I.I. Guseinov. Department of Physics, Faculty of Arts and Sciences, Onsekiz Mart University, Çanakkale, Turkey

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1 Epanion and one-range addiion heore for coplee orhonoral e of pinor wave funcion and Slaer pinor orbial of arbirary half-inegral pin in poiion oenu and four-dienional pace I.I. Gueinov Deparen of Phyic Faculy of Ar and Science Oneiz Mar Univeriy Çanaale Turey Abrac The analyical relaion in poiion oenu and four-dienional pace are eablihed for he epanion and one-range addiion heore of relaiviic coplee orhonoral e of eponenial ype pinor wave funcion and Slaer pinor orbial of arbirary half-inegral pin. Thee heore are epreed hrough he correponding nonrelaiviic epanion and one-range addiion heore of he pin-0 paricle inroduced by he auhor. The epanion and one-range addiion heore derived are epecially ueful for he copuaion of ulicener inegral over eponenial ype pinor orbial ariing in he generalized relaiviic Dirac-Harree-Foc-Roohaan heory when he poiion oenu and fourdienional pace are eployed. ey word: Eponenial ype pinor orbial Slaer ype pinor orbial Addiion heore Relaiviic Dirac-Harree-Foc-Roohaan heory 1. Inroducion The oluion of he Dirac equaion for hydrogen-lie ye play a ignifican role in heory and applicaion o relaiviic quanu echanic of ao olecule and nuclei. However he relaiviic hydrogen-lie poiion orbial and heir eenion o oenu and four-dienional pace canno be ued a bai e becaue hey are no coplee unle he coninuu i included [1-4]. In Ref. [5] we have conruced in poiion oenu and four-dienional pace he coplee orhonoral e of wo- and four-coponen relaiviic pinor wave funcion baed on he ue of coplee ohonoral e of nonrelaiviic orbial. By he ue of hi ehod in a previou wor [6] we inroduced he new coplee orhonoral e of relaiviic ETSO) and Ψ -eponenial ype pinor orbial ( Ψ - Χ -Slaer ype pinor orbial ( Χ -STSO) for paricle wih arbirary half-inegral pin in poiion oenu and four-dienional pace hrough he correponding

2 nonrelaiviic ψ -eponenial ype orbial ( ψ -ETO) [7] and χ -Slaer ype orbial (χ - STO). The elaboraion of algorih for he oluion of generalized Dirac equaion [8] in linear cobinaion of aoic pinor orbial (LCASO) approach neceiae progre in he developen of heory for one-range addiion heore of pinor orbial of uliple order. Addiion heore play a ore and ore iporan role in nonrelaiviic and relaiviic aoic and olecular elecronic rucure calculaion [9]. Two fundaenally differen ype of addiion heore occur in he lieraure. The fir ype of he addiion heore ha he wo-range for of Laplace epanion for he Coulob poenial. There i econd cla of addiion heore which can be conruced by epanding a funcion locaed a a cener a in er of a coplee orhonoral e locaed a a cener b. The ue of onerange addiion heore in elecronic rucure calculaion would be highly deirable ince hey are capable of producing uch beer approiaion han he wo-range addiion heore. In Ref.[10-13] we have developed he ehod for conrucing in poiion oenu and four-dienional pace he one-range addiion heore of coplee orhonoral e of nonrelaiviic ψ -ETO and χ -STO. The ai of hi wor i o derive he relevan epanion and one-range addiion heore of coplee orhonoral e of relaiviic Ψ -ETSO and Χ -STSO in poiion oenu and four-dienional pace hrough he correponding heore for nonrelaiviic orbial ψ -ETO and χ -STO. Thee heore igh be ueful for he calculaion of ulicener inegral which appear in relaiviic MO LCASO heory of arbirary half-inegral pin paricle when he pinor orbial bai e in poiion oenu and four-dienional pace are eployed. 2. Definiion and baic forula In order o derive he epanion and one-range addiion heore for 2(2+- coponen pinor orbial in poiion oenu and four-dienional pace we ue he following definiion: Coplee orhonoral e of nonrelaiviic orbial ( ζ ) ψ ( ζ ) φ ( ζ ) nl nl r nl z nl( ζ ω ) ( ( ζ ) ψ ( ζ r) φ ( ζ ) z ( ζ ω ) (2) nl nl nl nl

3 Slaer ype nonrelaiviic pinor orbial ( ζ ) χ ( ζ r) u ( ζ ) v ( ζ ω ) (3) nl nl nl nl Coplee orhonoral e of 2(2+-coponen relaiviic pinor orbial ( ζ ) Ψ ( ζ r) Φ ( ζ ) Z ( ζ ω ) (4a) nl nl nl nl ( ζ ) Ψ ( ζ r) Φ ( ζ ) Z ( ζ ω ) (4b) nl nl nl nl ( ζ ) Ψ ( ζ r) Φ ( ζ ) Z ( ζ ω ) (5a) nl nl nl nl ( ζ ) Ψ ( ζ r) Φ ( ζ ) Z ( ζ ω ) (5b) nl nl nl nl Slaer ype 2(2+-coponen relaiviic pinor orbial ( ζ ) Χ ( ζ r) U ( ζ ) V ( ζ ω ) (6a) nl nl nl nl ( ζ ) Χ ( ζ r) U ( ζ ) V ( ζ ω ) (6b) nl nl nl nl where r ω and ω βθϕ. See Ref.[6] and [14-15] for he eac definiion of quaniie occurring in Eq (-(6). We hall alo ue he following forula for 2(2+-coponen pinor orbial hrough he independen e of wo-coponen pinor defined a a produc of coplee orhonoral e of radial par of nonrelaiviic calar ψ -ETO and odified Clebch-Gordan coefficien appearing in wo-coponen enor pherical haronic (ee Ref.[6] and [14-15]): for ETSO 0 nl 2 nl ( ζ ) ( ζ ) ( ζ ) 2 1 nl nl ( ζ ) = nl 2 1 ( ζ nl ) 2 nl 0 nl ( ζ ) ( ζ ) (7a)

4 η a ( λ) ( ζ ) = η ( λ ( ζ ) a l λ nl ( ζ ) nl l + nl (7b) l ( ) λ λ ζ nl ( ζ ) = nl (7c) l (2 ( λ + ) ( ζ ) nl for ETSO 0 nl 2 nl ( ζ ) ( ζ ) ( ζ ) 2 1 nl nl ( ζ ) = nl 2 1 ( ζ nl ) 2 nl 0 nl ( ζ ) ( ζ ) η a ( λ) ( ζ ) = η ( λ ( ζ ) a l λ nl ( ζ ) nl l + nl (8a) (8b) l ( ) λ λ ζ nl ( ζ ) = nl (8c) l (2 ( λ + ) ( ζ ) nl for STSO 0 nl 2 nl ( ζ ) ( ζ ) ( ζ ) 2 1 nl nl ( ζ ) = nl 2 1 ( ζ nl ) 2 nl 0 nl ( ζ ) ( ζ ) (9a)

5 η a ( λ) ( ζ ) = η ( λ ( ζ ) a l λ nl ( ζ ) nl l + nl (9b) l ( ) λ λ ζ nl ( ζ ) = nl (9c) l (2 ( λ + ) ( ζ ) nl where λ= Epanion and one-range addiion heore for ETSO and STSO Wih he derivaion of epanion and one-range addiion heore for 2(2+-coponen pinor orbial in poiion oenu and four-dienional pace we ue he ehod e ou in previou paper [16-17] decribed for he nonrelaiviic cae. Then uing Eq. (7)-(9) and carrying hrough calculaion analogou o hoe for he nonrelaiviic bai e we obain he following relaion in er of nonrelaiviic cae: EXPANSION THEOREMS: for ETSO ( ζ ) ( ζ ) = F ( ζ ζ ; ) + F ( ζ ζ ; ) (10a) ' λ λ nl n l λ= 0 nl n l nl n l ; F a ( ) a ( ) ( ) ( ) ( ) λ l l ζ ζ = ηη λ λ ζ ζ nl nl n l n l l l + a ( ζ ) ( ζ ) a nl n l ; a (2 ) a (2 ) ( ) ( ) ( ) λ l l F ζ ζ = λ λ ζ ζ nl n l nl n l + a (2 ) a (2 ) ( ζ ) ( ζ ) l l nl n l (10b) (10c) for STSO λ λ ( ζ ) ( ζ ) = F ( ζ ζ ; ) + F ( ζ ζ ; ) (11a) ' nl n l λ= 0 nl n l nl n l F ζ ζ ; ηη a ( λ) a ( λ) ( ζ ) ( ζ ) ( ) = λ l l nl nl n l n l + λ+ λ+ ζ ζ l l a ( a ( ( ) ( ) nl n l (11b)

6 F ζ ζ ; a (2 λ) a (2 λ) ( ζ ) ( ζ ) l l ( ) = λ nl ( λ) nl n l n l + a + a + l l (2 (2 ( ζ ) ( ζ ). nl n l (11c) ONE-RANGE ADDITION THEOREMS: for ETSO η a ( λ) ( ζ y) η ( λ + ( ζ ) a y l λ nl ( ζ ) = nl l nl y (12a) l ( ) λ λ ζ nl y ( ζ ) = nl y (12b) l (2 ( λ + ) ( ζ ) nl y for STSO η a ( λ) ( ζ y) η ( λ + ( ζ ) a y l λ nl ( ζ ) = nl l nl y (13a) l ( ) λ λ ζ nl y ( ζ ) = nl y (13b) l (2 ( λ + ) ( ζ ) nl y where r ω and y R p ω. p The forula for he epanion and one-range addiion heore for quaniie ( ( ζ ) ( ζ nl n l ) nl( ζ )) y and( ( ζ ) ( ζ nl n l ) nl( ζ y )) occurring on he righ hand ide of hee equaion have been eablihed in previou wor [16 17] and [18 19] repecively. A can be een fro he forula of hi wor all of he epanion an one-range addiion heore of 2(2+-coponen ETSO and STSO defined in poiion oenu and fourdienional pace are epreed hrough he correponding nonrelaiviic epanion and one-range addiion heore. Thu he relaion of nonrelaiviic epanion and one-range addiion heore derived in previou paper [16-19] can be alo ued in he cae of 2(2+-coponen pinor orbial in poiion oenu and four-dienional pace.

7 Reference 1. I.P. Gran Relaiviic Quanu Theory of Ao and Molecule Springer G. Dyall.Fægri Inroducion o Relaiviic Quanu Cheiry Oford Univeriy Pre I.P. Gran H.M. Quiney Adv. A. Mol. Phy. 23 (1998) R. Szyowi J. Phy. A: Mah. Gen. 31 (1998) I.I. Gueinov J. Mah. Che. 47 (2010) I.I. Gueinov Copu. Phy. Coun.(ubied). 7. I.I. Gueinov In. J. Quanu Che. 90 (2002) I.I. Gueinov arxiv: v4. 9. I.N. Levine Quanu Cheiry 5 h ed. Prenice Hall New Jerey I.I. Gueinov J. Mol. Model. 9 (2003) I.I. Gueinov J. Mol. Model. 9 (2003) I.I. Gueinov J. Mol. Model. 11 (2005) I.I. Gueinov J. Mol. Model. 12 (2006) I.I. Gueinov Phy. Le. A 372 (2007) I.I. Gueinov Phy. Le. A 373 (2009) I.I. Gueinov J. Mah. Che. 42 (2007) I.I. Gueinov J. Mah. Che. 43(2008) I. I. Gueinov J. Theor. Copu. Che.7(2008) I. I. Gueinov Chin. Phy. Le. 25 (2008) 4240

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