The Three-Body Coulomb Potential Polynomials ABSTRACT

Transcription

1 Malaysa Joural of Mathematcal Sceces (): - (00) The Three-Body Coulomb Potetal Polyomals Agus Kartoo ad Mustafa Mamat Laboratory for Theoretcal ad Computatoal Physcs, Departmet of Physcs, Faculty of Mathematcal ad Natural Sceces, Isttut Pertaa Bogor, Kampus IPB Darmaga, Bogor 660, Idoesa. Departmet of Mathematcs, Faculty of Sceces ad Techology, Uverst Malaysa Tereggau, 00 K. Tereggau. Malaysa. E-mal: akartoo@pb.ac.d, mus@umt.edu.my ABSTRACT I ths paper, we preset a geeral aalyss of the three-body Coulomb potetal polyomals. We show why the three-body Coulomb wave fuctos expaso a o-orthogoal Laguerre-type fucto bass gves two modfed Pollaczek polyomals. The froze-core model s used to exame the three-body Coulomb Hamltoa. The resultg three-term recurrece relato s a specal case of the Pollaczek polyomals whch s a set of orthogoal polyomals havg a oempty cotuous spectrum addto to a fte dscrete spectrum. The completeess of the three-body Coulomb wave fuctos s further studed for dfferet Laguerre bass sze. Keywords: three-body Coulomb, o-orthogoal Laguerre, Pollaczek polyomal, froze-core model INTRODUCTION I atomc ad uclear scatterg, t s ofte desrable to use Slater or oscllator fucto as a bass fucto. Heller ad Yama [] have preseted a ew method for performg scatterg calculatos etrely wth squaretegrable (L ) fuctos. They developed techques whch they attempted to take full advatage of the aalytc propertes of a gve Hamltoa ad also of the L fucto bass whch was used to descrbe the wave fuctos. They developed the basc theory usg o-orthogoal Laguerre-type fucto bass approprate for s-wave scatterg. A L dscretzato of the radal two-body Coulomb problem has bee gve by Yama ad Rehardt []. They preseted the relatoshp betwee the matrx egevalues of the L operator wth a cotuous spectrum, ad the assocated Gaussa quadrature was dscussed for the radal ketc eergy ad for the repulsve ad attractve Coulomb Hamltoas. It was show that dscretzato of the radal ketc eergy

2 Agus Kartoo & Mustafa Mamat a o-orthogoal Laguerre-type fucto bass gave a Ultrasphercal (Gegebauer) polyomal, whle dscretzato a oscllator-type bass geerated a Laguerre polyomal. They showed that o-orthogoal Laguerre-type fucto bass dscretzato of the Coulomb problem gave a Pollaczek polyomal the repulsve case ad a ew modfed Pollaczek polyomal the attractve case. The Coulomb problem, wth a attractve potetal, s kow to have a fte umber of boud states as well as a oempty cotuous spectrum. These suggest that the polyomals correspodg to the attractve Coulomb potetal wll be orthogoal wth respect to a dstrbuto fucto havg a absolutely cotuous compoet ad ftely may dscotutes. Board [] proved that the equvalet quadrature Fredholm determat ad the J-matrx method are essetally equvalet, yeldg the same result from the same approprate treatmet of the potetal ad exact treatmet of ketc eergy operator. It was show that relatg the spacg of the pseudostate egevalues to the relatve ormalzato of the pseudostate ad actual cotuum matrx elemets provded a alteratve to the Steltjes magg method. Board [] has also appled the bass of Yama ad Rehardt [] ad Gaussa quadrature based Pollaczek polyomals for calculatg two-photo processes hydroge. A ew expaso of the radal two-body Coulomb wave fuctos a orthogoal Laguerre-type fucto bass was later preseted by Stelbovcs []. It was show [6, 7] that the orthogoal Laguerre-type fucto bass could be drectly appled to the coupled-chaels formulato of electro-hydroge ad electro-helum scatterg. The purpose of ths paper s to preset a ew expaso of the threebody Coulomb wave fuctos a o-orthogoal Laguerre-type fucto bass. The froze-core model [6, 7] s used to calculate the three-body Coulomb Hamltoa. It s show that dscretzato of the radal ketc eergy ad the Coulomb problem the attractve case for the helum groud state (s) gve the modfed Pollaczek polyomals of Yama ad Rehardt [], whereas the other dscretzato of the radal ketc eergy ad the Coulomb problem the attractve ad electro-electro potetal case for the helum exctato states gve a ew modfed Pollaczek polyomal. Ths paper s orgazed as follows. I secto, we preset the threebody Coulomb wave fuctos ad o-relatvstc Hamltoa. Noorthogoal Laguerre-type fucto bass dscretzato of the three-body Coulomb potetal gves two modfed Pollaczek polyomals ad the Malaysa Joural of Mathematcal Sceces

3 The Three-Body Coulomb Potetal Polyomals froze-core model s used to exame the three-body Coulomb Hamltoa are gve Secto. The resultg three-term recurrece relato s show to be a specal case of the Pollaczek polyomals whch s a set of orthogoal polyomals havg a oempty cotuous spectrum addto to a fte dscrete spectrum. I secto, the completeess of the threebody Coulomb wave fuctos s the examed by Gaussa quadrature. The umercal results are gve Secto. Fally Secto 6, we draw the cocludg remarks from ths work. Accordgly, we troduce the fudametals of orthogoalty ad the aalytcal study of orthogoal polyomals. A dstrbuto fucto ( x) s a fxed o-decreasg fucto wth ftely may pots of crease the fte or fte terval [a, b], ad the momets, x d ( x) Malaysa Joural of Mathematcal Sceces b a, exst ad are fte for = 0,,,.... A set of polyomals { P ( x )}, where ( ) P x s a polyomal of precse degree, s called orthogoal wth respect to provded b P ( x) Pj ( x) d ( x) = 0, j a. () Wth a terval [a, b] ad weght fucto, w ( x), we may assocate Eq. () b w( x) P ( x) Pj ( x) dx = 0, j a () whch s defed for all the orthogoal polyomals, that s, b w( x) x dx < 0 for all. If ( x) a d ( x) () wth w( x) =. O the other had, f ( x) s absolutely cotuous, () reduces to s a jump fucto, that dx s costat except for jumps of the magtude w at x = x, the () reduces to sum w P ( x ) Pj ( x ) = 0, for j = () ) whch s the approprate defto for fuctos of a dscrete varable. A set of (where { x : } s the set of jump dscotutes of ( x)

4 Agus Kartoo & Mustafa Mamat orthogoal polyomals { ( ) : 0} relato ( ) ( ) ( ) ( ) P x wll satsfy a three-term recurrece P x = A x + B P x C P x, =,,.... () Here A, B, ad C are costats, A > 0 ad C > 0, ad the postvty codtos A >, =,,..., () A C 0 are satsfed. For may of the classcal polyomals there are aalytc relatos betwee fucto ad ts frst ad/or secod dervatves dp ( x) : 0 whch may be used to geerate the dervatves f dx x= x eeded. I the absece of such relatoshps, t s trval to dfferetate the recurso ay umber of tmes to obta equatos useful for computg dervatves. For example, ( ) ( ) ( ) dp x dp x dp x = ( A x + B ) C + A P x dx dx dx ( ) x= x x= x x= x may be used for the frst dervatve oce the fuctos are kow. Startg from a three-term recurrece relato t s possble to determe two learly depedet sets of polyomals wth tal codtos, (6) P ( x ) =, ( ) 0 ( ) dp x dx x= x P x = A x + B, 0 0 = A. 0 dp0 dx ( x) x= x = 0, (7) The recurrece relato () ad the postvty codto () mply orthogoalty []. Malaysa Joural of Mathematcal Sceces

5 The Three-Body Coulomb Potetal Polyomals The Pollaczek polyomals are defed by three-term recurrece relato ( ) ( ) ( ) ( ) P x; a, b + + a x + b P x; a, b P + + = 0 () =,,, wth tal codtos: P ( x; a, b) = 0 ad P ( x a b) geeratg fucto = 0 θ ( ) ( ) ;, + φ θ θ = φ θ ( ) ( ) ( ) 0 ;, =. They have the P x a b z ze ze, z <, (9) acosθ + b where x = cosθ, 0 θ π, ad φ ( θ ) =.Whe a ad b real, sθ a b, >, the Pollaczek polyomals satsfy the orthogoalty relato the terval x + P ( x; a, b) P j ( x; a, b) w ( x; a, b) dx = jδ, j, = Γ ( + ) ( + + ), (0) a /! wth the weght fucto [9, 0] / w ( x; a, b) = exp{ ( θ π ) φ ( θ )}( x ) Γ ( + φ ( θ )). () π THE THREE-BODY COULOMB WAVE FUNCTIONS AND HAMILTONIAN We cosder frst a system of two electros LS couplg. We defe the orbtal fuctos radal, φ l, sphercal harmoc, Y ( ˆ lm r ), ad sp fucto, χ ( σ ), for a sgle-electro as ϕ ( x) = φ ( ) ( ˆ l r Ylm r) χ ( σ ). () r Malaysa Joural of Mathematcal Sceces

6 Agus Kartoo & Mustafa Mamat Here x s used to deote both the spatal ad sp coordates. The radal part of the sgle-partcle fuctos, ca the be wrtte usg the o-orthogoal Laguerre-type fucto bass (see Yama ad Rehardt []) l+ l + ( r) ( r ) exp( r / ) L ( r) φ =, () l l l l where the l + L ( r) l are the assocated Laguerre polyomals, l s the teracto parameter ad rages from to the bass sze. The twopartcle space s wrtte terms of the product of these orbtal for coordates r ad r. We may rearrage these products to lear combatos whch are egevalues of the total orbtal agular mometum ad total sp ϕ ϕ π υ = φ φ υ, () ( x ) ( x ) : lms ( r ) ( r ) l l : lm X ( s ) l l r r the otato ad are used to deote the frst ad secod electro, where ( ˆ ) ( ˆ ) l l lm = l l m m lm Y r Y r, () : l m l m m, m ad the two-electro sp fucto s defed by = µ µ. (6) ( υ ) µ µ sυ χ ( σ ) χ ( σ ) X s σ σ, The three-body Coulomb wave fuctos cofgurato teracto form are Φ x, x = C φ r φ r l l : lm X sυ, (7) ( ) π lmsυ ( ) π l l, r r ( ) ( ) ( ) where the cofgurato teractos are chose so that the selecto rules are satsfed for the combato () ad they are correctly at-symmetrzed two-electro states of party ( ) l + l ad wth total orbtal agular mometum egevalues l, m ad sp egevalues s,υ. Here the ( ) cofgurato teracto coeffcets C satsfy the symmetry property ( ) ( ) l + l l s ( ) C = C, () to esure at-symmetry of the two-electro system states. 6 Malaysa Joural of Mathematcal Sceces

7 The Three-Body Coulomb Potetal Polyomals as The o-relatvstc three-body Coulomb Hamltoa ca be wrtte H = H + H + V, (9) where Z H = K + V =, (0) r for =,, s the oe-electro Hamltoa of the He + o (Z = ), ad V = r r, () s the electro-electro potetal. Atomc uts (a.u.) are assumed throughout. THE THREE-BODY COULOMB POTENTIAL POLYNOMIALS We cosder frst a system of two electros LS couplg. Whereas the geeral Hamltoa formalsm Eq. (9) cludes two-electro exctato, practce we have foud that t s suffcet to use the frozecore model, where oe of the electros s a fxed orbtal (the groud state) whle the secod electro s descrbed by a set of depedet L fuctos, thus permttg t to spa the dscrete ad cotuum exctatos, whch all cofguratos have oe of the electros occupyg the lowest orbtal. I order to get a good descrpto of the groud state (s) polyomal, Φ x, x we must dagoalzed the groud state Hamltoa ( ) where m r Z Φ ε Φ = 0, () ε s the eergy assocated wth the s state of He + o. By usg the recurrece relatos, orthogoalty relato ad dfferetato formula of the Laguerre polyomal [, 0], Eq. () fally becomes l Z Z + l + ( ) l + P ( ) ( ) x l x P x l P ( x ) = + + +, l l () =,,.... Malaysa Joural of Mathematcal Sceces 7

8 To talze the recurrece oe sets Agus Kartoo & Mustafa Mamat l + l + ( x ) = 0 ; P ( x ) P 0 =, () where ad ( ) ( l ) Γ ( + ) ( ) l + Γ + + P x = C x, () x l ε = l ε +. (6) The eergy ε whch are obtaed from (6) are gve by l + x ε =. (7) x The exctato states polyomal for ( ) by solvg the equato m r r r Φ x, x ca ow be obtaed Z Φ + ε Φ = 0, () where ε s the eergy assocated wth the exctato states of the helum atom. The matrx elemets of the electro-electro potetal teracto for states where the orbtal agular mometa of the two o-equvalet electros (dfferet ad l) are coupled to a specfc l ad the sps to a specfc s. The expectato value of the electro-electro potetal teracto for ths state cossts of two drect terms for whch the orderg of the quatum umbers s the same o both sdes ad two exchage terms for whch the order s reversed. The electro-electro potetal matrx elemets may the be wrtte as the dfferece of a drect ad a exchage matrx elemet, for a LS cofgurato ( l = 0, l = l ), ths reduces to the smple expresso Malaysa Joural of Mathematcal Sceces

9 The Three-Body Coulomb Potetal Polyomals Φ Φ = ( ) ( ) lmsυ lmsυ r r φ φ φ φ + ( r ) ( r ) ( r ) ( r ) r ( ) ( l + ) 0 0 s l r< l + r> ( ) ( ) ( ) ( ) φ r φ r φ r φ r dr dr (9) whch r < stads for the smaller of the two dstaces r ad r, r s > greater of the two dstaces r ad r. Ths cofgurato has two LS terms: a trplet term wth s = ( l ) ad a sglet term wth s = 0 ( l ). These two terms are splt by the exchage part of the electro-electro potetal teracto. Thus, ths splttg of the two sp states s a cosequece of the at-symmetry of fuctos. I order to use ths formula for a real system, made up of dstgushable partcles, we must, of course, use properly atsymmetrzed fuctos. After the same step as the groud states equato ad the electroelectro potetal teracto calculato of a two-electro system, Eq. () ca be wrtte as Z Z l V x V P x ( ) l + + exc + exc l l l l l + l + ( + l ) P ( x ) P ( x ), =,,..., = + (0) to talze the recurrece oe sets where V exc l + l + ( x ) = 0 ; P ( x ) P ( ) ( l ) Γ ( + ) 0 ( ) Γ l P x = C x ( ) l s l r< l + r> ( r ), ( r ) ( r ), ( r ) =, (), () l φ ( r ) φ ( r ) φ ( r ) φ ( r ) dr dr l ( l ) =, () φ φ φ φ Malaysa Joural of Mathematcal Sceces 9

10 Agus Kartoo & Mustafa Mamat ad l ε x = l ε +. () The eergy ε whch s obtaed from equato () s gve by l + x ε =. () x It s clear that the resultg three-term recurrece relato () ad (0) are a specal case of recurrece relato () satsfed by the Pollaczek polyomals. The three-term recurrece relato () ad (0) are defed by the geeralty three-term recurrece relato ( ) ( ) ( ) ( ) ( ) P x + a x + a P x + + P x = 0, =,,, (6) where (the subscrpt of ad for () ad (0) respectvely), ad a Z = ; l a = Z Vexc, (7) l l = l + ; = l +, () are orthogoal polyomals whe the postvty codto () holds. The followg sequece of equaltes must be vald ( a)( a)( ) 0 < + + +, =,,.... (9) Case A: > 0. The equalty (9) wth = s ( a )( a)( ) > 0. (0) 0 Malaysa Joural of Mathematcal Sceces

11 The Three-Body Coulomb Potetal Polyomals Ths equalty wll be satsfed f ( a) > 0. Moreover, the remag equaltes (9) for =,,... wll also be satsfed. O the other had, f ( a) < 0, the there wll be a smallest teger k such that ( k + a) > 0. The kth equalty (9) wll fal, sce( k + a ) 0. Ths shows that f > 0, the the polyomals geerated by the recurrece relato (6) are orthogoal wth respect to a postve measure f ad oly f ( a) > 0. Case B: < < 0. Sce s egatve, (0) holds whe ( a ) ad have opposte sgs. I partcular, (0) holds ( a) whe ( a) > 0 ad ( a ) ( a) 0 < 0. It follows that ( a) > 0 ad + >. The equalty (9) for = s ( a)( a)( ) + > 0. () Ths hold because > 0 ths case. Moreover, each of the terms (9) wll be postve for the remag equaltes whe =,,. Whe ether ( a) 0 a > 0, (0) fals. Ths shows that for <, so ( ) < or ( ) < < 0 the polyomals geerated by recurrece (6) are orthogoal wth respect to a postve measure f ad oly f < a < 0. Case C: <. I ths case ( ) < 0 ad ( ) (0) would requre ( a ) ad ( a) () requres ( a) ad ( a) < 0. The valdty of to have opposte sgs. Now + to have opposte sgs. Ths s mpossble. Ths shows that whe < the recurrece (6) wll ever geerate a fte set of polyomals orthogoal wth respect to a postve measure. Malaysa Joural of Mathematcal Sceces

12 Agus Kartoo & Mustafa Mamat I summary, the postvty codto for () holds f ad oly f or > 0 ad ( a) > 0 (a) < < 0 ad < a < 0. (b) THE COMPLETENESS OF THE THREE-BODY COULOMB WAVE FUNCTIONS Wth the ad of the recurrece relatos ad orthogoalty relato of Laguerre polyomal ad Chrstoffel-Darboux relato [] satsfed by the Pollaczek polyomal, the ormalzato cofgurato teracto costats C ad C ca be expressed as follows ( ) ( ) Γ + l + l l d l C P l ( x ) P = x l Γ dx x x + = ad ( ) Γ ( + ) Γ + l + l l d l C P l ( x ) P = x dx l x= x (a). (b) The observato that for the Laguerre bass () the three-body Coulomb wave fuctos have L expaso coeffcets proportoal to the Pollaczek polyomal ca be exploted further to show a coecto wth Gaussa quadrature rules. Cosder the completeess relato for the true egefuctos folded betwee two arbtrary L wave fuctos f ad g f Φ Φ g + f Φ Φ g de = f g () l l El El 0 ad a fte bass represetato the space spaed by the frst bass states of Eq. () type f Φ Φ g = f g. () Malaysa Joural of Mathematcal Sceces

13 The Three-Body Coulomb Potetal Polyomals Geerally f ad g may be chose to possess a fte umber of ozero Fourer coeffcets so f g f g, (6) but f g f g as. (7) To derve a equvalet quadrature rule for ths covergece we wrte () as ( ) ( ) ( ) ( ) ( ), ( ) f g = C f Φ x Φ x g. () We see that the mpled quadrature rule for a fucto f(x) s b a = = ( ) ( ) ( ) ( ) ( ) ( ) w x f x dx C f x. (9) The terval lmt [a, b] comprse ay terval whch covers the pot ad cotuum mapped to x varable. The quadrature rule (9) ca be detfed as a specal case of Gaussa quadrature based Pollaczek polyomals. Ths ca be see by otg some stadard results that for geeral orthogoal polyomal there exsts a Gaussa quadrature formula gve by Chhara []. The assocated quadrature weghts, w, are gve by w ( ) πγ ( + l + ) = l Γ ( + ) l+ ( ) d l ( ) ( ) ( ) P x P x dx. (0) The cofgurato teracto coeffcet determed usg (0) s gve by ( ) C Eq. () whch are the l ( ) ( C ) = ( ) π l( ) ( ) ( x ) w ( ). () Malaysa Joural of Mathematcal Sceces

14 Agus Kartoo & Mustafa Mamat NUMERICAL RESULTS We state a umber of results that ca be obtaed by choosg subsets of the bass we have trucated the Fourer expaso. Ths s equvalet to mposg the boudary codto that the (+)th coeffcet s zero, amely that P l+ ( x ) = 0, = 0,,..., - () as the otato mples there are real roots to the Eqs. () ad (0). A good descrpto of the groud state ( ε = -.00 a.u.), we take l =.0 for = (). The secod electro ca be ay l state, we use the set of =, 0, ad 0 wth l = 0.9 for the, S states to obtaed a approxmato of egatve ad postve states (0). All of the roots ad resultg egevalues are preseted Table. All excted-state eerges are descrbed to a accuracy of better tha 0.% ad are preseted Table. We obtaed the eergy of -.6 a.u. for S the bass sze. If s show by creasg the umber of bass sze up to 0 we obtaed the covergece umber of -.70 a.u. whle the expermet measured a.u.[, ]. The dscrepacy betwee the calculated ad the expermet s 0.9%. I order to crease the covergece for S oe ca slghtly chage l ad results are preseted Table. The values of the cofgurato teracto coeffcet C ad weght w whch belog to the groud state (s) are.99 ad The values of ad C w are tabulated Tables ad for sglet ad trplet S-states wth dfferet bass sze ad l = 0.9 respectvely. CONCLUDING REMARKS As we dcated the Itroducto, the geeral Pollaczek P x; a, b are orthogoal wth respect to a absolutely { } polyomals ( ) dstrbuto fucto f a b, >. () Malaysa Joural of Mathematcal Sceces

15 The Three-Body Coulomb Potetal Polyomals Note that ( ;, ) ( ) ( ;, ) P x a b = P x a b, () whch easly follows from the geeratg fucto (9), dcates that there s o loss of geeralty cosderg oly oe of the cases b 0 or b 0. P x; a, b s depedet of b, { } The postvty codto () the case of ( ) sce b appears ether A or orthogoal f ad oly f (a) or (b) holds. C. Ths meas that P ( x; a, b) { } Usg the restrcted bass for the three-body Coulomb states whch oe of the electros s fxed orbtal (s) whle the secod electro s descrbed by a set of depedet L fuctos, we are able to produce the complete helum atom eerges whch are agree well wth the expermetal results of refereces [, ] ad other calculatos of Koovalov ad McCarthy [] ad Accad et al. []. The covergece of the eerges s show as the Laguerre bass sze creases. For example, we obtaed the eergy of a.u. for S the bass sze. If s show by creasg the umber of bass sze up to 0 we obtaed the covergece umber of a.u. whle the expermet measured -.70 a.u.[, ]. The completeess relato of the three-body Coulomb wave fuctos s calculated terms of the cofgurato teracto coeffcet va the Gaussa quadrature. It s show that the weghts ad cofgurato teracto coeffcets coverge to certa umber for dfferet bass sze. are ACKNOWLEDGMENTS The authors lke to thak Prof. Adrs Stelbovcs ad Prof. Igor Bray for ther helpful refereces they suppled. REFERENCES [] Heller, E.J. ad Yama, H.A. 97. Phys. Rev. A, 9:0, 09. [] Yama, H.A. ad Rehardt, W.P. 97. Phys. Rev. A, :. Malaysa Joural of Mathematcal Sceces

16 Agus Kartoo & Mustafa Mamat [] Board, J.T. 97. Phys. Rev. A, : 0. [] Board, J.T. 9. Phys. Rev. A, :9. [] Stelbovcs, A.T. 99. J. Phys. B : At. Mol. Phys., : L9. [6] Bray, I. ad Stelbovcs, A.T. 99. Adv. At. Mol. Phys., : 09. [7] Bray, I., Fursa, D.V., Khefets, A.S. ad Stelbovcs, A.T. 00. J. Phys. B, : R7. [] Szegö, G. 97. Orthogoal Polyomals, Vol. (Amerca Mathematcal Socety Colloquum Publcatos) th ed. (Amerca Mathematcal Socety, Provdece, RI). [9] Szegö, G. 90. Proc. Amer. Math. Soc., : 7. [0] Erdély, A., Magus, W., Oberhettger, F., Trcom F.G. 9. Hgher Trascedetal Fuctos, Vol. II., New York: McGraw- Hll Book Co., Ic. [] Chhara, T.S. 97. A Itroducto to Orthogoal Polyomals, Mathematcs ad Its Applcatos, Vol., New York: Gordo ad Breach. [] Radzg, A.A., Smrov, B.M. 9. Referece Data o Atoms, Molecules, ad Ios, Spger-verlag Berl, Chaps. 6 ad 7. [] NIST Atomc Spectra Database Levels Data He I, [] Koovalov, D.A., McCarthy, I.E. 99. J. Phys. B: At. Mol. Phys., : L9. [] Accad, Y., Pekers, C.L., Schff, B. 97. Phys. Rev. A, : 6. 6 Malaysa Joural of Mathematcal Sceces

17 The Three-Body Coulomb Potetal Polyomals TABLE : The roots ad pseudo-states eerges ( ε + ε ) whch are produced from oorthogoal Laguerre-L expasos are show for the groud states, l =.0 for =,, S excted states, l = 0.9 for =, 0, ad 0. Powers of te are deoted by the umber brackets. N x S x S (+) -0.07(+) (+) (+) 0.667(+) (+) (+) (+) (+) -0.(+) -0.0(+) (+) (-) (+) (+) (+) (+) -0.0(+) (+) (+) (+) -0.76(+) -0.66(+) -0.7(+) (+) (-) (+) (+) (+) -0.0(+) (+) -0.99(+) (+) Malaysa Joural of Mathematcal Sceces 7

18 Agus Kartoo & Mustafa Mamat TABLE (cotued) : The roots ad pseudo-states eerges ( ε + ε ) whch are produced from o-orthogoal Laguerre-L expasos are show for the groud states, l =.0 for =,, S excted states, l = 0.9 for =, 0, ad 0. Powers of te are deoted by the umber brackets. N x S x S (+) (+) -0.76(+) -0.96(+) -0.0(+) (+) (+) (-) (+) (+) (+) -0.0(+) (+) -0.00(+) -0.7(+) (+) (-) TABLE : The groud ad excted-states egevalues ( ε + ε ) of the o-relatvstc Hamltoa of the three-body Coulomb wave fuctos ( a.u.) are show as a fucto of umber of L bass fuctos =, 0, ad 0. The observato results by refereces [, ]. Hghly accurate o-relatvstc eergy levels of the helum atom by Koovalov ad McCarthy (KM) [] ad by Accad et al. []. 0 0 Observato KM Accad et al. [Preset work] [, ] [] [] State S S S S S S S Malaysa Joural of Mathematcal Sceces

19 The Three-Body Coulomb Potetal Polyomals TABLE : Covergece of the groud states ( S) egevalues ( ε + ε ) for the three-body Coulomb wave fuctos ( a.u.) are show as a fucto of umber of L bass fuctos =, 0, ad 0 ad l. l TABLE : The weghts of Gaussa quadrature ad cofgurato teracto coeffcet are show for l = 0.9, l = 0 (sglet) ad dfferet bass szes. Powers of te are deoted I x by the umber brackets. w w C 0.79(+) (+) (+) (+) -0.07(+) (+) (+) (+) -0.(+) -0.0(+) (+) (-) (+) (+) (-) 0.97(-) 0.0(-) (-) (-) (-) (-) 0.660(-) (+) (+) (-) 0.77(-) 0.99(-) Malaysa Joural of Mathematcal Sceces 9

20 Agus Kartoo & Mustafa Mamat TABLE (cotued): The weghts of Gaussa quadrature ad cofgurato teracto coeffcet are show for l = 0.9, l = 0 (sglet) ad dfferet bass szes. Powers of te are deoted by the umber brackets. I x (+) (+) -0.76(+) -0.66(+) -0.7(+) (+) (-) (+) (+) -0.76(+) -0.96(+) -0.0(+) (+) (+) (-) w w (+) (-) (-) (-) (-) (-) 0.9(-) (-) 0.996(-) 0.90(-) 0.0(-) 0.970(-) 0.760(-) 0.770(-) (+) (-) 0.97(-) (-) (-) 0.767(-) 0.907(-) 0.706(-) 0.090(-) 0.96(-) 0.766(-) 0.67(-) 0.66(-) 0.9(-) 0.990(-) (-) 0.797(-) 0.000(-) 0.(-) (+) (+) C (-) 0.69(-) 0.97(-) (-) (-) 0.906(-) (-) 0.69(-) 0.6(-) 0.0(-) 0.709(-) 0.97(-) (-) (-) 0.(-) 0.977(-) Malaysa Joural of Mathematcal Sceces

21 The Three-Body Coulomb Potetal Polyomals TABLE : The weghts of Gaussa quadrature ad cofgurato teracto coeffcet are show for l = 0.9, l = 0 (trplet) ad dfferet bass szes. Powers of te are deoted by I x the umber brackets. w w C (+) (+) 0.667(+) (+) (+) (+) (+) (+) (+) -0.0(+) (+) (-) 0.606(-) (-) (-) (-) 0.090(-) 0.067(-) 0.606(-) 0.676(-) 0.69(-) 0.990(-) 0.907(-) 0.70(-) (-) (-) 0.70(-) 0.00(-) (-) 0.760(-) 0.790(-) 0.0(-) 0.900(-) 0.690(-) 0.790(-) 0.700(-) (+) (+) (+) -0.0(+) (+) -0.99(+) (+) (-) 0.090(-) 0.0(-) 0.707(-) 0.999(-) (-) 0.99(-) (-) 0.9(-) 0.096(-) (-) (-) 0.0(-) (-) (-) 0.760(-) 0.790(-) 0.90(-) (-) (-) (-) 0.770(-) 0.700(-) 0.670(-) (-) (-) Malaysa Joural of Mathematcal Sceces

22 Agus Kartoo & Mustafa Mamat TABLE : The weghts of Gaussa quadrature ad cofgurato teracto coeffcet are show for l = 0.9, l = 0 (trplet) ad dfferet bass szes. Powers of te are deoted by the umber brackets. I x (+) (+) (+) -0.0(+) (+) -0.00(+) -0.7(+) (+) (-) w w (-) 0.090(-) 0.0(-) 0.79(-) (-) 0.976(-) 0.709(-) (-) 0.90(-) 0.69(-) (-) (-) 0.0(-) (-) 0.609(-) 0.77(-) (-) 0.677(-) 0.07(-) C (-) 0.760(-) 0.790(-) (-) (-) 0.900(-) 0.60(-) (-) 0.70(-) (-) 0.90(-) (-) 0.760(-) (-) 0.60(-) (-) Malaysa Joural of Mathematcal Sceces