A study of Fermi hole and partial Fermi hole for Li- and C-Atoms in different states 1s 2p, 1s 3p, 1s 3d and 2s 2p

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1 Iraq Joural of Physcs, 05 Vol.3, No.7, PP. 5-6 A study of Fer hole ad partal Fer hole for L- ad C-Atos dfferet states s p, s 3p, s 3d ad s p Al. A. Alzubad, Khall H. AL-Bayat, Naeeah Ch. M. Physcs Departet, College of Scece, Uversty of Baghdad. Physcs Departet, College of Scece for Woe, Uversty of Baghdad. E-al: a_ch_a@yahoo.co Abstract The electro correlato effect for ter-shell have bee aalysed ters of Fer hole ad partal Fer hole for L-ato the excted states (s 3p) ad (s 3d) usg Hartree-Fock approxato (HF). Fer hole f( ) ad partal Fer hole g(, ) were detered posto space. Each plot of the physcal propertes ths work s oralzed to uty. The calculato was perfored usg Mathcad 4 progra. Key words Fer hole, Partal Fer hole, electro correlato. Artcle fo. Receved: Oct. 04 Accepted: Dec. 04 Publshed: Sep. 05 دراسة فجوة فيرمي وفجوة فيرمي الجزئية لذرتي الليثيوم والكاربون في حاالت مختلفة s p و s p, s 3p,s 3d علي عبد اللطيف كريم الزبيدي, خليل هادي أحمد البياتي, نعيمة جيجان مذكور قسم الفيزياء, كلية العلوم, جامعة بغداد قسم الفيزياء, كلية العلوم للبنات, جامعة بغداد الخالصة تم تحليل الترابط األلكتروني لالغلفة البينية من خالل فجوة فيرمي وفجوة فيرمي الجزئية لذرة الليثيوم في الحاالت المتهيجة (3p (3d بأستخدام الدالة التقريبية لهارتي-فوك.(HF) تم تحديد فجوة فيرمي ) s) و f(s) وفجوة فيرمي الجزئية ) g( r, في الفضاء المكاني. كل رسم للخصائص الفيزيائية في هذا البحث يحقق شرط العيارية للواحد. الحسابات تم تنفيذها باستخدام برنامج.Mathcad 4 Itroducto The electro correlato a ato or olecule s of two kds, due to Coulob ad exchage teracto. I the frst of these, the classcal Coulob repulso betwee each par of electros whch have dfferet sps, leads to the forato of a hole the atoc or olecular charge cloud aroud each chose electro whch s called a Coulob hole []. The secod kd, the teracto of electros wth parallel sps whch keeps the electros apart ad lead to the Fer hole (t s also kow as the exchage hole) whch arses fro the ature of feros The frst uercal calculatos for the 5

2 Iraq Joural of Physcs, 05 Al. A. Alzubad, et al. Fer hole atos were reported by Masle (956) [] ad Sperber (97)[3]. The Fer hole was defed quattatvely ters of par dstrbuto fuctos by Boyd ad Coulso 974 [4]. Explct calculatos of Boyd ad Coulso 974, Moseyev et al. 975 [5] for the 3 s state of the helu soelectroc sequece ad the groud state of beryllu have provded a sple physcal pcture of the average Fer hole the charge cloud aroud ay chose electro sple atoc systes. The electro correlato has bee exaed wth tershell for a seres of L-lke os ther excted state (s 3s) coparg wth (s p) by AL-Bayat (985) [6]. Iter-shell descrptos were obtaed at a Hartree-Fock (HF) level fro Wess 963 [7]. The Fer ad Coulob holes that ca be used to descrbe the physcs of electro correlato are calculated ad aalysed for a uber of typcal cases, ragg fro prototype dyacal correlato to purely odyacal correlato by Bujse & Baereds (00) [8], ad Fer hole has studed for the cofguratos s p ad p for He ato where the Paul prcple was satsfed by Da Dll (003) [9]. Parttog the secod-order desty to ts parwse copoets, correlato effects were the aalyzed ters of Fer hole f( ) ad partal Fer hole g(, ) (s 3s) ad (s 3p) states by Al A. Alzubad 005 [0]. I the preset reseearch, the aalyss s exteded here to the thrd ad fourth excted states of the L-ato (s 3p) ad (s 3d). Moreover, sce the thrd ad fourth excted states have a oe-zero total agular oetu, a coparso of ts correlato propertes wth those for the groud state of S syetry, wll be of specal terest. Theory ad ethodology HF wave fucto s approprate for ay dfferet applcatos electro structure theory. It descrbe the teral dyacs of a N-electro ato the teral spatal coordates ad the sp coordates ad gve by A P r r,..., ra K rk P, () A! P K The total wave fucto of ay partcle systes s wrtte as a deterat of oeelectro fuctos (sp-orbtals) [7]. The the total wave fucto for three electro syste s gve by! 3 () 3 s s ad for sx electro syste l! (3) 6 s s s s p x py where φ s the spatal part of the sp-orbtal; ad α ad β refer to the two copoets of the sp part. The orbtals, tur, are wrtte as a expaso soe set of aalytc bass fuctos l C l, (4),l = (s, 3p) or (s, 3d). The bas s fuctos are stadard oralzed Slatertype orbtals (STO's) ad gve by [7] ( r,,, ) R ( r, ) Y (, ) (5), l, l ll Where ( r ) R ( r, ) r N e (6) 6

3 Iraq Joural of Physcs, 05 Vol.3, No.7, PP. 5-6 ad C represets the costat coeffcet yeld fro the self cosstet feld (SCF) ethod, s the prcple quatu uber, s the orbtal expoet, ad N s a oralzato costat gve by [] N /! / (7) If (,j) labels a par of occuped φ ad φ j the restrcted HF descrpto of the syste, the the chage, due to correlato the (,j) copoet of the secod-order desty Γ j (χ, χ ) whch derved fro the HF wave fuctos usg the parttog techque of Bayard ad Mashat (977) []. j, j j (8) To study the physcal propertes for a ay partcle systes, we ust fd the electropar desty for each dvdual electroc shell, so 3 ( x, x ) ( x, x ) (9) j j For ay N-electro atoc syste, the radal terpartcle dstrbuto fucto s defed ters of the sp-free secod order desty atrx as [3.4] f d d j ( r ) j ( r, r ) dr (0) where r represets the space coordates of electro. The eleet of volue d s defed as d r s d d dr. r r r r r cos () oe obtas rdr r r sd () provded that, r are kept fxed. The cobed volue eleet therefore becoes [3] dd r rr dr dr sddd (3) dr where ω deotes the agle of rotato gve Fg.. r z (r,θ,φ ) (0, ) r (,θ,φ ) ω θ θ θ (, 0) 0 φ φ Polar axs y x a. b. Fg. : a. The radal tegrato rage Equ.(3.0) s represeted by the shaded area. b. Coordate syste volved the deterato of f( ). Here,, r are posto vectors for electros ad relatve to the org pot [3]. 7

4 Iraq Joural of Physcs, 05 Al. A. Alzubad, et al. By substtutg Equ.(3) to Equ.(0), we get [5] f r r r r dr dr d d d j ( ) j (, ) s (4) For the sphercally syetrc case (s-orbtal) where the azuthal (or orbtal) agular oetu quatu uber 0, f( ) ca be evaluated fro [3]. f ( r r rr rr ) 8 r r dr ( r, r ) r dr r dr ( r, r r dr 0 rr r r r (5) ) I the preset aalyss the p, 3p ad 3d states are exaples of o-sphercally syetrc syste. The expresso for f obtaed fro [6] r r r r r f r r r dr r r r dr I r dr ( ) (, ) 0 rr r rr, r r r dr I (6) I the foregog, the dstrbuto fucto f s averaged over all postos of the two electros. But we ght ask what the dstrbuto fucto would look lke f oe of the electros s at a specfed dstace fro the ucleus. If the posto of electro s specfed as r the oe ca troduced a ew dstrbuto fucto whch depeds o rad r ad gve by [7] g( r, r ) r r rr rr j ( r, r ) r dr I (7) I represet the tegrals of the ost geeral type I Y ( , ) Y (, ) Y (, ) Y (, ) sdd d I ca be wrtte as [0] I ( ) L LL3L LL 4 C( ; ) C( 3 4 ; 3 4 ) C, (8) C P ( ;000) ( 3 4 ;000) ( ) (cos ) where C( ; ) s the Clebsch- Gorde coeffcet defed by [8], ad s soetes expressed tradtoal otato, L,, 3 4 ad xy s the Kroecker delta fucto. Eq.() ca be wrtte ter of Wger 3j sybols by usg the followg relato (9) 8

5 Iraq Joural of Physcs, 05 Vol.3, No.7, PP. 5-6 The portat propertes ad the values of Wger 3j sybols were gve ad tabulated Ref. [9] The assocated Fer hole ca be wrtte as a dfferece betwee the electro-electro dstrbuto fucto f( ) for trplet state ( 3 s) or (αα) ad these for sglet state ( s) or (αβ) [0] f r ) f 3 ( r ) f ( r ) (0) ( KM( s) KM( s) ad the partal Fer hole, Δg(, ) s defed as the dfferet betwee the partal dstrbuto fucto g(, ) for trplet ad sglet state g( r, r ) g 3 ( r, r ) g ( r, r ) KM( s) KM( s) () Partal Fer hole Δg(, ) agast (, ), eables us to study correlato effects whe the test electro s located at a gve radal dstace fro the ucleus; the tegral of Δg(, ) agast equals Δf( ). Results ad dscusso The probablty of fdg the ter-partcle dstrbuto fucto f( ) betwee electros ulke ad lke sps KM shell for L-ato the dfferet excted states ca be observed the curves (A) ad (B) preseted Fg.. For both states, at sall, the f( ) dstrbuto fucto s flueced aly by the electro par behavor, sce the outer electro caot peetrates the K-shell, so we see a flat rego. The absece of ths feature s due to the dfferece syetry of the s ad 3p ad 3d orbtals. I s 3p state these curves dcates two prcpal axa each curve refer to the probablty of fdg the outer electro oe p ad the other 3p. Fos 3p state the axu values of f( ) betwee electros wth lke sp s greater tha betwee electros wth ulke sp ad the locatos s less ad vce versa fos 3d state. The axu values of f( ) fos 3d > s 3p ad vce versa for ts locatos. We ow exae the terpartcle dstrbuto fucto f( ) for ter-shells as s α p α, s β p α, s α p α ad s β p α of the C-ato Fgs.(3) (a) ad (b) fos α p α, s β p α, s α p α ad s β p α whch show the p electros caot peetrated the s or s so we see a flat regos at sall ad ths regos for αα > βα because Fer effect ad the desty of two electros parallel state larger tha sglet state s p but vce versa for s p, the axu values of f( ) fos p > s p ad ts locatos for s p > s p. It ca be see that Fer hole s plotted as a dfferece betwee f( ) of K α M α shell (trplet state) ad K β M α shell (sglet state) for L-ato the dfferet excted states Fg. 4 ad for dfferet states of C-ato Fg. 5 whch shows the depth of Fer hole decreases ad ts radus creases whe the outer electro excted to farther orbtal L-ato ad a creasg the depth ad radus of Fer hole s p state copared wth s p state C-ato, due the large dstace betwee tow electros L-shell. 9

6 f(r ) Iraq Joural of Physcs, 05 Al. A. Alzubad, et al. (a) s α 3p α s β 3p α (b) s α 3d α s β 3d α Fg. : The ter-partcle dstrbuto fucto f( ) for L-ato the dfferet excted states (a) s α 3p α, s β 3p α ad (b) s α 3d α, s β 3d α. (a) (b) s α p α s β p α Fg. (3). The ter-partcle dstrbuto fucto f( ) for L- ato the dfferet excted states (a) s α p α, s β p α ad (b) s α p α, s β p α. (a) s 3p (b) s 3d f( ) Fg. 4: The Fer hole f( ) evaluated fro f αα (r)-f βα (r) for L- ato dfferet excted states (a) s 3p ad (b) s 3d. 0

7 Iraq Joural of Physcs, 05 Vol.3, No.7, PP. 5-6 f( ) - _ f( ) (a) (b) s p s p Fg. 5: The Fer hole f( ) evaluated fro f αα (r)-f βα (r) for C-ato dfferet excted states (a) s p ad (b) s p. The g(, ) dagras (surfaces ad cotours) for KM( 3 s), KM( s), KL( 3 s), KL( s), LL( 3 s) ad LL( s) shells for L- ad C-atos states s 3p, s 3d, s p ad s p Fgs. 6-9 respectvely, show the chage behavor of the ter-partcle probablty fuctos as the posto of the test electro s vared ad show the sae characterstcs the rego of the dagoal = axs, whe s large copared wth r K. Ths slarty also holds for the features parallel to the axs, whe r. Such features K are characterstc of a ter-shell dstrbuto. I s α 3p α - shell values represet the locato of the three axa for K-, L- ad M- shell. I the dagoal structure, specto of g, cofrs that at the axu, whe =r M, the value of does deed exceed. Furtherore, for a gve dagoal cotour, the spa of dcates the spreads of the L- ad M-shells destes whereas, for a fxed choce of, say r M, the rage of reflects the relatve copactess of the K-shell desty. Coversely, for the parallel feature, the rages of ad relate to the K-, L- ad M-shells dstrbutos, respectvely. Naturally, the Fer effect produces eglgble values of g(, ) at sall for all. Dstrbuto fos β 3p α -shell shows the characterstc dagoal ad parallel features see prevously fos α p α. However, sce Fer effect s ot preset ths stace, the K-shell effect does ot appear the g, surface due to the dfferet syetry. The sae behavor ca be see s 3d state but there dfferet exstece values represet the axa of two probabltes of fdg electro K- ad M-shell because outer electros caot peetrates L-shell. The g(, ) dagras (surfaces ad cotours) for s α p α ad s β p α sub-shells for C-ato have ther a characterstcs located about the dagoal le, ad the axu value for the desty where r p ad r sp. As there s a very sall effect parallel to the axs represet the probablty of pereato the s electro to s so s ad represet the dstace dstrbutos betwee s ad p sup-shell. I addto, ad as expected, for every ter-shell.

8 g(r,r) g(r,r) g(r,r) g(r,r) Iraq Joural of Physcs, 05 Al. A. Alzubad, et al. (a) s α 3p α (b) s β 3p α (a) s α 3p α (b) s β 3p α,r of L-ato excted state (s 3p), (a) s α 3p α ad (b) s β 3p α. Fg. 6: The surfaces ad cotours of the partal dstrbuto fucto g fos 3p state (a) s α 3d α (b) s β 3d α (a) s α 3d α (b) s β 3d α,r of L-ato excted state (s 3d), (a) s α 3d α ad (b) s β 3d α. Fg. 7: The surfaces ad cotours of the partal dstrbuto fucto g fos 3d state

9 g(r,r) g(r,r) g(r,r) g(r,r) Iraq Joural of Physcs, 05 Vol.3, No.7, PP. 5-6 (a) s α p α (b) s β p α (a) s α p α (b) s β p α,r Fg. 8: The surfaces ad cotours of the partal dstrbuto fucto g of C-ato (a) s α p α ad (b) s β p α. fos p state (a) s α p α (b) s β p α (a) s α p α (b) s β p α,r Fg. 9: The surfaces ad cotours of the partal dstrbuto fucto g of C-ato (a) s α p α ad (b) s β p α. for s p state 3

10 Δg(r,r) Δg(r,r) Iraq Joural of Physcs, 05 Al. A. Alzubad, et al. A partal Fer hole as a dfferece betwee αα ad βα for the selected systes of L- ad C-ato are plotted as surface ad cotour dagras Fgs. 0 ad respectvely. These Fgures show that the depth of partal Fer hole decreases ad the radus creases as the outer electro excted to 3d orbtal Lato ad the depth as well as the radus of g, fos p s p. (a) s 3p (b) s 3d (a) s 3p (b) s 3d Fg. 0: The surfaces ad cotours of the partal Fer hole g(, ) evaluated fro g αα (, )-g βα (, ) for L- ato dfferet excted states (a) s 3p ad (b) s 3d. 4

11 Δg(r,r) Δg(r,r) Iraq Joural of Physcs, 05 Vol.3, No.7, PP. 5-6 (a) s p (b) s p (a) s p (b) s p Fg. : The surfaces ad cotours of the partal Fer hole g(, ) evaluated fro g αα (, )-g βα (, ) for dfferet states of C-ato, (a) s p ad (b) s p. Coclusos Fro the preset research, we deduce soe otes for the ter-partcle dstrbuto fucto f( ), Fer hole f( ), the partal desty dstrbuto fucto g(, ) ad partal Fer hole g(, ). Whe coparg the f( ) ad g(, ) for L- ad C-atos the dfferet excted states, we see the axu values fos 3d are larger ad ther locato the earest to the ucleus fro s 3p L-ato ad fos p > s p C-ato, ad a flat rego the sall. The depth of Fer ad partal Fer hole decreases ad the radus creases whe the outer electro excted to further orbtal. The depth of Fer ad partal Fer hole for s p s > s p ad vce versa for radus. As well as, peetrato the orbtals by further orbtals electros, occur oly betwee orbtals are slar sphercal syetry. Refereces [] R. J. Boyd, C.A. Coulso, J. Phys. B: Ato. Molec. Phys. 6 (973) 78. [] V. W. Masle, Proc. Phys. Soc. A 69 (956) 734. [3] G. Sperber, It. J. Quatu Che. 5 (97)89. [4] R. J. Boyd, C. A. Coulso, J. Phys. B: At Mol. Phys. 7 (974) 805. [5] N. Moseyev, J. Katrel, R. J. Boyd, J. Phys. B: At. Mol. Phys. 8 (975) 30. [6] K. H. Al-Bayat, PhD. Thess, Lecester Uversty (984). [7] A. W. Wess, Astrophys. J. 38 (963) 6. [8] M. A. Bujse, E. J. Baereds, Mol. Phy. 00 (00) 40. 5

12 Iraq Joural of Physcs, 05 Al. A. Alzubad, et al. [9] D. Dll, Mechacs/ Fer Holes ad Heaps, Departet of Chestry Bosto Uversty, Bosto MA05 (003). [0] A. A. Alzubad, Ph.D Thess, Uversty of Techology, Iraq (005). [] J. Huag, Q. Zhao, G. Jag, Cou. Theor. Phys. 5 (00) 87. [] K. E. Bayard ad M. M. Mashat, J. Che. Phys. 67 (977) 405. [3] C.A. Coulso, A. H. Nelso, Proc. Phys. Soc. 78 (96) 83. [6] R. Beesch, It. J. Quat. Che. 6 (97)8. [7] R. J. Boyd, C. A. Coulso, J. Phys. B: Ato. Molec. Phys. 6 (973) 78 [8] M. E. Rosse, "Eleetary Theory of Agular Moetu" Joh Wley ad Sos, New York, (967). [9] R. D. Cowa, "The Theory of Atoc Structure Spectra" Uversty of Calfora Press, Calfora, (98). [4] S. Bub ad O. Prezhdo, Phys. Rev. Lett. (03) [5] P. R. Dressel ad F. W. Kg, J. Che. Phys. 00 (994)