Radial Atomic Properties of Excited States for Beryllium Atom (1s^{2} 2s ns) (^{1}s)
Ruqaya Jabir Hadi^{1}, Ali Abid Abojassim^{1}, Laith Najam^{2}
^{1}Department of Physics, College of Science, Kufa Univ., Kufa, Iraq
^{2}Department of Physics, College of Science, Mosul Univ., Mosul, Iraq
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To cite this article:
Ruqaya Jabir Hadi, Ali Abid Abojassim, Laith Najam. Radial Atomic Properties of Excited States for Beryllium Atom (1s^{2} 2s ns) (^{1}s). American Journal of Modern Energy. Vol. 2, No. 1, 2016, pp. 1-4. doi: 10.11648/j.ajme.20160201.11
Received: April 9, 2016; Accepted: June 3, 2016; Published: June 17, 2016
Abstract: Some radial atomic properties of Be-atom in different excited states (1s^{2} 2s 3s, 1s^{2} 2s 4s, 1s^{2} 2s 5s) (^{1}s) have been obtained using two electron density function (r_{1},r_{2}) in order to solve Hartree-Fock equations using slater type orbitals using partitioning technique within the individual electronic shells of different configuration of Be-atom in position space. Radial expectations values for one electron and two electrons , correlation coefficients , electron density at the nucleus , the nuclear magnetic shielding constant and The diamagnetic susceptibility have been calculated for these states of the same atom.
Keywords: Hartree-Fock-Roothaan Method, Slater Type Orbitals, Two Electron Density Function, Radial Expectation Values, The Nuclear Magnetic Shielding Constant
1. Introduction
The Hartree–Fock–Roothaan (HFR) is one of the approximated methods for solving multi electron systems problem and it is powerful method for studying atomic and molecular systems not only for ground state but also for excited states. In this literature the radial part of one electron basis is expanded in terms of slater type orbital (STOs) defined by [1]
(1)
Here, > 0 is the orbital exponent. The quantity n occurring in eq. (1) is a positive principal quantum number of (STO). The determination of nonlinear parameters n and is very important for describing the atomic orbitals.
2. Theory
Most of the physically properties of a multi electron system can be calculated from the two-particle reduced density matrix , without reference to the wavefunction . It is contains all the necessary information to compute the energy and many properties of the atoms. It can be gained by integrating the spin and spatial coordinates of all electrons except two electrons () from the N-particle density matrix, so the two-electron reduced density matrix given by [2, 3].
(2)
Where = combined space and spin variable with represents the number of electron pairs which can be obtained by integrating the second-order reduced density matrix, so is normalized to the number of independent electron pairs within the systemas [4,5].
(3)
The density for the individual electronic shells is obtained by partitioning into its pair-wise components (i,j), where (i,j) labels the occupied normalized spin-orbitals in the HF description, the two electron density can be written as [6,7].
(4)
(5)
For Be -atom the number of electron pairs is
(6)
3. Expectation Values
Expectation value for one-electron is determined by the following expression [8].
(7)
Where m integer number ( . From leads to evaluate the correlation coefficient. The radial expectation value leads to the electron-nuclear potential energy and the nuclear magnetic shielding constant , from , we determined the atomic diamagnetic susceptibility , represented the distance between nucleus and the electron.
is the one-electron radial density function which represents the probability density function to finding an electron at a distance between r_{1} and r_{1} + dr_{1} from the coordinate origin ( i.e nucleus). The single-electron radial density D(r_{1}) is obtained from integration over two–electron radial density function D(r_{1}, r_{2}) and defined as [9,10],
(8)
Where is the two-electron radial density function and represents the probability density that one electron is located at a radius r_{1} and the other electron at a radius r_{2} simultaneously. The two-electron radial density function is given by [11].
= ∫∫ (r_{1},r_{2}) (9)
Where is the solid angle.
Two particle expectation value is defined as [12].
(10)
3.1. Nuclear Magnetic Shielding Constant
The nuclear magnetic shielding constant is determined from the formula [13].
(11)
Where is the distance from the nucleus to the electron in unit of atomic unit. and () fine structure constant = in the atomic units the inner electronic shells shield the external ones thus giving a smaller effective nuclear charge for the outer electrons. the predominant contribution of the S atomic orbitals, and the decreasing relative contribution of the external orbitals as compared to the inner one[14,15].
3.2. The Diamagnetic Susceptibility
The magnetization of a substance is proportional to the magnetizing (external) magnetic field B, thus . The proportionality factor is the magnetic susceptibility (). The vast majority of substances have negative magnetic susceptibility (are diamagnetic),or positive, which can be paramagnetic substances [16]. The diamagnetic susceptibility is defined by relation[17]:
(12)
3.3. Electron Density at the Nucleus
The electron density at the nucleus can be evaluated using the following form [18].
(13)
3.4. Correlation Coefficients
The correlation coefficients are similar to those used in classical statistics.for two electron shell they are defined as [19].
(14)
This quantity is restricted by the value -1≤ ≤ +1,(,means prefect positive (negative) correlation and means either non-correlated variables or for independent variables. For the atomic systems non-correlated variables means that the position vectors of any pairs of particles are orthogonal while independent variables means that the diagonal term of the two particle density matrix is the product of the one particle distribution functions[20]. In these systems for inner shell (KαKβ )-shell for different excited states.
4. Results and Discussion
Different excited states (1s^{2} 2s ns) (^{1}s) of Be-atom where n=3,4,5 have been analyzed and some atomic properties have been calculated, the one electron expectation value , two electron expectation values the nuclear magnetic shielding constant , the diamagnetic susceptibility ,the electron density at the nucleus and radial correlation coefficient ,it should be mentioned that all numerical calculations have been performed in the Hartree-Fock frame work where in these systems there are six shells, the shells of state 1s^{2} 2s 3s are (KαKβ, LαMβ LαMβ KαLα, KαMβ, KαMβ, KβMβ ),the shells of state 1s^{2} 2s 4s are (KαKβ, LαNβ KαLα, KβLα, KαNβ,KβNβ)while the shells of state 1s^{2} 2s 5s are (KαKβ, LαOβ KαLα, KβLα, KαOβ, KβOβ). Table (1) shows the calculated one electron expectation values for studied systems. Table (2) represents expectation values for two electrons, table (3) represents the nuclear magnetic shielding constant , table (4)shows the diamagnetic susceptibility , table (5) represents the electron density at the nucleusand table (6) represents the radial correlation coefficient .
Table 1. One particle expectation value for Be atom in different excited states for each individual electronic shell and for total system.
Table 2. Two particles expectation values for Be atom in different excited states for each individual electronic shell and for total system.
5. Conclusion
This paper shows the effect of radial expectation values of Be-atom in its excited states (1s^{2} 2s ns) where n=3,4,5 to calculate some atomic properties for these systems. When m= -1 this leads to calculate the nuclear magnetic shielding constant , when m = 1 this means the expectation value for the distance between an electron and the nucleus and also leads to determine the correlation coefficients and when m = 2 this leads to calculate The diamagnetic susceptibilityusing Hartree-Fock approximation.
References