calculation the Hartree -Fock energy of 1s shell for some ions
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1 JOURNAL OF KUFA PHYSICS Vol.6/ No. (4) calculation the Hatee -Fock enegy of s shell fo some ions Depatment of Physics, College of Science, Kufa Univesity shaimanuclea@yahoo.com Abstact: In this eseach we calculated the impotant atomic popeties such as Hatee- Fock enegy of S shell fo the ions Si +, P +3,S +4,Cl +5,A +6 and K +7,Also we found the values the enegies in hatee unit ( , , , , ,-349.3) hatee espectively.all the studied atomic popeties wee nomalized and we used the atomic units in ou calculations. Keywods: Hatee-Fock, atomic popeties,atomic units حساب طاقت هارتزي فوك للقشزة S لبعض االيوناث شيماء عواد كاظم Si +, ( s الخالصت: ف هذا البحث حم حساب بعض الخىاص الذر ت المهمت مثل طاقت هارحزي فىك للغالف لال ىناث,-, , +7 K ) P +3,S +4,Cl +5,A +6 and فىجدناان ق م الطاقاث بىحدة الهارحزي كانج ( هارحزي على الخىال. جم ع الخىاص الذر ت ( , , المدروست كانج ع ار ه واسخخدمنا الىحداث الذر ت ف حساباحنا. كلماث مفتاحيت: هارتزي- فوك, الخصائص الذريت, الوحذاث الذريت 95
2 JOURNAL OF KUFA PHYSICS Vol.6/ No. (4) Intoduction: In 96 Ewin Schödinge developed an equation, called a wave equation, to descibe the behavio of matte waves. An acceptable solution to Schödinge s wave equation had called a wave function. By using Schödinge s equation scientists can find the wave function, which solves a paticula poblem in quantum mechanics. Unfotunately, it is usually impossible to find an exact solution to this equation, so cetain assumptions have been used in ode to obtain an appoximate answe fo the paticula poblem. Both Ewin Schödinge and Wene Heisenbeg (97) independently fomulated geneal quantum theoy. At fist sight, the two methods appeaed to be diffeent because Heisenbeg s method has fomulated in tems of matices wheeas Schödinge s method has fomulated in tems of patial diffeential equations. Just a yea late, howeve, Schödinge was able to show that the two fomulations ae mathematically equivalent[]. Theoy: The wave function ψ(,, 3, N ) of many paticles system is a mathematical expession that caies the infomation about the system popeties such as position, momentum, enegy and etc. Thee ae two main kinds of the wave function appoximation methods [] - The uncoelated wave function appoximation. - The coelated wave function appoximation. In the uncoelated wave function appoximation, each paticle is assumed to move in only the aveage field of all othe paticles in the system. This pocedue uses an appoximation method called the Hatee- Fock appoximation (HF). In the coelated wave function appoximation, one takes into account the coelation between all the paticles of the system by using the configuation inteaction appoximation (CI). The wave function fo the one paticle in the atomic system is defined as [] m ( ) R ( ) Y (, ) ( s) () nlm l m s nl l The adial function R nl () is elated to the distance of electon fom the nucleus and it depends on the ( the pincipal quantum m l numbe) and Y (, ) epesents function of spheical hamonic [4]. 96
3 JOURNAL OF KUFA PHYSICS Vol.6/ No. (4) Schödinge s equation cannot be solved exactly fo two-electon atoms o ions. So that an appoximation method must be used. Fo atoms, such as Helium and Lithium the vibational method is accuate enough, fo atoms of highe atomic numbe, the best appoach to find a good wave function lies in fist calculation an appoximate wave function using the Hatee-Fock pocedues which depends on the Cental Field appoximation. The basic idea of this appoximation is that each of the atomic electons moves in an effective spheically symmetic potential v() ceated by the nucleus and all the othe electons. The poblem is the detemination of the electon-electon epulsion tem. [5] i j. ij Thee ae two methods to calculate this tem; fist the simple semi-classical method of Thomas and Femi, second the moe pecise Hatee-Fock o self-consistent field appoach. Hatee wave function fo the atoms (ions) is not antisymmetic in the electon coodinates because it uses the simple poduct wave function equation (). The genealization of the Hatee method which takes into account this antisymmety equiement imposed by the Pauli Exclusion Pinciple was caied out in 93 by Fock and Slate[].This genealization of Hatee theoy, is known as the Hatee-Fock method. In the Hatee-Fock appoach, it is assumed, in accodance with the independent paticle appoximation and the Pauli exclusion pinciple, that the N-electon wave function is a Slate deteminant,o in othe wods an antisymmetic poduct of individual electon spin-obitals. The optimum Slate deteminant is then obtained by using the vibational method to detemine the best individual electon spin-obital. The Hatee-Fock method is theefoe a paticula case of the vibational method, in which the tial function fo the N-electon atom is a Slate deteminant whose individual spin-obitals ae optimized. It should be noted that the N-electon atom wave function ψ(,,3, N) solution of the Schödinge equation stated below N N Z ( i ) ( ) (,,... N) E (,,... N ) i ij i i j i () can only be epesented by an infinite sum of Slate deteminants. So that the Hatee-Fock method may be consideed as a fist step in the detemination of atomic wave function and enegies. We also emak that the Hatee- Fock method is also applied to othe systems such as molecules o a solids.[3] 97
4 JOURNAL OF KUFA PHYSICS Vol.6/ No. (4) In the Hatee-Fock method, the tial wave function is a Slate deteminant. Let us denote the gound state enegy of the system by E : Accoding to the vibational method E H (3) whee is a tial wave function which is assumed to be nomalized to unity Fo -electon systems.[6] s () s () (,) (5) s () s () The standad method fo detemining the optimal fom of the spatial pat of the spin obital in a deteminant wave function such as equation (5). D()s=R S() (6) We can wite the HF wave function fo two electon atom as, ) () ().. (4) ( s s Results and discussion: One paticle adial density distibution function D( ) : One Electon adial density distibution function is the pobability of finding the electon in a spheical shell at distance fom the nucleus, we calculated fom the equation(6).[8] Table():-values of positions and maximum values of the D()fo studied system fo s shell. Ions Si + P +3 S +4 Cl +5 A +6 K Dmax()
5 JOURNAL OF KUFA PHYSICS Vol.6/ No. (4) K + 7 A + 6 Cl + 5 Si + S + 4 P + 3 Fig():Relation between the one-paticle adial density distibution function D() and position() Fom table ()we noted, That the maximum values fo D( ) inceases when the atomic numbe incease, that means the pobability of finding an electon incease and the locations of these peaks ae contacted towad the nucleus.fom fig.() we obseved that: the distance equal to zeo fom the systems, the pobability of finding an electon equal to zeo {when =, D( ) = }. This means that the electon is not found in the nucleus and when the distance is fa fom nucleus, the pobability of finding an electon equal to zeo also {when =, D( ) = }. That means the electon is not found out the atom. One paticle expectation value < m > : The one-paticle expectation value and the standad deviation using equations.[8] D m wee calculated by < m > = ( ) d..(7).(8) 99
6 JOURNAL OF KUFA PHYSICS Vol.6/ No. (4) Consecutively fo diffeent values fo m whee m is intege numbe and takes the values m -,the esults of the adial expectation values of < m > and standad deviation ae tabulated in fig. (). Table(): Values of one-paticle expectation values fo diffeent values of (m) and stande deviation. Ions Si P S Cl A K fom table() we noted : - In all studied systems we found the onepaticle expectation value < m > incease when the atomic numbe incease when m take negative value -,- whee the elated to the attaction enegy expectation value < V en > = -Z[N. ], N epesent the numbe of electon in the shell and indicates how the density distibution in nea egions of the nucleus. - We obseved when Z incease the oneexpectation value < m > begins decease fom positive values +, +. Whee is elated to the stande deviation and epesented the distance between the nucleus and the electon. 3- When m equal to zeo, the one-paticle expectation value < m > equal to unity fo all studied systems,this epesents the nomalization condition because the ne-
7 JOURNAL OF KUFA PHYSICS Vol.6/ No. (4) paticle adial density distibution function ( m D( ) is nomalized D ) d 4- The standad deviation fo all studied systems decease with inceasing atomic numbe, because the distance between electon-nucleus become small. Inte-paticle Distibution Function f ( ) : The pai distibution function evaluated fom equation below[5]: RS ( ) RS ( ) d d f ( ) (S ).5 (9) ( ) ( ) R S R S d d It epesents coulomb epulsion between a pai of electon with.the esults ae tabulated fo all systems in the table(3).and fig.() indicates the elation between f ) and inte electonic distance. ( Table(3):The location and maximum values of the inte-paticle distibution function f ) fo studied systems. ( Ions Si + P +3 S +4 Cl +5 A +6 K f()
8 JOURNAL OF KUFA PHYSICS Vol.6/ No. (4) Fom fig(),as atomic numbe incease, the distance between two electon deceases, this behavio fom the fact that,electons shinkage towad the nucleus. inceasing the atomic numbe which lead to inceases the attaction foce between the nucleus and the electon then the distance between electons deceases. Fom the table(3) we noted the maximum value f ) of inceases as Z inceases and ( the inte-paticle distance deceases because Inte-paticle Expectation Value m We evaluated the inte-paticle expectation value m using equations below[8] and standad deviation by
9 JOURNAL OF KUFA PHYSICS Vol.6/ No. (4) m f ) m ( d () () Respectively and tabulated of the esults in table (4), which show the elationship between m with m. Table(4): expectation values m whee - m and standad deviation. Standad deviation Si P S Cl A K fom analyzed the inte-paticle expectation value m and standad deviation pesented in table (4) we noted : - The inte-paticle expectation value m incease when Z incease and when m takes values -,-, whee epesented epulsion enegy between twoelectons. - when m takes positive values +,+, the inte-paticle expectation value decease with Z incease. 3- The inte-paticle expectation value m equal to unity fo all systems when m =,this epesent the nomalization condition because the inte-paticle distibution function is nomalized f ( ) d 4- The standad deviation decease when Z incease fo all systems. The standad deviation contains much infomation concening the shape of the two-paticle 3
10 JOURNAL OF KUFA PHYSICS Vol.6/ No. (4) distibution function f ( ), the shape esults indicate f ) become less diffuse. ( V Z..() The expectation value of enegy E :- E V..(3) The potential enegy and total enegy wee These esults ae listed in table (5). calculate by using equations below[8,9] : Table(5):The expectation values fo attaction,epulsion,kinetic and Hatee- Fock enegies of the Ions. Ions Ven V ee V T EHF Si + pesent.wok Ref.wok[] P +3 pesent.wok Ref.wok[] S +4 pesent.wok Ref.wok[] Cl +5 pesent.wok Ref.wok[] A +6 pesent.wok Ref.wok[] K +7 pesent.wok Ref.wok[]
11 JOURNAL OF KUFA PHYSICS Vol.6/ No. (4) Fom the table (5): - we obseved the esults of the expectation values fo total potential enegy,kinetic enegy and the total enegy of Hataee-Fock ae lage ageement with published esults. - Both V en, V fo ions incease when ee atomic numbe incease. This esult can be undestood fom the fact that all shell shink towad the nucleus due to the incease attactive foce between the poton and the electons because the distance between nucleus-electon and electon-electon decease, this esult leads to incease in epulsion potential enegy and attaction potential enegy,the expectation values fo total potential enegy V incease in each ions(si +,P +3,S +4,Cl +5,A +6 and K +7 ) as follows : - when Z incease the maximum values fo one-paticle adial density distibution function D( ) and intepaticle distibution function f ) ( incease, and the position of these maximum values decease with Z incease. - Fo both one-paticle expectation m,and two-paticle expectation m incease when Z inceases and when m = -, - and both decease fo m = +, + when m = zeo, this value system because the incease of V en ae epesents the nomalization condition. lage than that of V. ee 3- The standad deviation of onepaticle and two-paticle 3- The expectation fo kinetic enegy T inceases when Z incease.the total enegy o decease fo all systems when the ion Hatee-Fock enegy become lage when Z incease also indicates the E as a HF 4- function of Z. numbe incease. All the expectation values of the enegies Conclusions V en, V ee, V, T and E HF incease Fom the pesent wok, we deduce some notes fo some atomic popeties fo some 5 when the atomic numbe incease and these values ae given lage ageement with the published esults.
12 JOURNAL OF KUFA PHYSICS Vol.6/ No. (4) Refeences []- E.F.Al-Kunani,M.Sc. Theoetical Study of Radial Coelation and Othe Atomic Popeties fo He Atom and He- like ions College of Science, Nahain Univesity(7). []-A Molecula Appoach, Physical Chemisty, Donald A.Mcquaie,John D.Simon (998). [3]-Physics of Atoms and Molecules, B.H.Bansden & C.J.Joachain, second edition,england (3). [4]- J.W. Hill, R. H. Petucci, T. W. Mcceay, and S. S. Pey Geneal Chemisty Peason Pentice Hall, Peason Education,Inc.,Uppe Saddle Rive,New Jesey,Fouth Edition (5). [5]- E. P. Lopez Physical Chemisty: A pactical appoach, P.99,Williamstown, MA 67(995). [6]- J. Avey The quantum Theoy of Atoms,Molecules, and Photons,New Yok, Mc Gaw Hill, (98). [7]-M.A.Mahmood.M.Sc.thesis Impoving Hatee-Fock wave function fo Helium and Lithum atoms the college of science fo women Univesity of Baghdad 8. [8]- B.H.Asaad,M.Sc. A study of the physical popetiesfo the electons oute shells fo some atoms College of Science fo women Baghdad Univesity (5). [9] E.F.Selman,M.Sc. Astudy of atomic popeties fo He-Like ions College of science,kufa univesity(). []- E.Clementi and Rotti,atomic data and nuclea data tables,vol. 4, p (974). 6
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