of the atomic number upon the electron density, where the density increases by increasing the atomic number Z is due to the Coulomb attraction forces.
Figure (1,2,3) show that the relation between the one-electron radial density distribution function(r1) with the position for 1s, 2s,3s,4s and 5s shells respectively ,where noted for each shell the maximum values of D(r1) increase when atomic number increase and location of these peaks contracted to ward of the nucleus ,also observed the value of D(r1) vanishes when the distance equal zero or infinity, in the figure(1,2,3) (B) noted two peaks ,the first represent the probability of finding the electron in small distance from the nucleus and the second represent the probability of finding the electron in 2sshell.Also noted in figure (1,2,3) (C) three peaks in (3s) shells, four peaks in (4s) shells and five peaks in (5s) shells

Iand expectation values k r 1
S S N 2347-3487 V o l u m e 1 1 , N u m b e r 8 J o u r n a l o f A d v a n c e s i n P h y s i c s c o u n c i l f o r I n n o v a t i v e R e s e a r c h M a y 2016 w w w .c i r w o r l d .c o mThe radial correlation contained within a correlated wave function may be investigated by evaluating the two-particle radial density distribution refer to solid angles, r is the distance between electrons and nucleus. .pq  defined as: denotes spin-part (  spin up,  spin down).

I
S S N 2347-3487 V o l u m e 1 1 , N u m b e r 8 J o u r n a l o f A d v a n c e s i n P h y s i c s c o u n c i l f o r I n n o v a t i v e R e s e a r c h M a y 2016 w w w .c i r w o r l d .c o m

I
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u m e 1 1 , N u m b e r 8 J o u r n a l o f A d v a n c e s i n P h y s i c s 3959 | P a g e c o u n c i l f o r I n n o v a t i v e R e s e a r c h M a y 2016 w w w .c i r w o r l d .c o m

Table ( 1): The one-particle expectation values and the standard deviation of the excited state in position space for
B -like ions(    ) .1.34937 c o u n c i l f o r I n n o v a t i v e R e s e a r c h M a y 2016 w w w .c i r w o r l d .c o m

Table ( 2): The one-particle expectation values and the standard deviation of the excited state in position space for B - like ions (𝟏𝐒 𝟐
) .

Table ( 3): The one-particle expectation values and the standard deviation of the excited state in position space for
B -like ions(    ) .5.20315 c o u n c i l f o r I n n o v a t i v e R e s e a r c h M a y 2016 w w w .c i r w o r l d .c o m

Table ( 4): The Two -particle expectation values and the standard deviation of the excited state in position space for B -like ions(𝟏𝐒 𝟐
) .
c o u n c i l f o r I n n o v a t i v e R e s e a r c h M a y 2016 w w w .c i r w o r l d .c o m

Table ( 5): The Two -particle expectation values and the standard deviation of the excited state in position space for B - like ions(𝟏𝐒 𝟐
) .

Table ( 6): The Two -particle expectation values and the standard deviation of the excited state in position space for B -like ions(𝟏𝐒 𝟐
) .