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
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
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)
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)
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= 0 2... 2 1. 2. 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) 2 1 + ' 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) 2 1 + ' 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)
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.
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