eclética química 35-3.indd ECLÉTICA química www.scielo.br/eq Volume 35, número 3, 2010 103 Artigo/Article Ecl. Quím., São Paulo, 35 - 3: 93 - 102, 2010102 Artigo Article Conclusions Simple spectrophotometric method for the determination of NVP have been developed and validated according to ICH guidelines. The me- thod is simple and easy to perform compared to other existing methods and do not entail any rigo- rous experimental variables which affect the relia- bility of the results. The ingredients usually pre- sent in the pharmaceutical formulations of these drugs seldom interfere in the proposed methods. The proposed method is simple, accurate and easy to perform and can be used for the routine deter- mination of NVP in bulk and in dosage forms. References [1] S. Staszewski, J. Morales-Ramirez, K. T. Tashima, A. Rachlis, D. Skiest, J. Stanford, R. Stryker, P. Johnson, D. F. Labriola, D. Farina, D. J. Manion, N. M. Ruiz, N. Engl. J. Med. 341 (1999) 1865. [2] J. S. G. Montaner, P. Reiss, D. Cooper, S. Vella, M. Harris, B. Conway, M. A. Wainberg, D. Smith, P. Robinson, D. Hall, M. Myers, J. M. A. Lange, JAMA, 279 (1998) 930. [3] D. Podzamczer, E. Ferrer, E. Consiglio, J. M. Gatell, P. Perez, J. L. Perez, E. Luna, A. Gonzalez, E. Pedrol, L. Lozano, I. Ocana, J. M. Llibre, A. Casiro, M. Aranda, P. Barrufet, J. Martinez-Lacasa, J. M. Miro, X. Badia, A. Casado, S. Lupo, P. Cahn, M. Manos, J. Estela, Antivir Ther. 7 (2002) 81. [4] L. I. Malaty, J. J. Kupper, Drug Sar. 20 (1999)147. [5] D. Back, S. Gibbons, S. Khoo, J. Acquir. Immune Defic. Syndr. 34 (2003) S8–14. [6] R. Panchagnula, S. Agrawal, Y. Ashokraj, M.V.S. Varma, K. Sateesh, V. Bhardwaj, S. Bedi, I. Gulati, J. Parmar, C. Kaul, B. Blomberg, B. Fourie, G. Roscigno, R. Wire, R. Laing , P. Evans, T. Moore, Methods Find. Exp. Clin. Pharmacol. 26 (2004) 703. [7] D. Rey, M. Partisani, H. -K. Georgette, V. Krantz, M. Priester, C. Christine, B. -H. Claudine, E. de Mautort, L. Decroix, J.-M. Lang, J. Acquir. Immune Defic. Syndr. 37 (2004) 1454. [8] C. C. J. Carpenter, M. A. Fischl, S. M. Hammer, M. S. Hirsch, D. M. Jacobsen, D. A. Katzenstein, J. S. G. Montaner, D. D. Richman, M. S. Saag, R. T. Schooley, M. A. Thompson, S. Vella, P. G. Yeni, P. A. Volberding, J. Am. Med. Assoc. 77 (1997) 1962. [9] R. M. Gulick, J. M. Mellors, D. Havlir, J. J. Eron, C. Gonzalez, D. McMahon, D. D. Richman, F. T. Valentine, L. Jonas, A. Meibohm, E. A. Emini, J. A. Chodekewitz, N. Engl. J. Med. 337 (1997) 734. [10] W. Cavert, D. W. Notermans, K. Staskus, W. W.Stephen, Z. Mary, G. Kristin, H. Keith, Z. –Q. Zhang, R. Mills, H. McDade, J. Goudsmit, S. A. Danner & T. H. Ashley, Science 276 (1997) 960. [11] A. Carr, D. A. Cooper, Adv. Exp. Med. Biol. 394 (1996) 299. [12] S. M. Hammer, J. Infect. Dis. 192 (2005) 1. [13] M. Hartmann, S. Witte, J. Brust, D. Schuster, F. Mosthaf, M. Procaccianti, J. A. Rump, H. Klinker, D. Petzoldt, Int. J. STD. AIDS 16 (2005) 404. [14] P. Barreiro, V. Soriano, F. Blanco, C. Casimiro, J. J. de la Cruz, J. Gonzalez-Lahoz, AIDS 14 (2000) 807. [15] N. L. Rezk, R. R. Tidwell, A. D. M. Kashuba, J. Chromatogr. B (2004) 805, 241. [16] C. F. Silverthorn, T. L. Parsons, Biomed. Chromatogr. 20 (2006) 23. [17] G. Ramachandran, A. K. Hemanthkumar, V. Kumaraswami, S. Swaminathan, J. Chromatogr. B 843 (2006) 339. [18] B. H. Chi, A. Lee, E. P. Acosta, L. E. Westerman, M. Sinkala, J. S. A. Stringer, HIV Clin. Trials 7 (2006) 263. [19] R. ter Heine, H. Rosing, E. C. M. van Gorp, J. W. Mulder, W. A. van der Steeg, J. H. Beijnen, A. D. R. Huitema, J. Chromatogr. B 867 (2008) 205. [20] R. Sekar, S. Azhaguvel, Chromatographia 67 (2008) 389. [21] S. Notari, C. Mancone, T. Alonzi, M. Tripodi, P. Narciso, P. Ascenzi, J. Chromatogr. B 863 (2008) 249. [22] G. Ramachandran, A. K. Hemanthkumar, V. Kumaraswami, S. Swaminathan, J. Chromatogr. B 843 (2006) 339. [23] G. R. Da Silva, G. P. Lages, G.A. Pianetti, E.D.A. Nunan, C. D. V. Soares, L. M. M. De Campos, Quimica Nova 29 (2006) 1159. [24] C. F. Silverthorn, T. L. Parsons, Biomed. Chromatogr. 20 (2006) 23. [25] P. Lemmer, S. Schneider, M. Schuman, C. Omes, V. Arendt, J.-C. Tayari, L. Fundira, R. Wennig, Therapeutic Drug Monitoring 27 (2005) 521. [26] U. S. Pharmacopeia 29 NF 24 (2006) 29 15-19-1520. BOUND STATE SOLUTIONS OF SCHRÖDINGER EQUATION FOR A MORE GENERAL EXPONENTIAL SCREENED COULOMB POTENTIAL VIA NIKIFOROV- UVAROV METHOD Benedict I. Ita, P. Ekuri Quantum Chemistry Group, Department of Pure and Applied Chemistry, University of Calabar, P. O. Box 3700, Calabar, CRS, Nigeria (Corresponding author: e-mail: iserom 2001@yahoo.com) Idongesit O. Isaac Department of Mathematics/Statistics and Computer Science, University of Calabar, Calabar, Cross River State, Nigeria. Abosede O. James Department of Pure and Industrial Chemistry, University of Port Harcourt, Nigeria Abstract: The arbitrary angular momentum solutions of the Schrödinger equation for a diatomic molecule with the general exponential screened coulomb potential of the form ( ) ( ) ( ){ }br2ebr11r/arV −++−= has been presented. The energy eigenvalues and the corresponding eigenfunctions are calculated analytically by the use of Nikiforov-Uvarov (NU) method which is related to the solutions in terms of Jacobi polynomials. The bounded state eigenvalues are calculated numerically for the 1s state of N2 CO and NO Keywords: Nikiforov-Uvarov method, Eigenvalues, Eigenfunctions, General Exponential Screened Cou- lomb Potential. Introduction The exact analytic solutions of the wave equations (non-relativistic and relativistic) are only possible for certain potentials of physical in- terest under consideration since they contain all the necessary information on the quantum system [1]. It is known that for certain potentials, the Schrödinger equation can be solved for the angu- lar momentum quantum numbers 0= [2]. How- ever, in some cases, like for the 0≠ states, some approximations are often used to obtain analytic solutions of the Schrödinger equation [3 – 5]. A more general exponential screened cou- lomb (MGESC) potential used in this paper is of the form [6]: ( ) ( ) ( ){ }br2expbr11 r a rV −++⎟ ⎠ ⎞⎜ ⎝ ⎛−= (1) where a is the strength coupling constant and b is the screened parameter. The potential in equation (1) is known to describe adequately the effective interaction in many-body environments of a variety of fi elds [6]. In this paper, we have de- cided to explore the possibility of also using it in obtaining bound state solutions of the Schrödinger equation for diatomics using Nikiforov-Uvarov (NU) method. Ecl. Quím., São Paulo, 35 - 3: 103 - 107, 2010 Ecl. Quím., São Paulo, 35 - 3: 103 - 107, 2010104 Ecl. Quím., São Paulo, 35 - 3: 103 - 107, 2010 105 Artigo Article Artigo Article Overview of Nikiforov-Uvarov (NU) Method The NU method is based on the solutions of general second order linear differential equa- tions with some orthogonal functions [7]. For the given potential, the Schrödinger equation in the spherical coordinates is reduced to a generalized equation of hyper-geometric type with an appro- priate )r(ss = coordinate transformation. Thus, it has the form [8]: ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0s s s s s s s 2 =ψ σ σ+ψ′ σ τ+ψ ′′ (2) , where ( )sσ and ( )sσ are polynomials, at most second-degree, and ( )sτ is a fi rst-degree polyno- mial. To fi nd a particular solution of equation (2), we use the following transformation [9]: ( ) ( ) ( )syss φ=ψ (3) This reduces Schrödinger equation (2) to an equation of hyper-geometric type: ( ) ( ) 0yysys =λ+′τ+′′σ (4) where ( )sφ satisfi es ( ) ( ) ( ) ( )s/ss/s σπ=φφ′ , y(s) is the hyper-geometric type function whose polynomial solutions are given by the Rodrigues relation: ( ) ( ) ( ) ( )[ ]ss ds d s B sy n n n n n ρσ ρ = (5) where nB is a normalization constant and the weight function ρ must satisfy the condition [9]: ( ) ( )[ ] ( ) ( )ssss ρτ=′ρσ (6) The function π and the parameter λ re- quired for this method are defi ned as: σ+σ−⎟ ⎠ ⎞⎜ ⎝ ⎛ τ−σ′ ±τ−σ′ =π k 22 2 (7) and π′+=λ k (8) Here, ( )sπ is a polynomial with the param- eter s and the determination of k is necessary for ( )sπ to be obtained. To fi nd k, the expression un- der the square root must be square of a polynomi- al. A new eigenvalue equation for the Schröding- er equation thus becomes: ( ) ( ),,2,1,0n, 2 1nn nn =σ′′−−τ′−=λ=λ (9) where ( ) ( ) ( )s2ss π+τ=τ (10) and ( )sτ′ must be negative. Bound State Solutions via Nikiforov-Uvarov (NU) Method The potential in equation (1) is substituted into the radial Schrödinger equation given as: ( ) ( ) ( ) ( ) ( ),rRrErRrV r2 h1 dr d r dr d r 1 2 h nnn2 2 2 2 2   = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + μ ++⎟ ⎠ ⎞⎜ ⎝ ⎛ μ − (11) where n denotes the radial quantum num- ber which together with  are both named as the vibration-rotation quantum numbers in molecular chemistry, r is the internuclear separation, nE is the exact bound state energy eigenvalues and V(r) is the internuclear potential energy function and we obtain: ( ) ( ) ( ) ( ) .0rR r2 h1 abee r a r a E h 2 dr rdR r 2 dr rRd n2 2 br2br2 n2 n 2 n 2 = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ μ +−+++μ++ −−    (12) Equation (12) can be rearranged to give: ( ) ( ) ( ) ( ) ( ) ( ) .0rR 2 h1 raearabeE h 2 r 1 dr rdR r 2 dr rRd n 2 br22br2 n22 n 2 n 2 = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ μ +−+++μ++ −−    (13) Introducing the following dimensional pa- rameters: [ ] ,abeE h 2 br2 n2 2 −+μ=ε  (14) [ ]br2 2 aea h 2 −+μ=β− (15) ( )1+=γ  (16) equation (13) is written as: ( ) ( ) ( ) ( ) .0rRrr r 1 dr rdR r 2 dr rRd n 22 2 n 2 n 2 =γ−β−ε++   (17) A comparison of equations (2) and (17) re- veals the following polynomials: ( ) ,2r =τ ( ) ,rr =σ ( ) γ−β−ε=σ rrr 22 (18) Substituting these polynomials into equa- tion (7), we get ( )rπ as: ( ) ( ) 14rk4r4 2 1 2 1 r 22 +γ+β++ε−±−=π (19) and ( )rσ′ is taken equal to 1. The discri- minant of the expression under the square root in equation (19) has to be zero for it to have equal roots. Therefore, we get: ( )[ ] ( )( ) 01444k4 22 =+γε−−β+ . (20) On solving equation (20) for k we get: 14ik +γε±β−=± , (21) where 14ik +γε−β−=− (22) and .14ik +γε+β−=+ (23) Substituting ±k into equation (19), gives the following four possible solutions obtained for ( )rπ as: ( ) ⎪ ⎪ ⎩ ⎪⎪ ⎨ ⎧ +γε+β−=+γ+ε +γε−β−=+γ−ε ±−=π + − .14ikfor,14 2 1 ri 14ikfor,14 2 1 ri 2 1 r (24) From the four possible forms of ( )rπ in equation (24), we select the one for which the function ( )sτ in equation (10) has a negative de- rivative. ( )sτ satisfi es these requirements with: ( ) 14ri21r +γ+ε−=τ (25) and ( ) 0i2r <ε−=τ′ (26) From equation (8), we obtain: ε−+γε−β−=λ i14i (27) and also .,3,2,1,0n,ni2n =ε=λ=λ (28) We then obtain the parameters 2ε as: . 14n21 2 2 ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ +γ++ β−=ε (29) Substituting the values of 2ε , β and γ from equations (14) – (16) into equation (29), yields: ( ) ( )[ ] 2 br2 2 br2 n 114n21 aea h 2 abeE ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ++++ +μ+−= − −   . (30) To fi nd y(r), we fi rst obtain ( )rρ from equa- tion (6) as: Ecl. Quím., São Paulo, 35 - 3: 103 - 107, 2010106 Ecl. Quím., São Paulo, 35 - 3: 103 - 107, 2010 107 Artigo Article Artigo Article ( ) ri2141 err ε−+γ+=ρ . (31) Substituting this into the Rodrigues rela- tion given in equation (5), we get: ( ) ( ) ( ) ⎥⎦ ⎤ ⎢⎣ ⎡= ε−+γ++ε+γ+− ri2141n n n ri2141 nn er ds d erBry  , (32) nB is the normalization constant. The polynomial solutions of ( )ryn in equation (32) are expressed in terms of the associated Laguerre polynomials, which is one of the orthogonal poly- nomials. We write: ( ) ( )vLry 141 nn +γ+= , (33) where ,ri2v ε= therefore, ( ) vi2r 1−ε= . (34) By substituting ( )rπ and ( )rσ into the ex- pression ( ) ( ) ( ) ( )r/rr/r σπ=φφ′ and solving the re- sulting differential equation, the other part of the wave function in equation (3) is obtained as: ( ) ri14 err 2 1 2 1 ε−−+γ=φ (35) or in terms of v, ( ) ( ) 2 v 2 1 2 1 2 1 2 3 evi2v 1414 −+γ+−+γ+−ε=φ . (36) Combining the Laguerre polynomials and ( )vφ in equation (3), enables the radial wave func- tion to be constructed as: ( ) ( )rArR nnn  ψ= (37) ( ) ( ) ( )vLevi2ArR 141 n 1414 nn 2 v 2 1 2 1 2 1 2 3 +γ+−+γ+−+γ+−ε=∴  . (38) If we introduce the variable 14 2 1 +γ=α , equation (38) becomes: ( ) ( ) ( )vLevi2ArR 21 nnn 2 v 2 1 2 3 α+−α+−α+−ε=  . (39) To fi nd nA , a new normalization constant, we write: ( )∫ ∞ = 0 2 n .1drrR  (40) Therefore, ( ) ( )[ ] .1dvvLevi2A 0 212 n v12322 n =ε ∫ ∞ +α−−α−α  (41) The above integral can be evaluated by us- ing the recursion relation for Laguerre polynomi- als and nA is found to be: ( ) ( ) ( )( ) 2 1 3 23 n !n22n2 i2!12n A ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ +α− ε+α−= α−  . (42) Therefore, ( )rRn becomes: ( ) ( ) ( ) ( )( ) ( ) ( ).vLevi2 !n22n2 i2!12n rR 12 n3 23 n 2 v 2 1 2 3 2 1 +α−−αα+− α− ε ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ +α− ε+α−= (43) Conclusion The analytical solutions of the Schröding- er equation for the general exponential screened coulomb potential has been presented. The Niki- forov-Uvarov method employed in the solutions enables us to explore an effective way of obtain- ing the eigenvalues and corresponding eigenfunc- tions of the Schrödinger equation for any  - state. Finally, we calculate the energies of the ex- ponential screened coulomb potential for diatomic molecules by means of equation (30) for the  - state. The explicit values of the energy at dif- ferent values of the screened parameter are shown in Table 1. Table 1. Bound State Eigenvalues for 05.0b0 ≤≤ for the 1s State of Diatomic Molecules in Atomic Units ( )1ah ==μ= b E1s(ev) 2N CO NO 0.01 0.2348302 0.2346764 0.2345716 0.02 0.2202668 0.2199730 0.2197791 0.03 0.2062752 0.2058684 0.2056002 0.04 0.1928410 0.1932419 0.1920133 0.05 0.1799455 0.17937370 0.1789975 Note: The r values for 2N (1.0940), CO (1.21282) and NO (1.1508) were adapted from M. Karplus and R. N. Porter, Atoms and Molecules: An Introduction for Student’s of Physical Chemistry, Benjamin, Menlo Park, CA, 1970. References [1] S. M. Ikhdair and R. Sever (2008). Improved analytical approximation to arbitrary  - state solutions of the Schrödinger equation for the hyperbolical potentials. Personal Communication. [2] C. Berkdemir and J. Han (2005). Chem. Phys. Lett. 409, 203 – 207. [3] E. Aydmer and C. Orta (2008). Quantum information entropies of the eigenstates of the eigenvalues of the Morse potential. Personal Communication. [4] H. Taseli (1997). Int. J. Quantum Chemistry, 63(5), 949 – 959. [5] M. W. Kermode, M. L. J. Allen, J. P. McTavish and A. Kervell (1984). J. Phys. G: Nucl. Phys. 773 – 783. [6] S. M. Ikhdair and R. Sever (2008). Bound states of a more general exponential screened coulomb potential. Personal Communication. [7] A. F. Nikiforov and U. B. Uvarov (1988). Special Functions of Mathematical Physics, Birkhauser: Basel. [8] C. Tezcan and R. Sever (2008). Quantum Physics. 15, 1 – 20. [9] S. Ikhdair and R. Sever (2008). Cent. Eur. J. Physics. 6, 141 – 152.