The rotating Morse potential model for diatomic molecules in the J-matrix representation: II. The S-matrix approach

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1 The rotatig Morse potetial model for diatomic molecules i the J-matrix represetatio: II. The S-matrix approach 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 secod article i which we study the rotatig Morse potetial model for diatomic molecules usig the tridiagoal J-matrix approach. Here, we improve further the accuracy of computig the boud states ad resoace eergies for this potetial model from the poles of the S-matrix for arbitrary agular mometum. The calculatio is performed usig a ifiite square itegrable basis that supports a tridiagoal matrix represetatio for the referece Hamiltoia, which is icluded i the computatios aalytically without trucatio. Our method has bee applied to both the regular ad iverted Morse potetial with favorable results i compariso with available umerical data. We have also show that the preset method adds few sigificat digits to the accuracy obtaied from the fiite dimesioal approach (e.g. the complex rotatio method). Moreover, it allows us to hadle easily both aalytic ad o-aalytic potetials as well as /r sigular potetials. PACS umbers: 3.65.Ge, 34..Cf, 34..Gj Keywords: Resoaces, Boud states, Rotatig Morse potetial, J-matrix. It is well kow that the Morse potetial model describes very well the vibratios of diatomic molecules []. This is because such systems ca be modeled by two positive charges (the two atomic uclei) that produce the Coulomb potetial i which the electroic cloud moves. A similar type of modelig was also applied to uclear molecules []. However, i this case there are o well defied ceters ad the potetial i which the ucleos move arises from the sum of iteractio betwee all ucleos. I the laguage of eergy scales, the electroic excitatio eergy, the vibratio eergy ad the rotatioal molecular eergy for diatomic molecules are widely separated eergy scales, hece makig the three degrees of freedom completely ucoupled. For a uclear molecule, o the other had, the sigle ucleo excitatio ad rotatioal eergies are comparable for eve the lowest agular mometum. Thus, it is expected that the eergy associated with vibratios of the iter-uclear coordiates will also be of the same order of magitude. This will lead to a more complex eergy spectrum reflectig the much stroger couplig betwee the itrisic, vibratioal ad rotatioal degrees of freedom of uclear molecules. The Morse potetial is a elemet i the class of potetials for which a aalytic solutio of the Schrödiger equatio exists for zero agular mometum [3]. It is writte as α( ) ( ) () x α x VM r = V e e ; x= r r () where V is the stregth of the potetial (or the dissociatio eergy i the cotext of diatomic molecules), r is the equilibrium itermolecular distace ad α is a positive umber cotrollig the decay legth of the potetial. For V >, this potetial has a

2 miimum of V at r = r ad it is called the regular Morse potetial. For V <, it has a maximum of V there ad it is called the iverted Morse potetial. Asymptotically (as r becomes very large) it goes to zero. The rotatig Morse potetial is the sum of the ( + ) repulsive cetrifugal potetial barrier,, ad the Morse potetials (). m r Geerally, this effective potetial shows a valley followed by a potetial barrier which will the have two major effects. For the iverted Morse potetials where V < some resoaces will be developed ad the umber of boud states will be reduced as icreases. There will also be a critical agular mometum value, c, beyod which o boud states are foud ad this critical value will deped o the potetial parameters V ad α. May umerical approaches usig various approximatio techiques have bee proposed for solvig the rotatig Morse potetial. These were extesively implemeted with varyig degrees of accuracy ad stability [4]. Other semi-aalytic methods have also bee used such as the Nikiforov-Uvarov method [5] ad the asymptotic iteratio methods [6]. I our previous work [7] we have used the tridiagoal represetatio approach ispired by the J-matrix method [8] to compute the boud state eergy spectrum of four differet types of diatomic molecules: H, LiH, HCl ad CO. Our approach provided a alterative method for obtaiig the boud states eergies for these diatomic molecules with improved accuracy ad could easily be exteded to other molecules. The umerical results have bee compared favorably with those obtaied usig other approximatio schemes [4]. I this work, we exted the approach to hadle ot oly the boud states but also resoaces associated with the rotatig Morse potetial model. The direct way to compute the resoaces is based o the accurate defiitio of the resoaces as beig the poles of the scatterig S-matrix i the complex eergy plae. Oe ca show that each elemet of the S-matrix is sigular at the complex resoace eergy E S ( E) = ; E = ER ± i EI. () This coditio is sufficiet for obtaiig the resoace positio E R ad width Γ= E I. Several methods to fid the complex resoace eergies of a give scatterig Hamiltoia are available. There are may techiques that eable us to evaluate the S- matrix. Oe such approach is to use the Jost fuctio ad its aalytic properties [9]. I aother approach, Yamai ad Abdelmoem [] showed how to calculate the S-matrix at the real Harris eergy eigevalues ad, subsequetly perform aalytic cotiuatio of the S-matrix to the complex eergy plae. The required resoace iformatio is the extracted from the aalytically cotiued S-matrix. I our preset work, we will evaluate the resoaces ad boud state eergies associated with the rotatig Morse potetial by combiig the properties of the S-matrix, the complex rotatio method [] ad the aalytical ad computatioal power of the J-matrix approach [8]. I our previous work [7], we have restricted our calculatio to a fiite dimesioal subspace spaed by a subset of the J-matrix square itegrable basis. Due to the fiiteess of the dimesioal space, the accuracy of our umerical results is somewhat limited. Ay additioal improvemet requires larger spaces, cosequetly more computatioal time ad less stability. Sice the objective is to icrease the accuracy ad improve the efficiecy i locatig the resoace positios ad widths without extedig too much the computig time or reducig umerical stability, we have opted i the preset work

3 to use the aalytic power of the J-matrix approach. It will eable us to iclude, without trucatio, all of the exactly solvable part of the Hamiltoia as a ifiite tridiagoal matrix tail (usually referred to as the H problem i the J-matrix literature). For the preset problem, H is just the kietic eergy operator (i.e., the sum of the secod order radial differetial operator ad the orbital term). Direct study of resoaces is usually doe i the complex eergy plae. Resoace eergies are the subset of the poles of the S-matrix which are located i the lower half of the complex eergy plae. Oe way to ucover these resoaces, which are hidde below the real lie i the complex E-plae, is to use the complex scalig (complex rotatio) method []. This method exposes the resoace poles ad makes their study ad maipulatio easier. The subset of eigevalues that correspods to resoace ad boud states spectra remai stable agaist variatios i all computatioal parameters. The time-idepedet Schrödiger wave equatio for a poit particle i a spherically symmetric potetial V(r) reads as follows d ( + ) ( H E) χ = + + V() r E χ =, (3) mdr mr where χ () r is the wavefuctio which is parameterized by the potetial parameters, ad E. We expad χ i a L complete basis set { φ } which is chose to make the matrix represetatio of the referece Hamiltoia, H H V, tridiagoal. The basis is parameterized by a positive legth scale parameter λ as { φ( λ r) }, which also allows for more computatioal freedom. The followig choice of basis fuctios [] is compatible with the domai of the Hamiltoia, satisfies the desired boudary coditios, ad results i a tridiagoal matrix represetatio for H + () ( ) x ν φ r = a λr e L() x ; =,,,.. (4) where x= λr ad ν = + i the Laguerre basis, whereas x= ( λr) ad ν = + ν i the oscillator basis. L ( x) is the Laguerre polyomial of degree ad a is the ormalizatio costat Γ ( + ) Γ ( + ν + ). The referece Hamiltoia H i this represetatio, which is at the heart of the J-matrix approach, is accouted for i full. O the other had, the short-rage Morse potetial V is approximated by its matrix elemets i a adequate subset of the basis usig the Gauss quadrature approach [3]. That is, the matrix represetatio of the full Hamiltoia becomes ( H ) + V ;, m N H. (5) ( H ) ;, m> N Such a represetatio is the fudametal uderlyig feature of the J-matrix [8] method. As it is obvious from (5), the referece Hamiltoia, which icludes the orbital term, is ot trucated as is usually doe i the fiite calculatio methods. This full accout of the referece Hamiltoia should result i a substatial improvemet o the accuracy of the results especially for large agular mometa. This costitutes the real power of the S-matrix approach that we are proposig. Moreover, i fidig resoaces ad boud state eergies we use the direct method based o the J-matrix calculatio of the scatterig matrix i the complex eergy plae. Boud states are associated with egative real poles of the S-matrix while resoaces are associated with complex poles 3

4 that have positive real parts ad egative imagiary parts. The S-matrix, i the J-matrix approach, is defied by [] + gn, N ( E) JN, N( E) RN( E) SE ( ) = TN ( E), where (6a) + + gn, N ( E) JN, N( E) RN( E) N N Λ N, N N N gn, N ( z) = Dν = D ν ( ε ) ( ) m m z ε z = = = ε z, ad (6b) c is ± c ± is T = ; R =. (6c) c + is c ± is g ( ) z is the iverse of the matrix (H z) where H is the N N fiite matrix give by the first lie i (5). The eigevalues ad the correspodig ormalized eigevectors of N the fiite matrix H are deoted by { ε } N ad { } N Λ =, respectively. { ε }, = are the = eigevalues of the trucated H obtaied by removig the last row ad last colum. I the two bases (the Laguerre ad oscillator), the matrix represetatio of the referece wave operator, whose elemets are defied by Jm, = φm ( H E) φ, is tridiagoal ad symmetric. The quatities s ad c are the expasio coefficiets of the two idepedet asymptotic solutios of the referece wave equatio ( H E) ψ =. I this J-matrix termiology, the two idepedet asymptotic solutios of the referece problem are writte as S = s φ ; C = c φ, (7) = = ad are usually called the sie-like ad cosie-like solutios, respectively. Table gives all the elemets eeded to calculate the S-matrix (6a). Usig the recursio relatio, we ca obtai ( ) ± TN E ad RN ( E) from T ( E ) ad R ± ( E) recursively i the form of a cotiued fractio as show i []. I the followig we give the recipe of our procedure for calculatig the boud ad resoace eergies for the rotatig Morse potetial for differet values of potetial parameters ad. Our calculatio strategy is based o the followig procedure: First we use the complex rotatio method [] to calculate these eergies for the give physical parameters. This step is very importat i our approach sice these values costitute the seed for the S-matrix computatios. Now we proceed to fid the roots of S ( E ) matrix. For this purpose, the built-i iteratio procedure i the computatioal software package MATHEMATICA (versio 5) was used [4]. Due to this iteratio, the accuracy of the S-matrix results depeds crucially o the accuracy of the seeds that were obtaied usig the complex rotatio approach. I the results of our calculatio, the umber of sigificat digits show is usually obtaied for the optimum values of the umerical parameters λ ad θ ad for a give choice of the dimesio parameter N. As a cotiuatio of our efforts deployed i paper I [7] we apply the strategy outlied above to the study of both resoaces ad boud states. However, umerical computatios for the diatomic molecules H, LiH, HCl ad CO with the associated model parameters give i [7] produced oly boud states. Therefore, we eed to choose appropriate potetial parameters that ca give rise to resoaces. As explaied i the itroductio the iverted Morse potetial gives rise to resoaces ad boud states eve for = ad it is for this reaso that it was cosidered i the literature [5]. 4

5 I Table a, we show the boud ad resoace eergies for the iverted Morse potetial obtaied usig the J-matrix for the same parameters ad uits used i [5]. That is, i the potetial model (), we took V = 6 fm, α =.3 fm, r = 4. fm, ad =. The eergies i the table are give i fm - as i [5]. Oe boud state ad 5 resoaces were obtaied for this potetial usig the S-matrix approach i the Laguerre basis. Our results complemet those i [5] where oly oe boud state ad two resoaces were give (it is worth metioig that the resoat at E R =.783 i [5] is ot cosistet with the k-value give i the same referece). Nie of the 5 resoaces are located i the fourth quadrat of the complex eergy plae while six are i the third quadrat. Except for the first two resoaces, which are shallow, all others resoaces are deep. It is to be oted that for deep resoaces, we eed larger values of N ad suitable values of λ ad θ to reach the desired accuracy. It is also worth metioig that egative eergy resoaces have recetly foud practical applicatio i ultra-cold atomic collisios at egative eergies [6]. Similar egative eergy resoaces were discussed i the past by oe of the Authors [7]. I Table b, we show the results of our calculatio usig the same parameters as i Table a but with ozero agular mometum ( = ). Additioally, i Table 3, we use the oscillator basis ad choose a differet set of model parameters (i atomic uits) ad for =. The umerical results for the resoaces ad boud states show i Tables ad 3, which are geerated by the S-matrix approach, are very close to those obtaied by complex rotatio. Hece the effect of the ifiite tail resultig from the J-matrix approach for this model has o substatial cotributio. O the other had, we expect that the ifiite tail origiatig from the H cotributio will be more importat for higher values of the agular mometum ad/or shorter potetial rage (large α). To study quatitatively the tail effect we use the followig umerical approximatio for the Hamiltoia ( H ) + V ;, m N H (8) ( H ) ; M + N, m > N which restricts the potetial cotributio to a N N subspace while the H part of the Hamiltoia is exteded to (N+M) (N+M) space to allow for a extra M-dimesioal tridiagoal tail cotributio from H. I the S-matrix approach, this tail goes to ifiity but we will see umerically that the tail cotributio i our model reaches its asymptotic cotributio very quickly. Specifically, the asymptotic limit is already reached for M =. Figure shows the results of resoace calculatios with the parameters (i atomic uits = m = ): V =, r =, α =.5 ad = 5 usig the Laguerre basis with λ = 5 ad θ =.9 radias. First, we computed the resoaces associated with this potetial for a large value of N, (N = ad M = ) show as crosses i Fig. where they serve as a referece for compariso to the values geerated with smaller values of N ad icreasig legth of the tail. We otice from the figure that the tail effect is more proouced for deep resoaces. A similar aalysis ca be doe by lookig at the umerical values i a tabular form for these resoaces i the presece of a tail, which shows that the tail effect is really cofied to improvig the last couple of digits. I summary, we exteded our approach i [7] to the computatio of the boud states ad resoace eergies, as beig the poles of the S-matrix, usig the power of the J-matrix techique that icludes a partial cotributio of the potetial but full aalytic cotributio of the referece Hamiltoia. Our approach could easily be geeralized to 5

6 hadle other short-rage ad eve /r sigular potetials such as the Yukawa ad Hulthé potetials, to metio oly a few [8]. These results suggest that our preset approach is of comparable accuracy to the complex rotatio approach. However, the J- matrix approach ca easily hadle o-aalytic ad /r sigular potetials, which caot be hadled with great accuracy usig other approaches ad caot be hadled easily usig the complex rotatio method. We have also studied the importace of the tail cotributio iheret i the J-matrix approach ad showed that the tail effect will improve the accuracy of the resoaces by few sigificat digits for the preset model. I the future third article of this series, we will ivestigate a geeralized threeparameter Morse potetial model that should be more suitable ad give a higher degree of freedom for the descriptio of various diatomic molecules. This three-parameter Morse potetial reads as follows () α( r r) r α( r r) r VGM r = V e βe, (9) where β is a ew dimesioless parameter whose value is uity for the regular Morse potetial. As show i paper I, this geeralized potetial results i a exact S-wave ( = ) solutio for the time-idepedet Schrödiger equatio. Ackowledgmet We ackowledge the support provided by the Physics Departmet at Kig Fahd Uiversity of Petroleum & Mierals uder project FT-8-. We are also grateful to Khaled Techical & Commercial Services (KTeCS) for the geerous support. We also appreciate the commets of the Referees that resulted i sigificat improvemets o the presetatio of the work. Refereces [] P. M. Morse, Phys. Rev. 34, 57 (99); S. H. Dog, R. Lemus, A. Frak, It. J. Quat. Phys. 86, 433 () ad refereces therei. [] R. R. Betts ad A. H. Wuosmaa, Rep. Prog. Phys. 6, 89 ( 997). [3] S. Flügge, Practical Quatum Mechaics, vol. I (Spriger-Verlag, Berli, 994). [4] C. L. Pekeris, Phys. Rev. 45, 98 (934); R. Herma, R. J. Rubi, Astrophys. J., 533 (955); M. Duff, H. Rabitz, Chem. Phys. Lett. 53, 5 (978); J. R. Elsum, G. Gordo, J. Chem. Phys. 76, 545 (98); E. D. Filho, R. M. Ricotta, Phys. Lett. A 69, 69 (); F. Cooper, A. Khare, ad U. Sukhatme, Phys. Rep. 5, 67 (995); D. A. Morales, Chem. Phys. Lett. 394, 68 (4); J. P. Killigbeck, A. Grosjea, ad G. Jolicard, J. Chem. Phys. 6, 447 (); T. Imbo ad U. Sukhatme, Phys. Rev. Lett. 54, 84 (985); M. Bag, M. M. Paja, R. Dutt ad Y. P. Varshi, Phys. Rev. A 46, 659 (99). [5] A. F. Nikiforov ad V. B. Uvarov, Special Fuctios of Mathematical Physics (Basel, Birkhausr, 988); C. Berkdemir ad J. Ha, Chem. Phys. Lett. 49, 3 (5); C. Berkdemir, Nucl. Phys. A 77, 3 (6). [6] O. Bayrak ad I. Boztosum, J. Phys. A 39, 6955 (6). [7] I. Nasser, M. S. Abdelmoem, H. Bahlouli ad A. D. Alhaidari, J. Phys. B 4, 445 (7). 6

7 [8] E. J. Heller ad H. A. Yamai, Phys. Rev. A 9, (974); H. A. Yamai ad L. Fishma, J. Math. Phys. 6, 4 (975); A. D. Alhaidari, E. J. Heller, H. A. Yamai, ad M. S. Abdelmoem (eds.), The J-matrix method: developmets ad applicatios (Spriger-Verlag, Dordrecht, 8). [9] S. A. Sofiaos ad S. A. Rakityasky, J. Phys. A 3, 375 (997); S. A. Rakityasky, S. A. Sofiaos ad N. Elader, J. Phys. A 4, 4857 (7); H. M. Nussezveig, Causality ad Dispersio Relatios (Academic, Lodo, 97). [] H. A. Yamai ad M. S. Abdelmoem, J. Phys. B 3, 633 (997); 3, 3743 (997). [] J. Aguilar ad J. M. Combes, Commu. Math. Phys., 69 (97); Y. K. Ho, Phys. Rep. 99, (983). [] A. D. Alhaidari, A. Phys. (NY) 37, 5 (5). [3] See, for example, Appedix A i: A. D. Alhaidari, H. A. Yamai, ad M. S. Abdelmoem, Phys. Rev. A 63, 678 (). [4] S. Wolfram, The Mathematica Book, Wolfram Media, 999. [5] G. Rawitscher, C. Merow, M. Nguye ad I. Simboti, Am. J. Phys 7, 9 (). [6] S. G. Bhogale, S. J. J. M. F. Kokkelmas ad Iva H. Deutsch, Phys. Rev. A 77, 57 (8). [7] A. D. Alhaidari, It. J. Mod. Phys. A, 657 (5). [8] A. D. Alhaidari, H. Bahlouli ad M. S. Abdelmoem, J. Phys. A 4, 3 (8). 7

8 Table Captios Table : The explicit form of the kiematic quatities T ( E ), R ± ( E), JN, N( E) ad N D ν i both Laguerre ad oscillator bases. The three-term recursio relatios for s ad c (collectively show as f ) i both bases are also give. Table a: Boud ad resoace eergies calculated usig the S-matrix approach i the Laguerre basis for the iverted Morse potetial with the parameters: V = 6 fm, α =.3 fm, r = 4. fm, ad =. Table b: Reproductio of Table a but for = Table 3: Boud ad resoace eergies calculated usig the S-matrix approach i the oscillator basis for the followig Morse potetial parameters (i the uits = m = ): V =, α =., r =., ad =. Table Laguerre basis Oscillator basis iθ T ( E ) iθ F(,; + ; e ) 8E λ e ; cosθ = iθ F(,; + ; e ) 8E+ λ R ± ( E) J N ( ) iθ (,; + 3; e ) iθ ( e ) F iθ e + F,; + ;, N( E) ( E λ 8 ) N ( N ν ) + + ( λ ) N D ν N + ν π Γ ( ν + ) π ν x/ (ν + ) 4 s = e x ; x= E λ c = e x F( ν, ν; x) x/ ( ν ) 4 π Γ ( ν + ) π νν ( + ) s = ( ν + x) e x x/ ( ν )/4 x/ (ν + )/4 c = e x F( ν, ν; x) N( N + ν) Recursio Relatio (cos θ )( + + ) f ( + + ) f + ( + )( + + ) f = ; 3 ( + + x) f ( + + ) f 3 + ( + )( + + ) f = ; 8

9 Table a N λ E = E i E (fm ) Rawitscher [5] R I i i i i i i i i i i i i i i i i.39 Table b N λ E = E i E (fm ) R I i i i i i i i i i i i i i

10 Table 3 N λ E = E i E (a.u.) R I i i i i

11 Figure Captio Figure : Resoace plots idetified with the pair (N,M), where N is the size of the potetial matrix ad N+M is the size of the H matrix. The crosses show the most accurate locatio of the resoaces (correspodig to N = ad M = ). The grid scale is. eergy uits ( V uits). Fig.