The rotating Morse potential model for diatomic molecules in the tridiagonal J-matrix representation: I. Bound states

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1 The rotatig Morse potetial model for diatomic molecules i the tridiagoal J-matrix represetatio: I. Boud states I. Nasser, M. S. Abdelmoem, ad H. Bahlouli Physics Departmet, Kig Fahd Uiversity of Petroleum & Mierals, Dhahra 36, Saudi Arabia A. D. Alhaidari Shura Coucil, Riyadh, Saudi Arabia This is the first i a series of articles i which we study the rotatig Morse potetial model for diatomic molecules i the tridiagoal J-matrix represetatio. Here, we compute the boud states eergy spectrum by diagoalizig the fiite dimesioal Hamiltoia matrix of H, LiH, HCl ad CO molecules for arbitrary agular mometum. The calculatio was performed usig the J-matrix basis that supports a tridiagoal matrix represetatio for the referece Hamiltoia. Our results for these diatomic molecules have bee compared with available umerical data satisfactorily. The proposed method is hady, very efficiet, ad it ehaces accuracy by combiig aalytic power with a coverget ad stable umerical techique. PACS umbers: 3.6.Ge, 3.6.Fd; 34..Cf, 34..Gj Keywords: Boud states, Rotatig Morse potetial, J-matrix, Morse potetial, Tridiagoal represetatio. I. INTRODUCTION The Morse potetial plays a domiat role i describig the iteractio of atoms i diatomic ad eve polyatomic molecules [-3]. A effective potetial, which is the sum of the cetrifugal potetial term that deped o the agular mometum ad the Morse potetial, has bee used as a model for such iteractios. It is referred to as the rotatig Morse potetial. For =, the radial Schrödiger equatio with this potetial has bee solved exactly []. However, for > oly umerical solutios are possible where several approximatio techiques have bee proposed ad extesively used with varyig degrees of accuracy ad stability [4-9]. Numerical solutios, with ad without Pekeris approximatio, have bee used to calculate the eergy spectrum [4-7]. The Pekeris approximatio [4] is based o the expasio of the cetrifugal part i expoetial terms with expoets that deped o a iter-uclear distace parameter. This is why Pekeris approximatio is valid oly for very small spatial variatios from the iter-uclear separatio (i.e. for low vibratioal eergy). Other methods that have also bee used iclude the variatioal method with Pekeris approximatio [8], supersymmetry (SUSY) [9,], the hyper-virial (HV) perturbatio method with the full potetial without Pekeris approximatio [], the shifted /N expasio (SE) [] ad the modified shifted large /N approach (MSE) [3]. I [3] it was stated that the MSE method leads to the exact wavefuctio for pure vibratioal states ad, as such, gives the most precise results. However, it is cumbersome ad requires a lot of computatioal time. Other methods, that are semi-aalytic, have also bee used. These iclude the Nikiforov Uvarov (NU) method [4,] ad the asymptotic iteratio methods (AIM)

2 [6]. I the AIM method, the eergy eigevalues are obtaied by simple trasformatio of the radial Schrödiger equatio ad the wavefuctio is calculated iteratively. The two-poit quasi-ratioal approximatio techique (TQA) [7], which is cosidered as a extesio of the Padé procedure, was used for the hydroge molecule. The exact quatizatio rule (EQR) [8] ad the Fourier grid Hamiltoia method (FGH) [9] have also bee used i this cotext. Our approach for the study of the rotatig Morse potetial model was ispired by the J-matrix method []. It is a algebraic method for extractig boud states ad scatterig iformatio usig computatioal tools devised etirely i square itegrable bases. Physically boud states are expadable i square itegrable (L ) fuctios that are compatible with the domai of the Hamiltoia. I fact, L fuctios arise aturally as egative eergy eigefuctios for the discrete boud states such as i the familiar hydroge atom. Hece it seems atural that the use of discrete basis sets offers cosiderable advatage i the calculatio of boud states sice this scheme requires oly stadard matrix techique rather tha the usual approach of umerical itegratio of the differetial equatios. I this approach, whe searchig for algebraic or umerical boud states solutios, the wave fuctio ψ spas the space of square itegrable fuctios with discrete basis elemets { φ} =. That is ψ ( re, ) = f ( E ) φ ( r ), where r is the set of coordiates for real space, ad E is the system s eergy. The basis fuctios must be compatible with the domai of the Hamiltoia ad satisfy the boudary coditios. I our case, this meas that the basis set must vaish at r = ad r =. Typically, whe calculatig the discrete spectrum, the choice of basis is limited to those that carry a diagoal represetatio of the Hamiltoia H. That is, oe looks for a L basis set { φ} such that = H φ = E φ givig the discrete spectrum of H. The cotiuous spectrum is obtaied from the aalysis of a ifiite sum of these complete basis fuctios. Solvig the eigevalue problem requires a efficiet method for computig matrix elemets of the Hamiltoia. I this article, we relax the restrictio of a diagoal matrix represetatio for the Hamiltoia. Istead, we split our origial Hamiltoia i two parts as H = H + V where H is the part of the Hamiltoia that could be treated aalytically while the remaiig part, V, is to be hadled umerically. I dealig with the matrix represetatio of H, we do ot require it to be diagoal but oly that it is hermitia ad tridiagoal. That is, the matrix elemets of the referece wave operator should take the followig form φ H E φ = ( a z) δ + b δ + b δ, (.) m, m, m, m+ where z ad the coefficiets { a, b} = are real ad, i geeral, fuctios of the eergy, agular mometum, ad potetial parameters. Thus, this approach embodies the aalytical power i dealig with the referece H problem as well as the very accurate umerical quadrature approach [] used i the evaluatio of the potetial matrix elemets. I additio, we beefit from the existece of two differet bases that support a tridiagoal matrix represetatio of the referece Hamiltoia. Each oe has two parameters which will be tued to ehace covergece, stability, ad of course accuracy. For a give potetial model, the choice of basis is made with a eye o these three computatioal characteristics.

3 The paper is orgaized as follows: I sectio II, we review ad summarize the theoretical formalism of our approach. The, i sectio III, we develop a umerical procedure for implemetig our approach o the S-wave Morse potetial. This eables us to measure the accuracy of our umerical results agaist aalytic oes for four differet types of biary molecules. I sectio IV, we apply our approach to the calculatio of the eergy eigevalues for the rotatig Morse potetial where ad compare our results with those obtaied by other methods elsewhere. This will be followed, i sectio V, by a short summary of our fidigs ad a coclusio of the work. I future articles, we will develop a efficiet method for locatig resoaces i these models usig the same represetatio together with the complex scalig method. We will also ivestigate geeralized three-parameter Morse potetial models that are more suitable ad give better descriptio of various molecules. Moreover, the scatterig matrix i the J-matrix method will be employed to ehace the accuracy of computatios at low basis dimesio. II. THEORETICAL FORMALISM As explaied i the itroductio the real power of our approach is that: () it allows for the aalytic computatio of the matrix elemets of the referece Hamiltoia, H, ad () it provides for a highly accurate evaluatio of the potetial matrix elemets usig Gauss quadrature approximatio. The computatio of the eigevalues of the resultig full Hamiltoia is performed to the desired accuracy usig two free parameters, a legth scale parameter λ ad the dimesio of the basis space N. I the atomic uits, = m =, the radial time-idepedet Schrödiger equatio for a system with reduced mass m i a spherically symmetric potetial V(r) reads as follows d ( + ) ( H E) ψ = + + V( r) E ψ =, (.) dr r where ψ is the wavefuctio which is parameterized by the agular mometum, eergy E, ad the potetial parameters. The cotiuum could be discretized by takig ψ as a elemet i a L space with a complete basis set { φ }. The itegratio measure i this space is dr. We parameterize the basis by a positive legth scale parameter λ as { φ( λ r) }. The followig choice of basis fuctios is compatible with the domai of the above Hamiltoia ad satisfies the boudary coditios (the vaishig of the wave fuctio at r = ad r ) α ( ) x ν φ x = Ax e L( x) ; =,,,.. (.) ν where x = λr, α >, ν >, L ( x) is the associated Laguerre polyomial of order, ad A is the ormalizatio costat λγ ( + ) Γ ( + ν + ). The matrix represetatio of the referece Hamiltoia H ( H V ) i this basis is writte as λ d λ ( + ) ( H ) = φ( x) + φ ( ) m m x. (.3) dx x I the maipulatio of (.3) we use the differetial equatio, differetial formulas, ad three-term recursio relatio of the Laguerre polyomials []. As a result, we obtai the 3

4 followig elemets of the matrix represetatio of the referece Hamiltoia, with α = ν + : ( H ) ( ) = λ + ν + δ 8, m + λ ( + ν) δ λ 8, m ( )( ν ) δ m , m (.4) This simple tridiagoal matrix represetatio for H holds for ν = +. The basis { φ }, o the other had, is ot orthogoal. Its overlap matrix, ( ), ( ), + ( )( ), φ φm Ω m= + ν + δm + ν δm + + ν + δm, (.) is tridiagoal. Now, the oly remaiig quatities that are eeded to perform the calculatio are the matrix elemets of the potetial V(r). These are writte as Vm = φ ( ) ( ) ( ) λr V r φm λr dr (.6) ν x ν ν = AA m xe L( ) ( )[ ( )] xlm x xv xλ dx The evaluatio of such a itegral is almost always doe umerically. I our case, we use the Gauss quadrature approximatio [] givig V N [ ε V ( ε λ )] (.7) Λ Λ m k mk k k k = for a adequately large iteger N, ad where ε k ad { Λ } k are the respective N = eigevalues ad eigevectors of the N N tridiagoal symmetric matrix J, whose elemets are J, = + ν +, J, + = ( + )( + ν + ). (.8) where ν = + for the Laguerre basis (.). There is aother basis (kow as the oscillator basis ) that carries a tridiagoal matrix represetatio for the same referece Hamiltoia. This alog with the correspodig matrix maipulatios are preseted i Appedix. A. The referece Hamiltoia H i this represetatio could therefore be fully accouted for aalytically, whereas the potetial V is approximated by its represetatio i a subset of the basis, such that ( H ) + V m ;, m N m H m. (.9) ; m, > N This represetatio is the fudametal uderlyig structure of the J-matrix method []. It is a algebraic method that has the advatage of takig ito accout the full cotributio of H exactly without ay trucatio. Nevertheless, we will cofie our implemetatio of the preset approach to the fiite matrix represetatio (i the N φ subspace { } ) for the potetial V as well as for the referece Hamiltoia H =. Takig ito accout the full referece Hamiltoia should result i a substatial improvemet o the accuracy of the results. This is curretly beig pursued i dealig with resoaces ad will be reported i the ear future. Now, i our umerical implemetatio we use the Morse potetial model for diatomic molecules defied by V() r = D e e where α( r re)/ re α( r re)/ re e (.) D e is the dissociatio eergy, e N r is the equilibrium iter-uclear distace, ad α is a parameter that cotrols the width of the potetial well. The umerical values of these parameters are show i Table for differet diatomic molecules alog with the sources from which these data were extracted. 4

5 I the followig sectios, we make explicit calculatios of the boud states for the Morse potetial with fiite agular mometum usig this L basis represetatio. Our calculatio strategy will be as follows: first we choose a basis set ad study the stability of the eigevalues as we vary the scalig parameter λ util we reach a plateau i λ. The, for a selected value of λ from withi the plateau, we compute the eigevalues of the full Hamiltoia ad compare them with available umerical results for the give molecule. For =, the compariso is made agaist well-kow exact aalytic results [3,8]. To improve the umerical results further we icrease the dimesio of the space N util the desired accuracy is reached. The iterplay betwee these two parameters, λ ad N, costitutes the power of our approach ad results i fast covergig computatios with high stability for relatively small values of N. III. S-WAVE MORSE POTENTIAL The S-wave ( = ) Morse potetial has bee solved aalytically [3,8] ad its boud states eergy is give i a closed form. A brief derivatio of this aalytic result will be give below for two reasos. O the oe had, it is a very valuable igrediet sice it costitutes a reliable referece ad calibratio tool to test the accuracy of our umerical approach. O the other had, it brigs out a alterative three-parameter model based o a geeralized Morse potetial that has a defiite advatage over the stadard two-parameter model. If we cosider the case = i equatio (.), the the S-wave Schrödiger equatio for the geeralized Morse potetial reads as follows r ( r d α r ) α( ) e r e + D e e βe E χ = m dr, (3.) where the potetial stregth D e >, the parameters α ad r e are positive ad β is a deformatio parameter of the classical Morse potetial. This equatio has a exact aalytic solutio for the boud states eergy spectrum. Its derivatio goes as follows. If α we let x = r / r e, E = mre, A = D e e E, ε = E E, the equatio (3.) becomes d α αx αx A e e βe + ε χ = dx (3.) Now, for a real parameter ξ, let y= ξe α x [,], the (3.) becomes α d d A e β ε y + y y y χ + = (3.3) dy dy α ξ ξ α μ Cosider the asatz [3] ( ) y ν ν χ y = y e L ( y), where L ( y) is the associated Laguerre polyomial, μ, ad ν >. It is a square itegrable fuctio that is compatible with the domai of H, satisfies the boudary coditios ad could support boud states. Isertig i (3.3) gives α d μ d μ A e β ε ν y y L + = (3.4) dy y dy y α ξ ξ α y Usig the differetial equatio of the Laguerre polyomials, this reduces to

6 α d Ae ε Aβ ν ( μ ν) + y μ μ L = (3.) dy 4 αξ y α αξ Now, sice all terms iside the curly bracket are liearly idepedet the we must impose the followig coditios ν = μ, ξ 4Ae α = α, μ = ( Aβ α ξ), ad ε = αμ (3.6) Givig μ = ξβe α ad sice μ, we must have ξβe α. Thus, fially the eergy spectrum is foud to be E ( ) = α E + ξβe α (3.7) which ca be writte more explicitly as follows E = ( ) ( ) re m α + γβ, =,,.., max (3.8) where ξ = γ e α re, γ = md α e, ad max is the umber of boud states. It is the maximum iteger that satisfies the coditio ( γβ ) max. For our umerical implemetatio, we use the parameters that are show i Table which were obtaied from [7] ad [8]. Usig the strategy outlied at the ed of sectio II, we have used both the Laguerre ad oscillator bases. The choice is made i favor of the basis that gives better covergece ad stability for the particular molecular model. As a example, we show i Fig. the behavior of a give eigevalue (i this case = ) as a fuctio of the basis scale parameter λ for the H molecule. I the Laguerre basis, a plateau is evidet for values of λ i the rage to 7 whereas this plateau arrows dow to betwee ad for the oscillator basis. Similarly, for LiH molecule we have see a plateau for λ i the rage 4 to i the Laguerre basis while i the oscillator basis it arrows dow to the rage 3 to 9. These observatios suggest that the Laguerre basis is more suitable for both H a LiH molecules. Noetheless, oe could still use the oscillator basis but i a arrower rage of values of the parameter λ. The diagoalizatio of the full Hamiltoia gives the umerical boud states show i Table a with the order of the eigevalues represeted by (the vibratioal quatum umber). These umerical computatios were performed usig the model parameters show i Table ad i the Laguerre basis with N = ad λ = 4 for H molecule ad N = ad λ = 6 for LiH molecule. We see that these eigevalues are i excellet agreemet with the exact results computed from (3.8) up to digits for low-level excitatios. I the Table we had to go to higher order boud states to reach a level at which our umerical results starts deviatig measurably from the exact oes. It is worth metioig that sigificat icrease i the accuracy ca still be achieved by moderately elargig the basis size N eve, say, up to N =. It is also worth otig that the fial choice of the parameter λ for a give molecule ad basis is made oly after studyig the effect of its variatio o the whole eergy spectrum. This is due to the fact that the plateau for λ arrows dow as the boud state idex icreases while goig over the whole spectrum. This behavior is demostrated i Fig. for the LiH molecule i the Laguerre basis ad for =,,,,, ad. Doig the same for the other two molecules, which are icluded i our curret study, we have the followig observatio. For the HCl molecule, o well-defied plateau exists for λ i the Laguerre basis with N = while a plateau betwee 3 ad 8 shows up 6

7 i the oscillatory basis. The same coclusio is reached for the CO molecule where a wider plateau of λ shows up i the oscillator basis betwee ad 6 with N =. Thus, it is clear that the oscillator basis is more suitable for HCl ad CO molecules whereas the Laguerre basis is more suitable for H ad HCl. Table b shows the umerically geerated boud states for HCl ad CO usig the oscillator basis as compared to the exact values geerated by the aalytic formula (3.8). The umerical parameters were N =, λ = for HCl ad N =, λ = for CO alog with the correspodig model parameters give i Table. The agreemet betwee the umerical ad exact results is demostrated up to 4 digits at low vibratios. IV. ROTATING MORSE POTENTIAL I this sectio, we will cosider the Morse potetial with o-vaishig agular mometum ad use our umerical procedure preseted i sectio II to evaluate the correspodig boud states eergy spectrum. The stability ad the accuracy of our results deped maily o two parameters, λ ad N. N represets the umber of basis elemets that we used (dimesio of our tridiagoal matrix) ad λ is the legth scale parameter of the basis. The Morse potetial with is ot exactly solvable ad hece our umerical results caot be checked agaist exact oes. However may umerical ad perturbative results have bee published i recet years [8-8]. The most widely used approximatio was devised by Pekeris [4] which is based o the expasio of the cetrifugal ( + ) r barrier i a series of expoetial terms aroud the equilibrium iter-uclear positio r e of the Morse potetial by keepig terms up to secod order i rr e (i.e., at low excitatio eergy, where r re ). Other approximatios have also bee devised but they require the umerical solutio of trascedetal equatios [7,,3]. I Table 3 we show the boud states eergy of the H molecule for differet values of the agular mometum geerated by usig the Laguerre basis with λ = 4, N = ad the model parameters i Table. We also list aalogous results obtaied by other approximatios such as the Hypervirial perturbatio method (HV) [], the Super- Symmetric method (SUSY) [9], the Asymptotic Iteratio Method (AIM) [6], the results of Duff (Duff) [6], the Modified Shifted /N expasio (MSE) [3], the Two-Poit Quasi-Ratioal Approximatio techique (TQA) [7], ad the variatioal method usig Pekeris approximatio [8]. I Table 4 we give the boud states eergies geerated by our method for differet values of the agular mometum for LiH i the Laguerre basis usig the parameters N =, λ = 6 ad the correspodig model parameters i Table. Table shows the umerically geerated boud states eergies for HCl usig the oscillator basis with N =, λ = 6 ad the model parameters i Table. Table 6 gives the boud states eergies for the CO molecule usig the oscillator basis with N =, λ = ad usig the model parameters i Table. The agreemet betwee our umerical results ad those geerated by other methods up to four sigificat digits is reassurig. Agai, higher accuracy could easily be achieved by icreasig N. V. CONCLUSION This work is the first i a series of articles i which we study the rotatig Morse potetial model for diatomic molecules i the tridiagoal J-matrix represetatio. Here, 7

8 we provided a alterative method for obtaiig the boud states eergies for four differet types of diatomic molecules: H, LiH, HCl ad CO. Our umerical results have bee compared favorably with those obtaied usig other approximatio schemes such as the variatioal method usig Pekeris approximatio, supersymmetry, hypervirial perturbatio, the shifted /N expasio, the modified shifted /N expasio, the asymptotic iteratio method, ad the two-poit quasi-ratioal approximatio techique. Furthermore, sice realistic diatomic potetials are more accurately modeled by the perturbed or deformed Morse potetials, we are cofidet that our approach will produce much more accurate iformatio about the structure ad dyamics of such molecules if we use these deformed potetial models. This will be the subject of ivestigatio i oe of our future articles i the series. We believe that our approach has the advatage of combiig aalytic ad umerical powers makig it very stable, rapidly covergig, ad highly accurate. Of course, the real advatage of our method will be exhibited whe we address the issue of eergy resoaces while takig ito accout the full cotributio of the referece Hamiltoia H (eve if it were to iclude log-rage solvable potetials such as the Coulomb potetial). As such, our method is systematic, highly accurate, ad could easily be exteded to other short-rage potetials of which the Morse is oly a example. ACKNOWLEDGMENTS The authors are grateful for the support provided by the Physics departmet at Kig Fahd Uiversity of Petroleum & Mierals. Appedix A I this Appedix, we complemet the theoretical formulatio preseted i sectio II with the oscillator basis defied by α ( ) x ν φ x = Ax e L( x) ; =,,,.. (A.) where x = ( λr ), α >, ν >, ad A is the ormalizatio costat to be chose below. The matrix represetatio of the referece Hamiltoia (.) H ( H V ) i this basis is writte as λ d d ( + ) ( H ) = m( x) x ( ) m φ + φ x. (A.) dx dx x I the maipulatio of (A.) we use the differetial equatio, differetial formulas to brig the matrix elemet i the followig form φ m ( H E ) φ = (α ν ½) ( α ¼) ( ½) 4 φm φ λ x (A.3) E A + α + φ ( )( ½) m φ φm x φ + + ν α ν φm φ. 4 λ 4 A x To make this matrix represetatio tridiagoal we must choose the parameters such that all idividual matrix terms are either tridiagoal or cacel each other for all. Thus, we require α = ν + ad ν = + givig 8

9 ( ), ( ), + ( )( ), φ x φ = + ν + δ + ν δ + + ν + δ φ φ = δ m m m m m m where the choice of ormalizatio, A λ Γ ( + ) Γ ( + ν + ), (A.4) =, was made to obtai a orthoormal basis. Usig the differetial equatio, differetial formulas, ad the three-term recursio relatio of the Laguerre polyomials [] we obtai the followig elemets of the matrix represetatio of the referece Hamiltoia ( H ) ( ) = λ + ν + δ, m + λ ( + ν) δ λ, m ( )( ν ) δ m , m (A.) Now, the oly remaiig quatity that is eeded to perform the calculatio is the matrix elemets of the potetial V(r). This is obtaied by evaluatig the itegral V m = φ ( ) ( ) ( ) λr V r φm λr dr. (A.6) ν x ν ν = AA m xe L( ) ( ) ( ) xlm x V x λ dx Usig Gauss quadrature itegral approximatio [], we obtai N V Λ Λ V ( ε λ) (A.7) m k mk k k = where N is a large eough iteger. ε k ad { Λ } k are the respective N eigevalues ad = eigevectors of the N N tridiagoal symmetric matrix, whose elemets are J, = + ν +, J, + = ( + )( + ν + ). (A.7) The differece betwee this quadrature matrix, which is associated with the oscillator basis, ad that i Eq. (.8), which is associated with the Laguerre basis, is that here we use ν = + ot ν = +. Moreover, oe should make ote of the marked differece i the term withi square brackets i Eq. (A.7) as opposed to that i Eq. (.7) for the Laguerre basis. N 9

10 REFERENCES: [] P. M. Morse, Phys. Rev. 34 (99) 7 [] S. H. Dog, R. Lemus, ad A. Frak, It. J. Quat. Phys. 86 () 433, ad refereces therei [3] S. Flügge, Practical Quatum Mechaics, vol. I (Spriger, Berli, 994) [4] C. L. Pekeris, Phys. Rev. 4 (934) 98 [] R. Herma ad R. J. Rubi, Astrophys. J. (9) 33 [6] M. Duff ad H. Rabitz, Chem. Phys. Lett. 3 (978) [7] J. R. Elsum ad G. Gordo, J. Chem. Phys. 76 (98) 4 [8] E. D. Filho ad R. M. Ricotta, Phys. Lett. A 69 () 69 [9] F. Cooper, A. Khare, ad U. Sukhatme, Phys. Rep. (99) 67 [] D. A. Morales, Chem. Phys. Lett. 394 (4) 68 [] J. P. Killigbeck, A. Grosjea, ad G. Jolicard, J. Chem. Phys. 6 () 447 [] T. Imbo ad U. Sukhatme, Phys. Rev. Lett. 4 (98) 84 [3] M. Bag, M. M. Paja, R. Dutt, ad Y. P. Varshi, Phys. Rev. A 46 (99) 69 [4] A. F. Nikiforov ad V. B. Uvarov, Special Fuctios of Mathematical Physics (Birkhausr, Basel, 988); C. Berkdemir ad J. Ha, Chem. Phys. Lett. 49 () 3 [] C. Berkdemir, Nucl. Phys. A 77 (6) 3 [6] O. Bayrak ad I. Boztosum, J. Phys. A 39 (6) 69 [7] E. Castro, J. L. Paz ad P. Marti, J. Mol. Str.: THEOCHEM 769 ( 6) [8] W. C. Qiag ad S. H. Dog, Phys. Lett. A 363 (6) 69 [9] C. C. Marsto ad G. G. Balit-Kurti, J. Chem. Phys. 9 (989) 37 [] E. J. Heller ad H. A. Yamai, Phys. Rev. A 9 (974) ; H. A. Yamai ad L. Fishma, J. Math. Phys. 6 (97) 4; A. D. Alhaidari, E. J. Heller, H. A. Yamai, ad M. S. Abdelmoem (eds.), The J-matrix Method: Recet developmets ad selected applicatios (Spriger, Heidelberg, 7) [] See, for example, Appedix A i: A. D. Alhaidari, H. A. Yamai, ad M. S. Abdelmoem, Phys. Rev. A 63 () 678 [] G. Szegö, Orthogoal polyomials, 4 th ed. (Am. Math. Soc., Providece, RI, 997); T. S. Chihara, A Itroductio to Orthogoal Polyomials (Gordo ad Breach, New York, 978); N. I. Akhiezer, The Classical Momet Problem (Oliver ad Boyd, Eiburgh, 96); R. W. Haymaker ad L. Schlessiger, The Padé Approximatio i Theoretical Physics, edited by G. A. Baker ad J. L. Gammel (Academic, New York, 97); D. G. Pettifor ad D. L. Weaire (eds), The Recursio Method ad its Applicatios (Spriger, Berli, 98) [3] A. D. Alhaidari, A. Phys. 37 () : Sec., Eq. (.7)

11 TABLE CAPTIONS: Table : Model parameters for the diatomic molecules i our study as obtaied from the cited sources. E is a derived quatity, which is calculated as E = mr. e Table : Numerical ad exact boud states eergies ( E) for the S-wave Morse potetial ( = ). The Laguerre (Oscillator) basis was used for H ad LiH (HCl ad CO) molecules. The iteger represets the eergy level i the spectrum. Numerically max is the maximum idex of the eergy level beyod which the eigevalues chage sig. Table 3: Boud states eergy eigevalues ( E) for the H molecule (i ev) for differet values of the rotatioal ad vibratioal quatum umbers i the Laguerre basis with N = ad λ = 4. Table 4: Boud states eergy eigevalues ( E) for the LiH molecule (i ev) for differet values of the rotatioal ad vibratioal quatum umbers i the Laguerre basis with N = ad λ = 6. Table : Boud states eergy eigevalues ( E) for the HCl molecule (i ev) for differet values of the rotatioal ad vibratioal quatum umbers i the Oscillator basis with N = ad λ = 6. Table 6: Boud states eergy eigevalues ( E) for the CO molecule (i ev) for differet values of the rotatioal ad vibratioal quatum umbers i the Oscillator basis with N = ad λ =.

12 FIGURE CAPTIONS: Fig. : Variatios i the computed value of oe of the boud states eergy levels of H molecule with the basis scale parameter λ. A plateau of computatioal stability is evidet for the rage λ = to 7 (λ = to ) i the Laguerre (Oscillator) basis. We took = ad basis size N =. Fig. : Same as Fig., but for a few eergy levels from across the whole spectrum. We took = ad used the Laguerre basis with basis size N =. For better display, we shifted each eergy level by a costat to brig it to lie with the rest. Fig. 3: A plot of the square root of the boud states eergy vs. the vibratioal quatum umber for the LiH molecule ad for differet agular mometa. The Laguerre basis was used with λ = 6 ad N =. The uits o the vertical axis is ev. Fig. 4: The boud states eergy vs. the agular mometum quatum umber for the LiH molecule ad for differet vibratioal umbers. The Laguerre basis was used with λ = 7 ad N =.

13 Table Molecule H [7] LiH [8] HCl [8] CO [8] D e (ev) r e(å) m (amu) α E (ev) Table a H LiH This work Exact This work Exact max N, λ Basis, 4 Laguerre, 6 Laguerre 3

14 Table b HCl CO This work Exact This work Exact max N, λ Basis, Oscillator, Oscillator 4

15 Table 3 This work HV [] SUSY [] AIM [6] Duff [6] MSE [3] Variatioal [8] TQA [7] Table 4 This work EQR [8] SUSY [] FGH [9] AIM [6] MSE [3] NU [] Variatioal [8]

16 6 Table This work EQR [8] SUSY [] FGH [9] AIM [6] MSE [3] Variatioal [8] Table 6 This work EQR [8] SUSY [] FGH [9] AIM [6] MSE [3] NU [] Variatioal [8]

17 Fig. Fig. 7

18 Fig. 3 Fig. 4 8