ψ - exponential type orbitals, Frictional

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1 ew develoment in theoy of Laguee olynomial I. I. Gueinov Deatment of Phyic, Faculty of At and Science, Onekiz Mat Univeity, Çanakkale, Tukey Abtact The new comlete othonomal et of L -Laguee tye olynomial ( L -LTP, =,,,,,... ae uggeted. Uing Schödinge euation fo comlete othonomal et of ψ -eonential tye obital ( ψ -ETO intoduced by the autho, it i hown that the oigin of thee olynomial i the centally ymmetic otential which contain the coe attaction otential and the uantum fictional otential of the field oduced by the aticle itelf. The uantum fictional foce ae the analog of adiation daming o fictional foce uggeted by Loentz in claical electodynamic. The new L -LTP ae comlete without the incluion of the continuum tate of hydogen like atom. It i hown that the nontandad and tandad convention of L -LTP and thei weight function ae the ame. A an alication, the et of infinite eanion fomula in tem of L -LTP and L-Genealized Laguee olynomial (L-GLP fo atomic nuclea attaction integal of Slate tye obital (STO and Coulomb-Yukawa like coelated inteaction otential (CIP with intege and nonintege indice ae obtained. The aange and eaanged owe eie of a geneal owe function ae alo invetigated. The convegence of thee eie i teted by calculating concete cae fo abitay value of aamete of obital and owe function. Key Wod: Centally ymmetic otential, ψ - eonential tye obital, Fictional uantum numbe, Genealized Laguee olynomial. Intoduction It i well known that the eigenfunction of the Schödinge euation fo the hydogen-like atom the adial at of which contain the L-GLP ae not comlete ue the continuum i included []. Becaue of thi, ome difficultie aie in the olution of diffeent hyical oblem when the L-GLP ae emloyed (ee Ref. [] and efeence uoted theein. Theefoe, the neceity aie fo the contuction of the new L -LTP uing comlete othonomal et of adial at of ψ -ETO []. A ha been hown in a eviou ae [4], the eigenfunction ψ -ETO coeond to the total centally ymmetic otential which contain the coe attaction otential and the Loentz fictional otential of the field oduced

2 by the aticle itelf. We notice that the Lambda and Coulomb-Stumian ETO intoduced in Ref. [5-8] ae the ecial cae of the ψ -ETO fo = and =, eectively.. Definition and baic fomula L The new L -LTP in non-tandad convention uggeted in thi wok ae defined a (! n l l k ( = ( L( nk ( n (! = Π k= l whee ( ( n ( ( k l!, ( k l F l! + kl Π = ( nk! F m ( n m ( n m ( a n! fo m n b =!! fo m < and m > n. c < (ee Ref. [4], = l +, Hee, i the fictional uantum numbe ( ( ( ( = n+ l + and L ae the non-tandad L-GLP. The L -LTP ae othonomal with eect to the weight function n : e ( n l ( d = δ nn L L ( n L ( = L (. (4 ow we take into account E. ( in the fomulae fo adial at of ψ -ETO in nontadad convention [] which ae the comlete othonomal et of eigenfunction of Schödinge euation fo hydogen-like atom. Then, we obtain: ( (, = ( R ζ ζ R ( (, = ( R ζ ζ R whee = ζ and (5a, (5b ( = L ( R e (6a

3 ( ( R = e L. (6b We note that the imila fomula can alo be deived in tandad convention uing the elation ( ( ( ( = ( = ( ( L! L! L, (7 whee L ( ( ae the tandad L-GLP. The non-tandad and tandad convention fo the L-GLP wee dicued in Ref. [9] and [], eectively (ee Ref. [] fo the definition of ψ -ETO in tandad convention. Taking into account E. ( and (7 we obtain fo the L -LTP in tandad convention the following elation: L ( l ( ( = (! L ( (8a n L ( ( ( = L (. (8b ( It i eay to how that the L ae othogonal with eect to the weight function ( ( e ( n'l ( d= δ nn L L. (9 The adial at of comlete otonomal et of ( the L ae defined a ( ( ( ( ζ, = ζ ( R R n ( ψ -ETO in tandad convention though : (a ( ( ( ( ζ, = ζ ( R R, (b whee ( ( ( = L ( R e (a ( ( ( ( R = e L. (b It i eay to how that the nontandad L and tandad ( ( L and L, eectively, ae the ame, i.e., ( L = L and L = L. ( L and thei weight function

4 The nontandad L - and tandad ( L -LTP fom the comlete othonomal et on the inteval (, with the weight function ( L and L, eectively. The comletene elation can be oved by the ue of method et out in Ref. [] (ee alo []. It i eay to how that + ( n=+ l n=+ l ( ( = ( ( = δ ( e L L R R ( ( ( ( ( ( ( ( L L = ( ( = δ ( + e R R n=+ l n=+ l It hould be noted that, in the ecial cae of the. ( ( L - and L -LTP fo = and =, the E. ( and ( decibe the comletene oetie of Lambda and Coulomb- Stumian function with nontandad and tandad convention, eectively (ee Ref.[5-8].. Diffeential euation of L -LTP In ode to deive the diffeential euation fo L -LTP we ue the Schödinge euation fo the adial at R ( in the fom ( + d d l l ζ V( R ( ( + + = εr d d ζ, (4 whee ε = ζ (ee Ref. [4]. Hee, V( ζ, denote the otential of the centally ymmetic field which coeond to the eigenfunction ψ -ETO. The ubtitution of E. (6a into (4 yield the following euation fo the L -LTP: ( + d L dl dl l l + ( ( V( + + L =. (5 d d d ζ et we ue the fomula ( in the euation dl dl ( ( L = (6 d d fo non-tandad L-GLP [] and the condition k d L k d = L. (7 + k Then, a imle algeba lead to ( + l( d L dl l l + ( ( n + + = d d L. (8 4

5 Thu, we obtained fo the L -LTP two kind of indeendent euation one of which, E. (5, contain the otential V( V ( ζ, n. The comaion of thee euation give V ( ζ, = U ( ζ, + U ( ζ,, (9 whee the fit and econd tem ae the coe attaction and fictional otential, eectively, ζ ( ζ, = ( Un n ( L ( ζ d l U ( ζ, = L a d ζ ( + = L L b whee = ζ. Hee, the function U ( ( ( (, ( i the Loentz fictional elf-otential of the field oduced by the aticle at the oint whee it i located. See Ref. [4] fo the decition of oetie of thee otential. Thu, we have etablihed a lage numbe ( =,,,,,... of indeendent comlete othonomal et of elation fo fictional otential of the field oduced by the aticle itelf. L -LTP the oigin of which i the coe attaction and 4. Ue of L -LTP and L-GLP in tudy of owe eie of a geneal owe function In thi ection we conide the aange and eaanged owe eie of a function n ( ξ, = = ξ + η ξ f e e whee, ( n i the intege at of * and < η* <. Fo thi uoe we utilize the following Laguee eie obtained with the hel of L -LTP and L-GLP: fo L -LTP ( ξ ( η e ξ A η = + fo L- GLP = L, ( ( ξ ( η e ξ B L η = =, (4 whee ξ <, =,,,,,..., =,,,... and 5

6 ( ξ ( ( η + + ( + ξ = Π η η = A B ( ξ = Γ + + (5 ( -! Γ ( η β (! = ( + ξ η η. (6 Hee, β ae the Laguee coefficient detemined by ( = β (7 = L + ( ( F ( F ( ( 8 β =!. + ow we obtain the owe eie. Fo thi uoe, we ue the method et out in eviou ae [,4]. Then, taking into account the oetie (b and (c in E. ( and (7 we obtain fo the aange, E. (9a and (a, and eaanged, E. (9b and (b, owe eie the following fomulae: fo L -LTP η ξ = ( ξ ( η Π = η = + = = + = e A lim A ξ Π (9a ( ξ = lim Q,, (9b = fo L-GLP η ( ξ β ( ξ β η ξ e = B = lim B η η = = = = = ( ξ (a = lim D,, (b whee η Q ( ( A, ξ = η ξ Π ( η = + D ( ( B, ξ = η ξ β. ( η = A an eamle of alication of E. (9a, (9b, (a and (b we calculate the atomic nuclea attaction integal of STO and Coulomb-Yukawa like CIP with intege and nonintege indice defined by [5] 6

7 ,, =,,, d, ( ( ζ ζ ξ χ ( ζ χ ( ζ ( ξ I f whee n lm, n lm, σ and χ ( ζ, = R ( ζ, Slm ( θϕ, m n + n n (, = ( Γ ( + R ζ ζ n e ζ n (4 (5 4 σ π ( = ( ( f ξ, f ξ, S θ, ϕ (6 σ + ( ξ = ξ f, e. (7 It i eay to deive fo integal ( the elation σ σ ( ζ, ζ, ξ = (, ( ζ, ζ, ξ I C lm lm A I mm n n σ whee C ( lm, lm = nn n n ae the genealized Gaunt coefficient [6] and ( ζ ζ ξ ( ζ ( ζ ( ξ, (8 I,, R, R, f, d. (9 Thee integal ae detemined fom the following analytical elation: I ( ζ ζ ξ = ( ζ ζ ( + + ( ε + ξ Γ + +, ',, ', (4 nn' nn' whee ε = ζ + ζ ', * = n* + n'* and nn' ( ζ ζ n + / n' + / ( ζ ( ζ' ( n ( n, ' =. (4 Γ + Γ + ow we evaluate the integal (9 uing elation (9a, (9b, (a and (b. Then, it i eay to deive fo the atomic nuclea attaction integal the following lage numbe of aange, E. (4a and (4a, and eaanged, E. (4b and (4b, owe eie eanion elation: fo L -LTP I A J lim A J nn Π Π = + = = + = n ( ζ, ζ, ξ n = ( ξ + ( ζ ζ ( ξ + ( ζ ζ ( 4 nn nn η, = η, a n+ (, ξ ( ζ, ζ, ( = lim Q J 4b = η nn 7

8 fo L-GLP I B J lim B J nn = = = = n ( ζζ,, ξ = n η ( ξ + ( ζ ζ ( ( ( 4a β nn η ξ +, = ζ ζ β nn, = ( ζ ζ ( n+ = lim D (, ξ J,, 4b η nn whee < and <. Hee, the uantitie J κ ( ζ ζ nn, ae detemined by ( κ Γ + + J, R, R, d, ', (44 κ κ+ ( ζζ = ( ζ ( ζ ( ζζ nn = n n nn' + κ + ε whee κ = n+ and κ = n umeical eult and dicuion The alicability of the aange and eaanged owe eie obtained fom the ue of comlete othonomal et of L -LTP and L-GLP i teted by calculating the atomic nuclea attaction integal detemined by E. (4a, (4b and (4a, (4b, eectively. On the bai of thee fomulae we contucted the ogam which ae efomed in the Mathematica 7. language ackage. The convegence oetie of Coulomb (fo ξ = and Yukawa (fo ξ = 5. like nuclea attaction integal fo and, eectively. = and ae hown in Figue The Figue and how a good ate ageement of value obtained fom the aange and eaanged eie eanion elation. Thu, the E. (4a, (4b, (4a and (4b dilay the mot aid convegence a a function of ummation limit fo = 4. We notice that the geate accuacy i attainable by the ue of moe tem in eie eanion elation obtained in thi wok. 6. Concluion We have demontated that the detemination of aange and eaanged owe eie of a function f ( ξ, obtained by the ue of L -LTP and L-GLP i legitimate fo the intege and nonintege value of indice *. A we ee fom ou tet that the owe eie deived in thi wok with the hel of L -LTP and L-GLP can be ueful tool fo evaluation of the multicente nuclea attaction integal when aange and eaanged one-ange addition theoem fo STO and Coulomb-Yukawa like CIP eented in ou eviou ae ae emloyed. 8

9 Refeence. I.. Levine, Quantum Chemity, 5 th ed. (Pentice Hall, ew Jeey,.. E. J. Wenige, J. Math. Chem., 5 ( 7.. I. I. Gueinov, Int. J. Quantum Chem., 9 ( I. I. Gueinov, 6 th Intenational Confeence of the Balkan Phyic Union, Ameican Intitute of Phyic Confeence Poceeding, 899 ( E.A. Hylleaa, Z. Phy., 48 ( E.A. Hylleaa, Z. Phy., 54 ( H. Shull, P.O. Lowdin, J. Chem. Phy., ( P.O. Lowdin, H. Shull, J. Chem. Phy., ( P. Kaije, V.H. Smith, Adv. Quantum Chem., ( W. Magnu, F. Obehettinge, R.P. Soni, Fomula and Theoem fo the Secial Function of Mathematical Phyic (Singe, ew Yok, I. I. Gueinov, J. Math. Chem., 4 ( R. Szmytkowki, J. Phy. B, (997 85, Aendi E.. J. C. Slate, Quantum theoy of atomic tuctue, Vol., Mc Gaw-Hill, ew Yok, I.I Gueinov, Phy. Rev. A, ( I. I. Gueinov, J. Theo. Comut. Chem., 7 ( I. I. Gueinov, J. Phy. B, (

10 a I..,4 H.56,4.65,L ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò æ ò ò ò ò òæ ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç æ æ analytical a= a= ò a= a=- ç a=- Fig.. Convegence of aange (4a and eaanged (4b owe eie fo Coulomb like. integal I., 4 (.56, 4.65, fo a I..,4 H.56,4.65,5.L ç ò æ æ ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç æ ò ò ò ò ò ò ò ò ò ò ò ò ò ç ò ò ò ò ò ç ò òç æ analytical a= a= ò a= a=- ç a= Fig.. Convegence of aange (4a and eaanged (4b owe eie fo Yukawa like. integal I.,4 (.56,4.65,5. fo

Basic propositional and. The fundamentals of deduction

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