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A Theoretical Study of the Atomic Properties for Subshells of N+ and O⁺² Using Hartree-Fock Approximation

Hayder Ali Abd Alabas*, Qassim Shamkhi AL-Khafaji, Abbas Hassan Raheem

Department of physics, Faculty of Sciences, Kufa University, Iraq

Abstract: In this research, we calculated the atomic properties of systems have been studied (N⁺ and O⁺² ) for intra-shells (1s, 2s and 2p) using Hartree-Fock wave function. These properties included, one-particle radial density function, one-particle and inter-particle expectation values, inter-particle density function and expectation values of energies. All these atomic properties increase with atomic number, have highest values in 1s shell and lowest values in 2p shell. All results are obtained numerically by using the computer program (MathCad 14) because it able to calculation and plot functions. All atomic properties are calculated in atomic units.

Keywords: Hamiltonian operator, wave function , Approximation methods, multi-electron systems and Hund's rules

Citation: Hayder Ali Abd Alabas, Qassim Shamkhi AL-Khafaji, Abbas Hassan Raheem, A Theoretical Study of the Atomic Properties for Subshells of N+ and O⁺² Using Hartree-Fock Approximation. International Journal of Physics, 4 (4). 74-77. January 2016. doi:10.12691/ijp-4-4-1

1. Introduction

The Hartree-Fock Self-Consistent Field approximation (HF-SCF), it is a good approximation to many-electron systems, which is described by wave function. The essence of HF-SCF approximation is to replace the complicated many-electron problem by a one-electron problem in which electron-electron repulsion is treated in an average way [1]. The approximation is based on two grounds, first, each electron moves in the potential field of the nucleus plus the N-1 other electrons (central field approximation) that mean the electrons move independently [2]. The second must on initial wave function consistent with final it when inter in the calculation. The wave functions where spin orbitals included four quantum number , using in the calculation obey on Pauli exclusion principle, so consequently the wave function antisymmetric when two electrons exchange their locations.

2. Theory

In order to wave function satisfy the antisymmetric principle have to have written as slater determinant which named after John C. Slater [3].

[picture] | (1)

Slater created such a basis set of functions known as the slater-type orbitals (STO's), which written [4].

[picture] | (2)

Where represented radial part of the wave function and its given as [5].

[picture] | (3)

Normalized constant and written as

[picture] | (4)

[picture] | (5)

Where n principle quantum number, is the distance of the electron from the atomic nucleus, is spherical harmonic, the orbital exponent.

The Hartree-Fock spin orbitals can be described as a linear combination of slater orbitals from the function called basis functions written as [6].

[picture] | (6)

Where represent the constant coefficient and is the slater orbitals.

The two-particle density contains all of the information necessary to calculate the energy and many properties of the atom or ion [7]. Written as

[picture] | (7)

Where represents the combined space and spin coordinate of electron and indicates integration summation over all N-electron except and .

The two-particle radial density function it is probability density of finding the electron 1 at and electron 2 at from nucleus simultaneously written as [8]

[picture] | (8)

The one-particle radial density function it is the probability density function of finding an electron at a distance and from the coordinate origin (i.e. nucleus) written as [9].

[picture] | (9)

The one-particle expectation value can be calculated from the following equation [10].

[picture] | (10)

Standard deviation it is spead out or difference in the expectation value written as [11]

[picture] | (11)

The inter-particle distribution function it is the probability density function of finding the electron 1 and electron 2 at the distance between and written as [12]

[picture] | (12)

The inter–particle expectation value It is given by [13]

[picture] | (13)

Standard deviation it is defined as [14].

[picture] | (14)

The expectation value of total energy for the system written by equation

[picture] | (15)

Where kinetic energy, electron-nucleus attraction energy and electron-electron repulsion energy.

[picture] | (16)

[picture] | (17)

From condition of the virial theorem [15]. The energy expectation value of total energy is related to expectation value of potential energy.

[picture] | (18)

3. Results and Discussion

Table 1 and Table 2 have contained the results of one-particle distribution function and the inter- particle distribution function respectively. and increases when atomic number Z increase because the distance between electrons and nucleus in 1s shell is smallest as well as the distance between electrons as a Figure 1, Figure 2. The greatest value of in 1s shell and smallest value in 2p shell. From Figure 1 when =0 or , =0, that means the probability of finding the electron inside the nucleus or far away from it equal zero. We noted two peaks of for 2s shell, the first peak represented the probability of finding the electron in 1s shell due to penetration phenomenon and the second peak represented the probability of finding the electron in 2s shell. The largest value of in 1s shell for each system as a Figure 2. From Figure 2 when or , =0 that means the probability of finding two electrons in the same position or too far away from each other equal zero.

Figure 1. The relation between one-particle radial density distribution function and location for each system

Figure 2. The relation between inter-particle distribution function and location for each system

Table 1. The maximum values of the one-particle distribution function and corresponding location r₁ for each system

Table 2. The maximum values of the inter-particle distribution function and corresponding location for each system

Table 3. The one-particle expectation values rn₁ where (n=-1,1,2) and standard deviation for each system

Table 3 and Table 4 have contained the one-particle and inter-particle expectation values and standard deviation. When the expectation values increase when the atomic number increase and the highest value of in 1s shell and lowest value in 2p shell. When the expectation values decrease when Z increase. The highest value in 2s shell and lowest value in 1s shell.

Table 4. the inter-particle expectation values rn₁₂ where (n=-1,1, 2) and standard deviation for each system

The standard deviation and decrease when atomic number increase because decrease the distance between electrons and between electrons and nucleus. in addition, the largest value of in 2p shell and smallest value in 1s shell for each system.

Table 5. The expectation values for all attraction, repulsion, kinetic and total energies of intra-shells for each system

Table 5 contained the results of expectation of energies which increase when atomic number increase. They have highest value in 1s shell and lowest value in 2p shell.

4. Conclusions

When to increase the atomic number, the one-particle distribution function, the inter-particle distribution function and the expectation values of energies are increased. These properties have highest values in 1s shell and lowest values in 2p shell. When =0 or and =0 or , =0 and ƒ(r₁₂)=0 respectively. For expectation values , , where increase when Z increase and the highest of in 1s shell. While when the expectation values decrease when Z increase and lowest value in 1s shell.

References

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