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Hilum atom system

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الكلية كلية العلوم     القسم قسم الكيمياء     المرحلة 4
أستاذ المادة عباس عبد علي دريع الصالحي       13/12/2017 07:16:45
University of Babylon Undergraduate Studies
College of Science
Department of Chemistry
Course No. Chsc 424 Physical chemistry
Fourth year - Semester 1
Credit Hour: 3 hrs.

Lectures of Quantum mechanics
Scholar year 2017-2018
Prof. Dr. Abbas A-Ali Draea
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Lecture No. Thirteen: System of Helium atom
1-Introudction.
Helium atom is the simplest multi electronic system. To conform the Hamiltonian operators for free spin, that’s give up largest value of energy at ground state. Operator of helium atom system must be consisted from kinetic energy of two electrons and the expressions of potential energy for interaction of nuclei with electron at the same time of interaction of electron with electron. High precession calculation methods involved spin coupling –orbital phenomena of angular momentum for both two electrons.
Electronic spin didn’t give up a difference in calculation of energy in this system.








There are three terms for operator:-
First term is kinetic energy of electrons
Second term is potential energy of nuclei and electrons.
Third term is potential energy of electron interaction with each other.
??=E?
?=-h^2/(8?^2 m).(?_1^2+?_2^2 )+(-(2e^2)/r1-(2e^2)/r2+e^2/r12) -----1
Schr?dinger equation for Helium atom is
-h^2/(8?^2 m).(?_1^2+?_2^2 )?+(-(2e^2)/r_1 -(2e^2)/r_2 +e^2/r_12 )?=E? ------2
At approximate state will ignore the repulsion term of two electrons with each other (?e^2/r?_12), at this time equation 2 can be separated into two parts to became as two equations of hydrogen atoms.
Therefore
-h^2/(8?^2 m).(?_1^2 )?°1(-(2e^2)/r_1 )=E°1?°1------3
-h^2/(8?^2 m).(?_2^2 )?°2(-(2e^2)/r_2 )=E°2?°2------4
Since equation 3 and 4 represent electron 1 and electron 2 respectively. If we supposed that both wave functions 1&2 are the wave for electrons of helium atoms, so according to the independent particle model we get on
?=??1+??2 -------5
The approximate total energy system is
E=E?1+E?2----------------6
The real total energy is E=E?1+E?2+E3--------7, since E3 is repulsion energy.
Schr?dinger equation for first and second electron becomes as follow:-
?_1^2 ?°1+h^2/(8?^2 m).(E°1+(2e^2)/r1)?°1=0------8
?_2^2 ?°2+h^2/(8?^2 m).(E°2+(2e^2)/r2)?°2=0------9

Equation 8 is multiple by ?°2 and Equation 9 is multiple by ?°1, to getting up approximate wave function ? in each state.
Energy value of hydrogen atom is
En= (-2?^2 me^4 Z^2)/(n^2 h^2 ) since Z2 and n2 equal to one for hydrogen atom therefore
E1= (-2?^2 me^4 )/h^2 =EH =-1/2 h (h= hartree) (Every 1h=27.21 ev)
Energy value in helium atom becomes
E?1= (-2?^2 me^4 Z^2)/(n^2 h^2 ) = -2h because (Z2/n2 =4) and
E?2= (-2?^2 me^4 Z^2)/(n^2 h^2 ) = -2h So that E?1+ E?2= -4h --------10
The calculated energy value of helium atom according to independent particle model equal to (-4h), but the practical value (experimental value) equal to (-2.909h) therefore must be developed the calculated value.
Depending on the values of quantum numbers can be calculating the wave function as following:- ??1=??1s1= 2^(3?2)/(??.?a??^(3?2) ).e^(-(2r_1)?(a?)) --------11
??2=??1s2= 2^(3?2)/(??.?a??^(3?2) ).e^(-(2r_2)?(a?)) --------12
The complete solution represented by
? =??1+ ??2 = 2^3/(?.?a??^3 ).e^(-(2(r_1+r_2))?(a?)) --------13
By using Approximation wave function and repulsion term in equation of energy calculation.
E=??1s1. ?1s2.?. ?1s1. ?1s2 ????------14
That s mean
E=??1s1.?1s2.[- 1/2 (?_1^2+?_2^2 )+(-2/r1-2/r2+1/r12)]. ?1s1. ?1s2 ????-----15
Separation the terms of equation 15
E=- 1/2 ??_1 s_1. ?_1 s_2.(?_1^2+?_2^2 ) ?_1 s_1.?_1 s_2 ??- 2 ??_1 s_1.?_1 s_2 (1/r1+1/r2). ?_1 s_1. ?_1 s_2????
+ ??_1 s_1.?_1 s_2 (1/r_12 )?_1 s_1.?_1 s_2 ??------16
In equation 16 ,values of first integral and second integral according to independent particle model equal to (-4h), since third terms values equal to (1.25h) and its solution is very complex. The total approximation value of energy equal (-2.75h) compare with the practical value (-2.905) this closer in value is not good for energy value.
Now must be recognized the effect of electrons on each other from nuclei according different distance from each other s. Therefore must be calculating the effective nuclear charge to become the energy value equal to -2.848h.


المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .