Original article iq.unesp.br/ecletica | Vol. 45 | n. 4 | 2020 | 40 Eclética Química Journal, vol. 45, n. 4, 2020, 40-56 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v45.4.2020.p40-56 ABSTRACT: Within the framework of the conventional Nikiforov- Uvarov method and a new form of Greene-Aldrich approximation scheme, we solved the Schrödinger equation with the energy- dependent screened Coulomb potential. Energy eigenvalues and energy eigenfunctions were obtained both approximately and numerically at different dimensions. The energy variations with different potential parameters, quantum numbers and energy slope parameter, respectively were also discussed graphically. The major finding of this research is the effect of the energy slope parameter on the energy spectra, which is seen in the existence of two simultaneous energy values for a particular quantum state. Our special cases also agree with the results obtained from literature, when the energy slope parameter is zero. Approximate solutions of the Schrödinger equation with energy- dependent screened Coulomb potential in D – dimensions Uduakobong Sunday Okorie1,3+ , Akpan Ndem Ikot2,3 , Precious Ogbonda Amadi3 , Alalibo Thompson Ngiangia3 , Etebong Emmanuel Ibekwe1,3 1. Department of Physics, Akwa Ibom State University, Ikot Akpaden P. M. B. 1167, Uyo, Nigeria 2. Department of Physics, University of South Africa, Florida 1710, Johannesburg, South Africa 3. Theoretical Physics Group, Department of Physics, University of Port Harcourt, P. M. B. 5323 Choba, Nigeria +Corresponding author: Uduakobong Sunday Okorie, Phone: +2347081545195 Email address: uduakobongokorie@aksu.edu.ng ARTICLE INFO Article history: Received: October 26, 2019 Accepted: April 07, 2020 Published: October 01, 2020 Keywords: 1. Schrödinger equation 2. energy-dependent screened Coulomb Potential 3. Nikiforov-Uvarov method 4. Greene-Aldrich approximation scheme 1. Introduction Quantum mechanics came into existence many decades ago to salvage the failure of classical mechanics, not being able to explain some of the physical phenomena such as Compton effects, specific heat capacity, blackbody radiation. From this point of view, many theoretical physicists have been investigating the exact and approximate solutions of the Schrödinger equation for some potentials of physical interests1,2. The solutions of the Schrödinger equation play a vital role in many branches of modern physics and chemistry3. This is because it contains all the necessary information needed for the full description of a quantum state such as the probability density and entropy of the system4. The Schrödinger equation with many physical potentials model have been investigated in recent times with different analytical methods such as Nikiforov-Uvarov (NU) method5-8, asymptotic iteration method (AIM)9-14, supersymmetric quantum mechanics(SUSYQM)15-18 among others19-22. One of such potential models is the screened Coulomb potential, which is given by Eq. 123. 𝑉(𝑟) = − 𝐴𝑒−𝛼𝑟 𝑟 (1) The screened Coulomb potential, also known as the Yukawa potential is greatly important, with applications cutting across nuclear Physics and Condensed-matter Physics23. Here, its usage is involved in short-ranged interactions24-26. The screened-Coulomb potential is known to be the http://revista.iq.unesp.br/ojs/index.php/ecletica/index https://doi.org/10.26850/1678-4618eqj.v45.4.2020.p40-56 mailto:uduakobongokorie@aksu.edu.ng mailto:uduakobongokorie@aksu.edu.ng http://orcid.org/0000-0002-5660-0289 http://orcid.org/0000-0002-1078-262X http://orcid.org/0000-0003-3691-3891 https://orcid.org/0000-0001-7642-2419 https://orcid.org/0000-0002-1754-6523 Original article 41 Eclética Química Journal, vol. 45, n. 4, 2020, 40-56 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v45.4.2020.p40-56 potential of a charged particle in a weakly non- ideal plasma. It also describes the charged particle effects in a sea of conduction electrons in solid- state physics27. An approximate solution of the Schrödinger equation interacting with an inversely quadratic Yukawa potential has been obtained using SUSYQM15, where the screened Coulomb potential was obtained as a special case by varying the potential strength. Also, an approximate analytical solution of the radial Schrödinger equation for the screened Coulomb potential has been obtained, with energy eigenvalues and its corresponding eigenfunctions computed in closed forms28. Several researchers have also devoted great attention to investigate the quantum systems of the energy dependence of different potentials29-31. Hassanabadi et al.32,33 studied the exact solutions of D-dimensional Schrödinger and Klein-Gordon equations using the Nikiforov-Uvarov method. Also, Lombard et al.34 investigated the wave equation energy-dependent potential for confined systems. Numerous applications of the energy- dependent potential of wave equations have been seen in the spectrum of confined systems and heavy quark confinement in nuclear and molecular physics35,36. Recently, Budaca37 studied an energy-dependent Coulomb-like potential within the framework of Bohr Hamiltonian. The author further reported that the energy dependence on the coupling constant of the potential drastically changes the analytical properties of wave function and the corresponding eigenvalues of the system. Also, Boumali and Labidi38 investigated the Shannon and Fisher information in the Klein-Gordon equation with energy- dependent potential. In this research, we seek to investigate the influence of the energy-dependent screened Coulomb potential defined as in Eq. 2, 𝑉(𝑟, 𝐸𝑛ℓ) = − 𝐴(1+𝑔𝐸𝑛ℓ)𝑒 −𝛼𝑟 𝑟 (2) where 𝑔 is the energy slope parameter, 𝐴 is the depth of the potential, and 𝛼 is the range of the potential. The effects of the energy dependence on the screened Coulomb potential have not been considered before in any literature, to the best of our knowledge. It can be deduced that when 𝑔 = 0, the potential of Eq. 2 reduces to the Screened Coulomb potential. When 𝑔 = 0 as 𝛼 → 0, the potential of Eq. 2 reduces to the Coulomb potential. Using the conventional NU method, we will derive the ℓ −wave bound state solutions and their eigenfunctions of the Schrödinger equation for the energy-dependent screened Coulomb potential, both analytically and numerically. Special cases are also considered and our results are compared with existing literature for confirmation sake. The organization of this work is as follows: In section 2, we determine the eigensolutions of the energy-dependent screened Coulomb potential by employing a new form of Greene-Aldrich approximation scheme and Nikiforov-Uvarov method. Section 3 is devoted to discuss the results obtained and compare to results in relevant literature. The conclusion of the work is presented in section 4. 2. Bound state solution of the energy-dependent screened Coulomb potential The radial part of the Schrödinger equation in D-dimension39 is given by Eq. 3. 𝜓″(𝑟) + 2𝜇 ℏ 2 (𝐸𝑛ℓ − 𝑉(𝑟))𝜓(𝑟) + 1 𝑟2 [ (𝐷−1)(𝐷−3) 4 + ℓ(ℓ+ 𝐷 − 2)]𝜓(𝑟) = 0 (3) where 𝜇 is the reduced mass, 𝐸𝑛ℓ is the non-relativistic energy eigenvalues to be determined. Substituting Eq. 2 into Eq. 3 gives Eq. 4. 𝜓″(𝑟) + 2𝜇 ℏ 2 (𝐸𝑛ℓ + 𝐴(1+𝑔𝐸𝑛ℓ)𝑒 −𝛼𝑟 𝑟 )𝜓(𝑟) + 1 𝑟2 [ (𝐷−1)(𝐷−3) 4 + ℓ(ℓ + 𝐷 − 2)]𝜓(𝑟) = 0 (4) In solving Eq. 4, we invoke a new form of Greene-Aldrich approximation scheme23 to deal with the centrifugal term since ℓ ≠ 0. The approximations schemes are given by Eq. 5 and 6. https://doi.org/10.26850/1678-4618eqj.v45.4.2020.p40-56 Original article 42 Eclética Química Journal, vol. 45, n. 4, 2020, 40-56 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v45.4.2020.p40-56 𝑓1(𝑟) = 1 𝑟2 ≈ 4𝛼2𝑒−2𝛼𝑟 (1−𝑒−2𝛼𝑟)2 = 𝑓2(𝑟) (5) 1 𝑟2 ≈ 4𝛼2𝑒−2𝛼𝑟 (1−𝑒−2𝛼𝑟)2 + 𝛼2 3 = 𝑓3(𝑟) (6) The plots explaining the rationality and validity of the above Eq. 5 and 6 and their expansions are given in Fig. 1(a–d). Figure 1a shows the plots of 𝑓1(𝑟), 𝑓2(𝑟)and𝑓3(𝑟) as they vary with 𝑟, when the screening parameter 𝛼 is taken to be 0.5. In addition, we employ 1 𝑟 from 𝑓1(𝑟) as given in Eq. 7. 𝑓4(𝑟) = 1 𝑟 ≈ 2𝛼𝑒−𝛼𝑟 1−𝑒−2𝛼𝑟 ≡ 𝑓5(𝑟) (7) Figure 1. The plots of expressions (a) 𝑓1(𝑟), 𝑓2(𝑟), 𝑓3(𝑟) as functions of 𝑟 with 𝛼 = 0.5; (b) 𝑓4(𝑟), 𝑓5(𝑟) as functions of 𝑟 with 𝛼 = 0.5; (c) 𝑓1(𝛼), 𝑓2(𝛼), 𝑓3(𝛼) as functions of 𝛼 with 𝑟 = 5; (d) 𝑓4(𝛼), 𝑓5(𝛼) as functions of 𝛼 with 𝑟 = 5𝑛𝑚. https://doi.org/10.26850/1678-4618eqj.v45.4.2020.p40-56 Original article 43 Eclética Química Journal, vol. 45, n. 4, 2020, 40-56 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v45.4.2020.p40-56 Here, we have ignored the second expansion term 𝛼(1−𝑒−2𝛼𝑟) (12𝑒−𝛼𝑟) when expanding Eq. 6. Furthermore, the rationality and validity of this expansion is given in Fig. 1b, with 𝛼 = 0.5. Considering the plots of the above expressions as function ns of the screening parameter 𝛼 as shown in Figs. 1c and 1d, we observe that the approximation 𝑓3(𝛼) corresponds to that of 𝑓1(𝛼) when 𝛼 ≤ 0.05. As such, we see that the approximation 𝑓3(𝛼) is better than that of 𝑓2(𝛼). With the above approximation schemes, Eq. 4 becomes Eq. 8. 𝜓″(𝑟) + [ 2𝜇𝐸𝑛ℓ ℏ 2 + 4𝜇𝐴(1+𝑔𝐸𝑛ℓ)𝑒 −2𝛼𝑟 ℏ 2(1−𝑒−2𝛼𝑟) − 4𝛼2𝛾𝑒−2𝛼𝑟 ℏ 2(1−𝑒−2𝛼𝑟)2 − 𝛾𝛼2 3 ]𝜓(𝑟) = 0 (8) where  is given by Eq. 9. 𝛾 = [ (𝐷−1)(𝐷−3) 4 + ℓ(ℓ + 𝐷 − 2)] (9) By using the coordinate transformation of Eq. 10. 𝑧 = 𝑒−2𝛼𝑟 (10) Eq. 8 becomes the differential equation of the form given in Eq. 11. 𝜓″(𝑧) + (1−𝑧) 𝑧(1−𝑧) 𝜓′(𝑧) + 1 𝑧2(1−𝑧)2 [−(𝜀2 + 𝛽2)𝑧2 + (2𝜀2 + 𝛽2 − 𝛾)𝑧 − 𝜀2]𝜓(𝑧) = 0 (11) where 2 is given by Eq. 12. 𝜀2 = −( 𝜇𝐸𝑛ℓ 2ℏ2𝛼2 − 𝛾 12 ) ; 𝛽2 = 𝜇𝐴(1+𝑔𝐸𝑛ℓ) ℏ 2𝛼 (12) By comparing Eq. 11 and Eq. A1 (see Appendix), we have the following parameters, Eq. 13: �̃�(𝑧) = 1 − 𝑧 𝜎(𝑧) = 𝑧(1 − 𝑧) �̃�(𝑧) = −(𝜀2 + 𝛽2)𝑧2 + (2𝜀2 + 𝛽2 − 𝛾)𝑧 − 𝜀2 (13) Substituting Eq. 13 into Eq. A8 (see Appendix), we get 𝜋(𝑧), Eq. 14: 𝜋(𝑧) = − 𝑧 2 ±√(𝑎 − 𝑘)𝑧2 + (𝑘 + 𝑏)𝑧 + 𝑐 (14) where a, b and c are given by Eq. 15. 𝑎 = 1 4 + 𝜀2 + 𝛽2 𝑏 = 𝛾 − 2𝜀2 − 𝛽2 (15) 𝑐 = 𝜀2 We can obtain the constant 𝑘, by expressing the discriminant under the square root of Eq. 14 to be equal to zero. As such, we have Eq. 16 https://doi.org/10.26850/1678-4618eqj.v45.4.2020.p40-56 Original article 44 Eclética Química Journal, vol. 45, n. 4, 2020, 40-56 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v45.4.2020.p40-56 𝑘± = −(𝛾 − 𝛽 2) ± 2√𝜀2√( 1 4 + 𝛾) (16) Substituting Eq. 16 into Eq. 14 yields Eq. 17: 𝜋(𝑧) = − 𝑧 2 ± { (√𝜀2 −√ 1 4 + 𝛾) 𝑧 − √𝜀2; 𝑓𝑜𝑟𝑘+ = −(𝛾 − 𝛽 2) + 2√𝜀2√( 1 4 + 𝛾) (√𝜀2 −√ 1 4 + 𝛾) 𝑧 + √𝜀2; 𝑓𝑜𝑟𝑘− = −(𝛾 − 𝛽 2) − 2√𝜀2√( 1 4 + 𝛾) (17) According to NU method40, we choose the expression 𝜋(𝑧)−which the function 𝜏(𝑧) has a negative derivative. This is given by Eq. 18 𝜋(𝑧)− = −( 1 2 + √𝜀2 +√ 1 4 + 𝛾) 𝑧 − √𝜀2 (18) with 𝜏(𝑧) being obtained with Eq. 19. 𝜏(𝑧) = 1 − 2√𝜀2 − 2(1 + 2√𝜀2 +√ 1 4 + 𝛾) 𝑧 (19) By recalling Eq. A9 (see Appendix), we define the constant 𝜆 (Eq. 20) as 𝜆 = −(𝛾 − 𝛽2) − 2√𝜀2√ 1 4 + 𝛾 − ( 1 2 + √𝜀2 +√ 1 4 + 𝛾) (20) Substituting Eq. 20 into Eq. A10 (see Appendix) and carrying out algebraic simplifications, where ’(z) is given by Eq. 21 𝜏 ′(𝑧) = −2(1 + √𝜀2 +√ 1 4 + 𝛾) (21) and ”(z), Eq. 22 ( ) 2z  = − (22) we obtain Eq. 23 𝜀2 = 1 4 [ (𝑛+𝜒)2−𝛽2 (𝑛+𝜒) ] 2 (23) where  is given by Eq. 24 𝜒 = 1 2 (1 + √1 + 4𝛾) (24) Substituting Eq. 12 into Eq. 23 yields a complicated transcendental energy eigenvalue equation of the energy dependent screened Coulomb potential in D-dimensions as (Eq. 25): [(𝑛 + 𝜒) − 𝜇𝐴(1+𝑔𝐸𝑛ℓ) ℏ 2𝛼(𝑛+𝜒) ] 2 + 2𝜇𝐸𝑛ℓ ℏ 2𝛼2 − 𝛾 3 = 0 (25) https://doi.org/10.26850/1678-4618eqj.v45.4.2020.p40-56 Original article 45 Eclética Química Journal, vol. 45, n. 4, 2020, 40-56 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v45.4.2020.p40-56 Eq. 25 can also be expressed by Eq. 26. 𝐸𝑛ℓ = −ℏ2𝛼2 2𝜇 [((𝑛 + 𝜒) − 𝜇𝐴(1+𝑔𝐸𝑛ℓ) ℏ 2𝛼(𝑛+𝜒) ) 2 − 𝛾 3 ] (26) where 𝛾 and 𝜒 are given in Eqs. 9 and 24, respectively. To obtain the special case, we first rewrite Eq. 26 to the form of Eq. 27: 𝐸𝑛ℓ = −ℏ2 2𝜇 [(𝛼(𝑛 + 𝜒) − 𝜇𝐴(1+𝑔𝐸𝑛ℓ) ℏ 2(𝑛+𝜒) ) 2 − 𝛼2𝛾 3 ] (27) As 𝛼 → 0𝑎𝑛𝑑𝑔 → 0, Eq. 2 reduces to the standard Coulomb potential of the form of Eq. 28. 𝑉(𝑟) = − 𝐴 𝑟 (28) Setting the parameters 𝐷 = 3 as 𝛼 → 0𝑎𝑛𝑑𝑔 → 0, we obtain the energy eigenvalue equation (Eq. 29): 𝐸𝑛ℓ = − 𝜇𝐴2 2ℏ2(𝑛+ℓ+1)2 (29) This result is very consistent with the result obtained in Eq. 101 of Birkdemir et al.41. Also, taking the natural units (ℏ2 = 𝜇 = 1) and setting𝐷 = 3𝑎𝑛𝑑𝑔 = 0, the energy eigenvalue expression of Eq. 26 can be reduced to Eq. 30. 𝐸𝑛ℓ = −𝛼2 2 [((𝑛 + ℓ + 1) − 𝐴 𝛼(𝑛+ℓ+1) ) 2 − ℓ(ℓ+1) 3 ] (30) The result of Eq. 30 is consistent with the result obtained in Eq. 18 of Dong et al.23. To obtain the corresponding wave functions, we substitute 𝜋(𝑧)−and𝜎(𝑧) from Eqs. 18 and 13, respectively into Eq. A4 (see Appendix) and solve the first-order differential equation. This gives Eq. 31. 𝛷(𝑧) = 𝑧√𝜀 2 (1 − 𝑧) 1 2 +√ 1 4 +𝛾 (31) The weight function 𝜌(𝑧) from Eq. A6 (see Appendix) can be obtained Eq. 32 𝜌(𝑧) = 𝑧2 √𝜀2 (1 − 𝑧)2 √ 1 4 +𝛾 (32) From the Rodrigues relation of Eq. A5 (see Appendix), we obtain Eq. 33 and Eq. 34. 𝑦𝑛(𝑧) = 𝐵𝑛𝑧 −2√𝜀2(1 − 𝑧)−2 √ 1 4 +𝛾 𝑑𝑛 𝑑𝑧𝑛 [𝑧𝑛+2√𝜀 2 (1 − 𝑧)𝑛+2 √ 1 4 +𝛾 ] (33) 𝑦𝑛(𝑧) ≡ 𝐵𝑛𝑃𝑛 (2√𝜀2,2√ 1 4 +𝛾) (1 − 2𝑧) (34) where 𝑃𝑛 (𝜃,𝜗) is the Jacobi Polynomial. Substituting 𝛷(𝑧)and𝑦𝑛(𝑧) from Eqs. 31 and 34, respectively into Eq. A2 (see Appendix), we obtain Eq. 35: https://doi.org/10.26850/1678-4618eqj.v45.4.2020.p40-56 Original article 46 Eclética Química Journal, vol. 45, n. 4, 2020, 40-56 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v45.4.2020.p40-56 𝜓(𝑧) = 𝐵𝑛𝑧 √𝜀2(1 − 𝑧)𝐺𝑃𝑛 (2√𝜀2,2𝐺−1) (1 − 2𝑧) (35) where G is given by Eq. 36. 𝐺 = 1 2 +√ 1 4 + 𝛾 (36) From the definition of the Jacobi Polynomials42 results Eq. 37. ( ) ( ) ( ) , 2 1 1 1 ( ) , 1, 1; ! 1 2 n n P F n n n           + + −  = − + + + +   +   (37) In terms of hypergeometric Polynomials, Eq. 35 can be written as Eq. 38. ( ) ( ) ( ) 2 2 2 2 2 1 2 2 1 ( ) (1 ) , 2 2 , 2 1; . ! 2 1 G n n z B z z F n G n z n        + + = − − + + +  + (38) 3. Results and Discussion In this study, the energy eigenvalues for the energy-dependent screened Coulomb potential as they vary with the screening parameter were computed for different quantum states as shown in Tab. 1. The existence of the energy slope parameter in Eq. 26 results in two different energy spectra for a particular quantum state and a specific screening parameter. The screening parameter varies in an inverse version with the duo energy spectra. The energy eigenvalues for the energy-dependent screened Coulomb potential as a function of screening parameters in higher dimensions were also computed, as shown in Tabs. 2 and 3, respectively. By employing Eq. 30, we have also computed the energy eigenvalues in three dimensions and in the absence of the energy slope parameter, as shown in Tab. 4. For the two different potential depths considered, we obtained the bound state energy eigenvalues for different quantum states of 𝟐𝒑-𝟒𝒇, as they vary with different screening parameters. It can be seen that the energy eigenvalues for the different potential depths decrease as the screening parameter increases, at each quantum state. Our analytical result of Eq. 30 and its corresponding numerical results of Tab. 4 are very consistent with the results obtained by Dong et al.23. We have also computed the energy eigenvalues of Eq. 26 for higher dimensions, as shown in Tabs. 5 and 6, respectively. The level of effects imposed by the potential parameters of Eq. 2 on the energy eigenvalues of Eq. 26 are shown in Figs. 2-7. For different dimensions, the energy slope parameter causes an interwoven interaction of the curves, as compared to the situation when the energy slope parameter is zero. Figs. 8 and 9 show the variation of energy eigenvalues for the energy-dependent screened Coulomb potential with the different quantum numbers, for different values energy slope parameter considered. The trend of the relationship shows that the energy eigenvalues decrease as the quantum numbers increase. In Fig. 10, we also plotted the variation of the energy eigenvalues with the energy slope parameter for different dimensions. Table 1. Eigenvalues (𝐸𝑛ℓ)for the energy-dependent screened Coulomb potential as a function of the parameter 𝛼 for 𝟐𝒑-𝟒𝒇 states, with 𝐷 = 3. States 𝜶 𝑬𝒏𝓵(𝒈 = −𝟏, 𝑨 = 𝟐) 𝑬𝒏𝓵(𝒈 = 𝟏, 𝑨 = 𝟐) 2𝑝 0.025 −0.05000000000, − 0.9484636700 −0.2469390303, − 3.653060970 0.050 −0.1000000000,− 0.8934950102 −0.2261820091, − 3.573817991 0.075 −0.1499999989,− 0.8344159637 −0.2056916349, − 3.494308363 Continue... https://doi.org/10.26850/1678-4618eqj.v45.4.2020.p40-56 Original article 47 Eclética Química Journal, vol. 45, n. 4, 2020, 40-56 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v45.4.2020.p40-56 3𝑝 0.025 −0.3282430205, − 2.396756978 −0.1279838429, − 6.147016156 0.050 −0.2185168124, − 2.731483190 −0.1003205303, − 5.949679472 0.075 −0.1419248550, − 3.033075144 −0.07486233817, − 5.750137661 3𝑑 0.025 −0.3273369696, − 2.397663029 −0.1276723472, − 6.147327652 0.050 −0.2155358280, − 2.734464174 −0.09903861948, − 5.950961383 0.075 −0.1360998141, − 3.038900185 −0.07189047036, − 5.753109529 4𝑝 0.025 −0.1013443775, − 6.298655622 −0.06696010299, − 9.533039897 0.050 −0.05236401820, − 6.747635982 −0.03856747648, − 9.161432524 0.075 −0.02019553609, − 7.179804460 −0.01650824116, − 8.783491756 4𝑑 0.025 −0.1008065565, − 6.299193443 −0.06660798158, − 9.533392018 0.050 −0.05037315511, − 6.749626845 −0.03710618138, − 9.162893819 0.075 −0.01600781253, − 7.183992184 −0.01308764621, − 8.786912351 4𝑓 0.025 −0.1000000000, − 6.300000000 −0.06607984858, − 9.533920151 0.050 −0.04738907715, − 6.752610923 −0.03491511579, − 9.165084884 0.075 −0.009735385820, − 7.190264611 −0.007961748818, − 8.792038248 Table 2. Eigenvalues (𝐸𝑛ℓ)for the energy-dependent screened Coulomb potential as a function of the parameter 𝛼 for 𝟐𝒑-𝟒𝒇 states, with 𝐷 = 4. States 𝜶 𝑬𝒏𝓵(𝒈 = −𝟏, 𝑨 = 𝟐) 𝑬𝒏𝓵(𝒈 = 𝟏, 𝑨 = 𝟐) 2𝑝 0.025 −0.6406250000, − 0.6619912551 −0.1771068474, − 4.791643153 0.050 −0.7187500000, − 0.4363825014 −0.1516973143, − 4.660802686 0.075 −0.5523050676, − 1.041444930 −0.1269955847, − 4.529254413 3𝑝 0.025 −0.1676529658, − 4.263597033 −0.09251657939, − 7.726233419 0.050 −0.1017568786, − 4.635743120 −0.06332496524, − 7.449175033 0.075 −0.05426252476, − 4.989487474 −0.03776841503, − 7.168481584 3𝑑 0.025 −0.1668742692, − 4.264375729 −0.09209870569, − 7.726651293 0.050 −0.09894423101, − 4.638555767 −0.06159768496, − 7.450902314 0.075 −0.04845181230, − 4.995298186 −0.03374430925, − 7.172505689 4𝑝 0.025 −0.06465429287, − 8.566595705 −0.04786732380, − 11.57088267 0.050 −0.02501719398, − 9.112482806 −0.02055263540, − 11.09194736 0.075 −0.002313419520, − 9.641436577 −0.002103392985, − 10.60414660 4𝑑 0.025 −0.06403407537, − 8.567215922 −0.04740969809, − 11.57134030 0.050 −0.02269659485, − 9.114803405 −0.01864771539, − 11.09385228 0.075 0.002607849543, − 9.646357846 0.002371302471, − 10.60862130 4𝑓 0.025 −0.06316592284, − 8.568084075 −0.04676908315, − 11.57198091 0.050 −0.01944974474, − 9.118050255 −0.01598192774, − 11.09651807 0.075 −0.009489204657, − 9.653239201 −0.008629543278, − 10.61487954 Table 3. Eigenvalues (𝐸𝑛ℓ)for the energy-dependent screened Coulomb potential as a function of the parameter 𝛼 for 𝟐𝒑-𝟒𝒇 states, with 𝐷 = 5. States 𝜶 𝑬𝒏𝓵(𝒈 = −𝟏, 𝑨 = 𝟐) 𝑬𝒏𝓵(𝒈 = 𝟏, 𝑨 = 𝟐) 2𝑝 0.025 −0.3273369696, − 2.397663029 −0.1276723472, − 6.147327652 0.050 −0.2155358280, − 2.734464174 −0.09903861948, − 5.950961383 0.075 −0.1360998141, − 3.038900185 −0.07189047036, − 5.753109529 3𝑝 0.025 −0.1008065565, − 6.299193443 −0.06660798158, − 9.533392018 0.050 −0.05037315511, − 6.749626845 −0.03710618138, − 9.162893819 0.075 −0.01600781253, − 7.183992184 −0.01308764621, − 8.786912351 3𝑑 0.025 −0.1000000000, − 6.300000000 −0.06607984858, − 9.533920151 0.050 −0.04738907715, − 6.752610923 −0.03491511579, − 9.165084884 0.075 −0.009735385820, − 7.190264611 −0.007961748818, − 8.792038248 4𝑝 0.025 −0.4194183076, − 11.08305817 −0.03358353903, − 13.84141646 0.050 −0.009315896675, − 11.74068410 −0.008259866067, − 13.24174013 0.075 0.005363836706, − 12.38036383 0.005257711402, − 12.63025771 4𝑑 0.025 −0.04123429366, − 11.08376571 −0.03301776018, − 13.84198224 0.050 −0.006652702848, − 11.74334730 −0.005898852562, − 13.24410115 0.075 0.01103813411, − 12.38603813 0.01081993497, − 12.63581993 4𝑓 0.025 −0.04029105188, − 11.08470895 −0.03226346048, − 13.84273654 0.050 −0.003103656681, − 11.74689634 −0.002752143970, − 13.24724786 0.075 0.01859579365, − 12.39359579 0.01822863105, − 12.64322863 https://doi.org/10.26850/1678-4618eqj.v45.4.2020.p40-56 Original article 48 Eclética Química Journal, vol. 45, n. 4, 2020, 40-56 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v45.4.2020.p40-56 Table 4. Eigenvalues (𝐸𝑛ℓ)of Eq. 29 as a function of the parameter 𝛼 for 𝟐𝒑-𝟒𝒇 states, with 𝐷 = 3𝑎𝑛𝑑𝑔 = 0. States 𝜶 𝑬𝒏𝓵(𝑨 = 𝟐. 𝟎) 𝑬𝒏𝓵(𝑨 = 𝟓. 𝟎) 2𝑝 0.025 −0.4510416666 −3.001041666 0.050 −0.4041666666 −2.879166666 0.075 −0.3593750004 −2.759375001 3𝑝 0.025 −0.1748263890 −1.266493056 0.050 −0.1326388888 −1.149305555 0.075 −0.09565972225 −1.037326388 3𝑑 0.025 −0.1744097223 −1.266076389 0.050 −0.1309722221 −1.147638888 0.075 −0.09190972225 −1.033576388 4𝑝 0.025 −0.07979166665 −0.6610416665 0.050 −0.04416666666 −0.5504166665 0.075 −0.01812500002 −0.4493750000 4𝑑 0.025 −0.07937500000 −0.6606250000 0.050 −0.04250000000 −0.5487500000 0.075 −0.01437500002 −0.4456250002 4𝑓 0.025 −0.07875000000 −0.6600000000 0.050 −0.04000000000 −0.5462500000 0.075 −0.008750000020 −0.4400000002 Table 5. Eigenvalues (𝐸𝑛ℓ)of Eq. 25 as a function of the parameter 𝛼 for 𝟐𝒑-𝟒𝒇 states, with 𝐷 = 4𝑎𝑛𝑑𝑔 = 0. States 𝜶 𝑬𝒏𝓵(𝑨 = 𝟐. 𝟎) 𝑬𝒏𝓵(𝑨 = 𝟓. 𝟎) 2𝑝 0.025 −0.2715625000 −1.876562500 0.050 −0.2262500000 −1.756250000 0.075 −0.1840625002 −1.639062500 3𝑝 0.025 −0.1167028062 −0.8988456630 0.050 −0.07701530615 −0.7841581630 0.075 −0.04420280616 −0.6763456635 3𝑑 0.025 −0.1161819728 −0.8983248295 0.050 −0.07493197280 −0.7820748300 0.075 −0.03951530614 −0.6716581635 4𝑝 0.025 −0.05470293210 −0.4982214506 0.050 −0.02251543210 −0.3910339505 0.075 −0.002202932106 −0.2957214510 4𝑑 0.025 −0.05418209880 −0.4977006172 0.050 −0.02043209876 −0.3889506171 0.075 0.002484567894 −0.2910339509 4𝑓 0.025 −0.05345293210 −0.4969714506 0.050 −0.01751543210 −0.3860339505 0.075 0.009047067895 −0.2844714510 Table 6. Eigenvalues (𝐸𝑛ℓ)of Eq. 26 as a function of the parameter 𝛼 for 𝟐𝒑-𝟒𝒇 states, with 𝐷 = 5and𝑔 = 0. States 𝜶 𝑬𝒏𝓵(𝑨 = 𝟐. 𝟎) 𝑬𝒏𝓵(𝑨 = 𝟓. 𝟎) 2𝑝 0.025 −0.1744097223 −1.266076389 0.050 −0.1309722221 −1.147638888 0.075 −0.09190972225 −1.033576388 3𝑝 0.025 −0.07937500000 −0.6606250000 0.050 −0.04250000000 −0.5487500000 0.075 −0.01437500002 −0.4456250002 3𝑑 0.025 −0.07875000000 −0.6600000000 0.050 −0.04000000000 −0.5462500000 0.075 −0.008750000020 −0.4400000002 4𝑝 0.025 −0.03718750000 −0.3821875000 0.050 −0.008750000000 −0.2787500000 0.075 0.005312500000 −0.1896874998 4𝑑 0.025 −0.03656250000 −0.3815625000 0.050 −0.006250000000 −0.2762500000 0.075 0.01093750000 −0.1840624998 Continue... https://doi.org/10.26850/1678-4618eqj.v45.4.2020.p40-56 Original article 49 Eclética Química Journal, vol. 45, n. 4, 2020, 40-56 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v45.4.2020.p40-56 4𝑓 0.025 −0.03572916666 −0.3807291666 0.050 −0.002916666666 −0.2729166666 0.075 0.01843750000 −0.1765624998 Figure 2. Energy Eigenvalue Variation with 𝛼for different dimensions with 𝑔 = 0, 𝐴 = 5, 𝑛 = 2, ℓ = 1. Figure 3. Energy Eigenvalue variation with 𝛼for different dimensions with 𝑔 = −1, 𝐴 = 5, 𝑛 = 2, ℓ = 1. https://doi.org/10.26850/1678-4618eqj.v45.4.2020.p40-56 Original article 50 Eclética Química Journal, vol. 45, n. 4, 2020, 40-56 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v45.4.2020.p40-56 Figure 4. Energy Eigenvalue Variation with 𝛼for different dimensions with 𝑔 = 1, 𝐴 = 5, 𝑛 = 2, ℓ = 1. Figure 5. Energy Eigenvalue Variation with 𝐴for different dimensions with 𝑔 = 0, 𝛼 = 100, 𝑛 = 2, ℓ = 1. https://doi.org/10.26850/1678-4618eqj.v45.4.2020.p40-56 Original article 51 Eclética Química Journal, vol. 45, n. 4, 2020, 40-56 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v45.4.2020.p40-56 Figure 6. Energy Eigenvalue Variation with 𝐴for different dimensions with 𝑔 = −1, 𝛼 = 100, 𝑛 = 2, ℓ = 1. Figure 7. Energy Eigenvalue Variation with 𝐴for different dimensions with 𝑔 = 1, 𝛼 = 100, 𝑛 = 2, ℓ = 1. https://doi.org/10.26850/1678-4618eqj.v45.4.2020.p40-56 Original article 52 Eclética Química Journal, vol. 45, n. 4, 2020, 40-56 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v45.4.2020.p40-56 Figure 8. Energy Eigenvalue Variation with ℓ for different values of 𝑔 with 𝐷 = 3, 𝛼 = 1, 𝐴 = 5, 𝑛 = 0. Figure 9. Energy Eigenvalue Variation with 𝑛 for different values of 𝑔 with 𝐷 = 3, 𝛼 = 1, 𝐴 = 5, ℓ = 1. https://doi.org/10.26850/1678-4618eqj.v45.4.2020.p40-56 Original article 53 Eclética Química Journal, vol. 45, n. 4, 2020, 40-56 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v45.4.2020.p40-56 Figure 10. Energy Eigenvalue Variation with 𝑔 for different dimensions, with 𝛼 = 10, 𝐴 = 100, 𝑛 = 2, ℓ = 1 4. Conclusions We have solved the Schrödinger equation with the energy-dependent screened Coulomb potential in higher dimensions. With the use of the conventional Nikiforov-Uvarov method and a new form of Greene-Aldrich approximation, the energy eigenvalues and its corresponding eigenfunctions were obtained approximately. Numerical values of the energy eigenvalues were also obtained with natural units, in three dimensions. We have also elucidated the variation of these energy spectra with different potential parameters and the energy slope parameter. The effect of the energy slope parameter is clearly seen in the existence of duo energy spectra, as compared to when the energy slope parameter is diminished. Special cases have been deduced, and these results agree perfectly with available literature. 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A1. 𝜓 ′′ �̃�(𝑧) 𝜎(𝑧) ′ �̃�(𝑧) 𝜎2(𝑧) (A1) Here, �̃�(𝑧), 𝜎(𝑧) are polynomials of at most second degree, and �̃�(𝑧) is a first-degree polynomial. By employing the transformation (Eq. A2), 𝜓(𝑧) = 𝛷(𝑧)𝑦𝑛(𝑧) (A2) we obtain the exact solution of Eq. A1 in a form of hypergeometric-type equation given by Eq. A3. ( ) ( ) ( ) ( ) ( ) 0n n nz y z z y z y z   + + = (A3) Let us define the logarithm derivative function 𝛷(𝑧) as40 (Eq. A4). 𝛷′(𝑧) 𝛷(𝑧) = 𝜋(𝑧) 𝜎(𝑧) (A4) where 𝜋(𝑧) is at most a first-degree polynomial. The second part of 𝜓(𝑧) being 𝑦𝑛(𝑧) in Eq. A2, is the hypergeometric function with its polynomial solution given by Rodrigues relation (Eq. A5) 𝑦𝑛(𝑧) = 𝐵𝑛 𝜌(𝑧) 𝑑𝑛 𝑑𝑧𝑛 [𝜎𝑛(𝑧)𝜌(𝑧)] (A5) Let us mention here that 𝐵𝑛 is the normalization constant and 𝜌(𝑧) is the weight function which must satisfy the condition expressed by Eq. A6. 𝑑 𝑑𝑧 [𝜎(𝑧)𝜌(𝑧)] = 𝜏(𝑧)𝜌(𝑧) (A6) with Eq. A7 𝜏(𝑧) = �̃�(𝑧) + 2𝜋(𝑧) (A7) The eigenfunctions and eigenvalues can be obtained using the definition of the following function 𝜋(𝑧) and parameter 𝜆as shown (Eq. A8): 𝜋(𝑧) = 𝜎′(𝑧)−�̃�(𝑧) 2 ±√( 𝜎′(𝑧)−�̃�(𝑧) 2 ) 2 − �̃�(𝑧) + 𝑘𝜎(𝑧) (A8) and (Eq. A9) 𝜆 = 𝑘 + 𝜋 ′(𝑧) (A9) To obtain the value of 𝑘, we set the discriminant of the square root in Eq. A8 equal to zero. As such, the new eigenvalue equation (Eq. A10) is obtained as ( 1) ( ) ( ) 0, ( 0,1,2,...) 2 n n n z z n   −  + + = = (A10) https://doi.org/10.26850/1678-4618eqj.v45.4.2020.p40-56