Original article iq.unesp.br/ecletica | Vol. 46 | n. 4 | 2021 | 60 Eclética Química Journal, vol. 46, n. 4, 2021, 60-73 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v46.4.2021.p60-73 Effect of the deformation parameter on the nonrelativistic energy spectra of the q-deformed Hulthen-quadratic exponential-type potential Ushie Patrick Obogo1 , Ofem Egbe Ubi1 , Collins Okon Edet2+ , Akpan Ndem Ikot2 1. Cross River University of Technology, Department of Physics, Faculty of Physical Sciences, Calabar, Nigeria. 2. University of Port Harcourt, Department of Physics, Rivers State, Nigeria. +Corresponding author: Collins Okon Edet, Phone: +2349016715883, Email address: collinsokonedet@gmail.com ARTICLE INFO Article history: Received: October 20, 2020 Accepted: May 27, 2021 Published: October 01, 2021 Section Editor: Assis Vicente Benedetti Keywords 1. Schrödinger equation 2. potential model 3. bound state 4. parametric Nikiforov–Uvarov (NU) method ABSTRACT: In this study, an approximate solution of the Schrödinger equation for the q- deformed Hulthen-quadratic exponential-type potential model within the framework of the Nikiforov–Uvarov method was obtained. The bound state energy equation and the corresponding eigenfunction was obtained. The energy spectrum is applied to study H2, HCl, CO and LiH diatomic molecules. The effect of the deformation parameters and other potential parameters on the energy spectra of the system were graphically and numerically analyzed in detail. Special cases were considered when the potential parameters were altered, resulting in deformed Hulthen potential, Hulthen potential, deformed quadratic exponential-type potential and quadratic exponential-type potential. The energy eigenvalues expressions agreed with what obtained in literature. Finally, the results can find many applications in quantum chemistry, atomic and molecular physics. http://revista.iq.unesp.br/ojs/index.php/ecletica/index https://doi.org/10.26850/1678-4618eqj.v46.4.2021.p60-73 mailto:collinsokonedet@gmail.com http://orcid.org/0000-0003-4972-8361 http://orcid.org/0000-0002-9185-4241 http://orcid.org/0000-0001-7762-731X http://orcid.org/0000-0002-1078-262X Original article iq.unesp.br/ecletica 61 Eclética Química Journal, vol. 46, n. 4, 2021, 60-73 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v46.4.2021.p60-73 1. Introduction Since the early days of quantum mechanics (QM), the study of a particle confined by a potential field has been of utmost importance (Edet and Okoi, 2019; Edet et al., 2020a; b; Landau, 1977; Schiff, 1995). Studies of this nature are carried out by solving the Schrödinger equation with different interaction potentials of interest. The solutions of the Schrödinger equation with different potential models of interest have been employed by many researchers to give insights, explanations, and predictions into the behavior of diatomic molecules, quarks, etc (Edet et al., 2020c; 2021a; Greiner, 2000; Okoi et al., 2020; Okorie et al., 2019). Recently, numerous researchers have proffered solutions to the Schrödinger equation for some potential of interest. The analytical solution of the Schrödinger equation with ℓ = 0 and ℓ ≠ 0 for some potentials has been addressed by many researchers in non-relativistic and relativistic quantum mechanics for bound states (Durmus and Yasuk, 2007; Edet et al., 2020d; 2021b; Louis et al., 2018a; 2018b). Some of these potentials include Deng-Fan potential (Falaye et al., 2015), Hyperbolic potential (Onate et al., 2018b), Eckart potential (Onate et al., 2017), generalized trigonometric Pöschl–Teller potential (Edet et al., 2020e), and screened Kratzer Potential (Ikot et al., 2020a). Also, several methods have been employed to obtain the solutions of the nonrelativistic wave equations with some potential models of interest, some of these methods include: the factorization method (Dong, 2007), formula method (Falaye et al., 2015), supersymmetry quantum mechanics (SUSYQM) (Falaye et al., 2014), Nikiforov-Uvarov method (NU) (Nikiforov and Uvarov, 1988) asymptotic iteration method (AIM) (Ciftci et al., 2003; 2005; Falaye, 2012), exact quantization Rule (Gu and Dong, 2011; Ma and Xu, 2005), proper quantization rule (Qiang and Dong, 2007), WKBJ (Ita et al., 2018), etc. Moreover, it is the goal in the present consideration to propose a potential of the form: 𝑉𝑞(𝑟) = 𝑉0𝑒−𝛼𝑟 1−𝑞𝑒−𝛼𝑟 + 𝑉1(𝑎+𝑏𝑒−𝛼𝑟+𝑐𝑒−2𝛼𝑟) (1−𝑞𝑒−𝛼𝑟)2 (1) Where V0 and V1 are the potential strengths, α is the screening parameter, a, b and c are the adjusted parameter and q is deformation parameter. We call the potential q-deformed Hulthen-quadratic exponential- type potential (q-HQEP). With respect to what was obtained in previous studies of the molecular potential, the potential to allow for more physical application and a comparative analysis to existing studies of the molecular potential was modified. In addition, in molecular physics, it has also been established that potential energy functions with more parameters tend to fit experimental data than those with fewer parameters and researchers have recently paid great attention to obtaining modified version of potential functions by employing dissociation energy, and equilibrium bond length for molecular systems as explicit parameters. This model will be an important tool for spectroscopists to represent experimental data, verify measurements, and make predictions. The potential is a superposition of the Hulthen (Ikhdair, 2009; Ikhdair and Sever, 2007; Onate et al., 2018a) and Quadratic exponential-type potentials (Okorie et al., 2018). These potentials have been individually applied to carry out studies extensively by several researchers (Ikot et al., 2014). Hence, the motivation to combine them. In this research article, the goal is in two-fold. First, the Schrödinger wave equation (SWE) is solved with the 𝑞-HQEP via parametric Nikiforov–Uvarov method. The effect of the deformation parameter on the energy spectra of some diatomic molecules is analyzed with the aid of some graphical representation and numerical analysis. The outline of the paper is as follows: section 2 provides brief a description of the parametric Nikiforov–Uvarov method. In section 3, the solutions of the three-dimensional (3D) Schrödinger equation (SE) with the q-HQEP via parametric NU. In Section 4, special cases of the potential understudy were discussed. In section 5, the results of this study are presented and discussed. Finally, in section 6, the concluding remarks are given. 2. The parametric NU method The parametric form of the NU method takes the form (Tezcan and Sever, 2009): 𝑑2𝜓 𝑑𝑠2 + 𝛼1−𝛼2𝑠 𝑠(1−𝛼3𝑠) 𝑑𝜓 𝑑𝑠 + 1 𝑠2(1−𝛼3𝑠)2 {−𝜉1𝑠 2 + 𝜉2𝑠 − 𝜉3}𝜓(𝑠) = 0 (2) http://revista.iq.unesp.br/ojs/index.php/ecletica/index https://doi.org/10.26850/1678-4618eqj.v46.4.2021.p60-73 Original article iq.unesp.br/ecletica 62 Eclética Química Journal, vol. 46, n. 4, 2021, 60-73 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v46.4.2021.p60-73 The energy eigenvalues equation and eigenfunctions satisfy the following sets of equations, respectively: 𝛼2𝑛 − (2𝑛 + 1)𝛼5 + (2𝑛 + 1)(√𝛼9 + 𝛼3√𝛼8) + 𝑛(𝑛 − 1)𝛼3 + 𝛼7 + 2𝛼3𝛼8 + 2√𝛼8𝛼9 = 0 (3) 𝜓(𝑠) = 𝑠𝛼12(1 − 𝛼3𝑠) −𝛼12− 𝛼13 𝛼3 𝑃𝑛 (𝛼10−1, 𝛼11 𝛼3 −𝛼10−1) (1 − 2𝛼3𝑠) (4) where 𝛼4 = 1 2 (1 − 𝛼1), 𝛼5 = 1 2 (𝛼2 − 2𝛼3), 𝛼6 = 𝛼5 2 + 𝜉1 𝛼7 = 2𝛼4𝛼5 − 𝜉2, 𝛼8 = 𝛼4 2 + 𝜉3, 𝛼9 = 𝛼3𝛼7 + 𝛼3 2𝛼8 + 𝛼6 (5) 𝛼10 = 𝛼1 + 2𝛼4 + 2√𝛼8, 𝛼11 = 𝛼2 − 2𝛼5 + 2(√𝛼9 + 𝛼3√𝛼8) 𝛼12 = 𝛼4 + √𝛼8, 𝛼13 = 𝛼5 − (√𝛼9 + 𝛼3√𝛼8) and Pn is the orthogonal Jacobi polynomial, which is defined as 𝛲𝑛 (𝛼,𝛽) (𝑥) = 𝛤(𝛼+𝑛+1) 𝑛!𝛤(𝛼+𝛽+𝑛+1) ∑ ( 𝑛 𝑚 )𝑛 𝑚=0 𝛤(𝛼+𝛽+𝑛+𝑚+1) 𝛤(𝛼+𝑚+1) ( 𝑥−1 2 )𝑚 (6) 3. Bound-state solutions of q-deformed Hulthen plus quadratic exponential-type potential The radial Schrödinger equation in arbitrary dimensions (Rampho et al., 2021; Ebomwonyi et al., 2017) can be given as: 𝑑2𝑅𝑛ℓ(𝑟) 𝑑𝑟2 + 2𝜇 ℏ2 [𝐸𝑛ℓ − 𝑉(𝑟) − ℓ(ℓ+1)ℏ2 2𝜇𝑟2 ] 𝑅𝑛ℓ(𝑟) = 0 (7) where 𝜇 is the reduced mass, 𝐸𝑛ℓ is the energy spectrum, ℏ is the reduced Planck’s constant and 𝑛 and ℓ are the principal and orbital angular momentum quantum numbers, respectively (or vibration-rotation quantum numbers in quantum chemistry) (Onate et al., 2018c). Substituting Eq. 1 into Eq. 7 gives: 𝑑2𝑅𝑛ℓ(𝑟) 𝑑𝑟2 + [ 2𝜇𝐸𝑛ℓ ℏ2 − 2𝜇 ℏ2 (− 𝑉0𝑒−𝛼𝑟 1−𝑞𝑒−𝛼𝑟 + 𝑉1(𝑎+𝑏𝑒−𝛼𝑟+𝑐𝑒−2𝛼𝑟) (1−𝑞𝑒−𝛼𝑟)2 ) − ℓ(ℓ+1) 𝑟2 ]𝑅𝑛ℓ(𝑟) = 0 (8) The radial Schrödinger equation with this potential can be solved exactly for ℓ = 0 (s-wave) but cannot be solved with this potential for ℓ ≠ 0. To obtain the solution for ℓ ≠ 0, the approximation scheme proposed by Greene and Aldrich (1976) is employed to deal with the centrifugal term, which is given as: 1 𝑟2 ≈ 𝛼2𝑒−𝛼𝑟 (1−𝑞𝑒−𝛼𝑟)2 (9) It is noted that for a short-range potential, the relation in Eq. 9 is a good approximation to 1 𝑟2, as proposed by Greene and Aldrich (1976), Ikot et al. (2020b) and Okorie et al. (2020). This implies that Eq. 9 is not a good approximation to the centrifugal barrier when the screening parameter α becomes large. Thus, the approximation is valid when 𝛼 ≪ 1. Substituting the approximation Eq. 8 into Eq. 9, an equation of the form is obtained: 𝑑2𝑅𝑛ℓ(𝑟) 𝑑𝑟2 + [ 2𝜇𝐸𝑛ℓ ℏ2 − 2𝜇 ℏ2 ( 𝑉0𝑒−𝛼𝑟 1−𝑞𝑒−𝛼𝑟 + 𝑉1(𝑎+𝑏𝑒−𝛼𝑟+𝑐𝑒−2𝛼𝑟) (1−𝑞𝑒−𝛼𝑟)2 ) − ℓ(ℓ+1)𝛼2𝑒−2𝛼𝑟 (1−𝑞𝑒−𝛼𝑟)2 ] 𝑅𝑛ℓ(𝑟) = 0 (10) To solve the differential equation above, the transformation 𝑠 = 𝑒−𝛼𝑟 is used so as to enable to apply the NU method as a solution technique to the hypergeometric-type differential equation. Hence, this transforms 𝑑2𝑅𝑛ℓ(𝑟) 𝑑𝑟2 into the form: http://revista.iq.unesp.br/ojs/index.php/ecletica/index https://doi.org/10.26850/1678-4618eqj.v46.4.2021.p60-73 Original article iq.unesp.br/ecletica 63 Eclética Química Journal, vol. 46, n. 4, 2021, 60-73 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v46.4.2021.p60-73 𝑑2𝑅𝑛ℓ(𝑟) 𝑑𝑟2 = 𝛼2𝑠2 𝑑2𝑅𝑛ℓ(𝑠) 𝑑𝑠2 + 𝛼2𝑠 𝑑𝑅𝑛ℓ(𝑠) 𝑑𝑠 (11) 𝛼2𝑠2 𝑑2𝑅𝑛ℓ(𝑠) 𝑑𝑠2 + 𝛼2𝑠 𝑑𝑅𝑛ℓ(𝑠) 𝑑𝑠 + [ 2𝜇𝐸𝑛ℓ ℏ2 − 2𝜇 ℏ2 ( 𝑉0𝑠 1−𝑞𝑠 + 𝑉1(𝑎+𝑏𝑠+𝑐𝑠2) (1−𝑞𝑠)2 ) − ℓ(ℓ+1)𝛼2𝑠 (1−𝑞𝑠)2 ] 𝑅𝑛ℓ(𝑠) = 0 (12) In view of the above, the differential equation of the form is obtained: 𝑑2𝑅𝑛ℓ(𝑠) 𝑑𝑠2 + 1−𝑞𝑠 𝑠(1−𝑞𝑠) 𝑑𝑅𝑛ℓ(𝑠) 𝑑𝑠 + 1 𝑠2(1−𝑞𝑠)2 [ −(휀𝑛ℓ𝑞 2 + 𝛽𝑞 − 𝛿3)𝑠 2 + (2휀𝑛ℓ𝑞 + 𝛽 − 𝛿3 − 𝛾)𝑠 − (휀𝑛ℓ − 𝛿3) ] 𝑅𝑛ℓ(𝑠) = 0 (13) where the following dimensionless abbreviations have been introduced for mathematical convenience: −휀𝑛ℓ = 2𝜇𝐸𝑛ℓ ℏ2𝛼2 ,𝛽 = 2𝜇𝑉0 ℏ2𝛼2,𝛿1 = 2𝜇𝑉1𝑎 ℏ2𝛼2 ,𝛿2 = 2𝜇𝑉1𝑏 ℏ2𝛼2 ,𝛿3 = 2𝜇𝑉1𝑐 ℏ2𝛼2 𝛾 = ℓ(ℓ + 1) (14) Equation 13 is of the form that is solvable by the NU method. Therefore, on comparison to Eqs. 2 and 13, the following parameters are obtained: 𝜉1 = 휀𝑛ℓ𝑞 2 + 𝛽𝑞 − 𝛿3 𝜉2 = 2휀𝑛ℓ𝑞 + 𝛽 − 𝛿3 − 𝛾 (15a) 𝜉3 = 휀𝑛ℓ − 𝛿3 and 𝛼1 = 1, 𝛼2 = 𝑞, 𝛼3 = 𝑞 𝜉1 = 휀𝑛ℓ𝑞 2 + 𝛽𝑞 − 𝛿3, 𝜉2 = 2휀𝑛ℓ𝑞 + 𝛽 − 𝛿3 − 𝛾, 𝜉3 = 휀𝑛ℓ − 𝛿3 𝛼4 = 0,𝛼5 = − 𝑞 2 , 𝛼6 = 𝑞2 4 + 휀𝑛ℓ𝑞 2 + 𝛽𝑞 − 𝛿3, 𝛼7 = −2휀𝑛ℓ𝑞 − 𝛽 + 𝛿3 + 𝛾 𝛼8 = 휀𝑛ℓ + 𝛿1, 𝛼9 = 𝑞2 4 − 𝛿2𝑞 + 𝛾𝑞 + 𝛿1𝑞 2 − 𝛿3 (15b) 𝛼10 = 1 + 2√휀𝑛ℓ − 𝛿1, 𝛼11 = 2𝑞 + 2(√ 𝑞2 4 − 𝛿2𝑞 + 𝛾𝑞 + 𝛿1𝑞 2 − 𝛿3 + 𝑞√휀𝑛ℓ + 𝛿1) 𝛼12 = √휀𝑛ℓ + 𝛿1, 𝛼13 = − 𝑞 2 − (√ 𝑞2 4 − 𝛿2𝑞 + 𝛾𝑞 + 𝛿1𝑞 2 − 𝛿3 + 𝑞√휀𝑛ℓ + 𝛿1) Substituting these polynomials into Eq. 3, it is possible to obtain 𝑞 (𝑛 + 1 2 ) 2 + 2(𝑛 + 1 2 ) (√ 𝑞2 4 − 𝛿2𝑞 + 𝛾𝑞 + 𝛿1𝑞 2 − 𝛿3 + 𝑞√휀𝑛ℓ + 𝛿1) − 𝛽 + 𝛿2 + 𝛾 + 2𝛿1𝑞 + 𝑞 4 + 2√(휀𝑛ℓ + 𝛿1) ( 𝑞2 4 − 𝛿2𝑞 + 𝛾𝑞 + 𝛿1𝑞 2 − 𝛿3) = 0 (16) By carrying out some algebraic manipulation, the following equation is obtained: 휀𝑛ℓ = −𝛿1 + 1 4 [ (𝑛+ 1 2 +√ 1 4 − 𝛿2 𝑞 + 𝛾 𝑞 +𝛿1− 𝛿3 𝑞2) 2 + 𝛿3 𝑞2+𝛿1− 𝛽 𝑞 (𝑛+ 1 2 +√ 1 4 − 𝛿2 𝑞 + 𝛾 𝑞 +𝛿1− 𝛿3 𝑞2) ] 2 (17) Substituting Eq. 14 into Eq. 17 and carrying some simple manipulative algebra, it is possible to arrive at the energy eigenvalue equation of the q-deformed Hulthen plus quadratic exponential-type potential in the form: http://revista.iq.unesp.br/ojs/index.php/ecletica/index https://doi.org/10.26850/1678-4618eqj.v46.4.2021.p60-73 Original article iq.unesp.br/ecletica 64 Eclética Química Journal, vol. 46, n. 4, 2021, 60-73 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v46.4.2021.p60-73 𝐸𝑛ℓ = 𝑉1𝑎 − ℏ2𝛼2 8𝜇 [ (𝑛+ 1 2 +√ 1 4 − 2𝜇𝑉1𝑏 ℏ2𝛼2𝑞 + ℓ(ℓ+1) 𝑞 + 2𝜇𝑉1𝑎 ℏ2𝛼2 − 2𝜇𝑉1𝑐 ℏ2𝛼2𝑞2) 2 + 2𝜇𝑉1𝑐 ℏ2𝛼2𝑞2+ 2𝜇𝑉1𝑎 ℏ2𝛼2 − 2𝜇𝑉0 ℏ2𝛼2𝑞 (𝑛+ 1 2 +√ 1 4 − 2𝜇𝑉1𝑏 ℏ2𝛼2𝑞 + ℓ(ℓ+1) 𝑞 + 2𝜇𝑉1𝑎 ℏ2𝛼2 − 2𝜇𝑉1𝑐 ℏ2𝛼2𝑞2) ] 2 (18) The corresponding wave functions can be evaluated from Eq. 4 as follows: 𝑅𝑛ℓ(𝑠) = 𝑁𝑛ℓ𝑠 √ 𝑛ℓ+𝛿1(1 − 𝑠) 1 2 +√ 1 4 − 𝛿2 𝑞 + 𝛾 𝑞 +𝛿1− 𝛿3 𝑞2 𝑃𝑛 (2√ 𝑛ℓ+𝛿1,2√ 1 4 − 𝛿2 𝑞 + 𝛾 𝑞 +𝛿1− 𝛿3 𝑞2) (1 − 2𝑠) (19) From the definition of the Jacobi polynomials (Edet et al., 2020a; b). 𝑃𝑛 (2𝜛,2𝜒) (1 − 2𝑠) = 𝛤(𝑛+2𝜛+1) 𝑛!𝛤(2𝜛+1) 𝐹2 1(−𝑛, 2𝜛 + 2𝜒 + 𝑛 + 1,2𝜛 + 1; 𝑠) (20) 𝜛 = √휀𝑛ℓ + 𝛿1 𝜒 = √ 1 4 − 𝛿2 𝑞 + 𝛾 𝑞 + 𝛿1 − 𝛿3 𝑞2 (21) In terms of hypergeometric polynomials, Eq. 21 can be written as 𝑅𝑛ℓ(𝑠) = 𝑁𝑛ℓ𝑠 𝜛(1 − 𝑠) 1 2 +𝜒 𝛤(𝑛+2𝜛+1) 𝑛!𝛤(2𝜛+1) 𝐹2 1(−𝑛, 2𝜛 + 2𝜒 + 𝑛 + 1,2𝜛 + 1; 𝑠) (22) 4. Special cases 4.1 Deformed Hulthen potential If 𝑉1 is set as 𝑉1 = 𝑎 = 𝑏 = 𝑐 = 0, the potential model (1) reduces to deformed Hulthen potential (Hall et al., 2018): 𝑉𝑞(𝑟) = 𝑉0𝑒−𝛼𝑟 1−𝑞𝑒−𝛼𝑟 (23) and the energy equation as follows: 𝐸𝑛ℓ = − ℏ2𝛼2 8𝜇 [ (𝑛+ 1 2 +√ 1 4 + ℓ(ℓ+1) 𝑞 ) 2 − 2𝜇𝑉0 ℏ2𝛼2𝑞 (𝑛+ 1 2 +√ 1 4 + ℓ(ℓ+1) 𝑞 ) ] 2 (24) This is in agreement with Eq. 44 (Edet and Okoi, 2019). 4.2 Hulthen potential If 𝑉1 is set as 𝑉1 = 𝑎 = 𝑏 = 𝑐 = 0 and 𝑞 → 1, the potential model (1) reduces to the Hulthen potential: http://revista.iq.unesp.br/ojs/index.php/ecletica/index https://doi.org/10.26850/1678-4618eqj.v46.4.2021.p60-73 Original article iq.unesp.br/ecletica 65 Eclética Química Journal, vol. 46, n. 4, 2021, 60-73 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v46.4.2021.p60-73 𝑉(𝑟) = 𝑉0𝑒−𝛼𝑟 1−𝑒−𝛼𝑟` (25) and the energy equation as follows: 𝐸𝑛ℓ = − ℏ 2𝛼2 8𝜇 [ (𝑛+ 1 2 +√ 1 4 +ℓ(ℓ+1)) 2 − 2𝜇𝑉0 ℏ2𝛼2 (𝑛+ 1 2 +√ 1 4 +ℓ(ℓ+1)) ] 2 (26) Equation 26 is in agreement with the energy equation given in Eq. 39 (Ikhdair, 2009), Eq. 31 (Ikhdair and Sever, 2007), Eq. 34 (Agboola, 2009), Eq. 35 (Bayrak et al., 2006) and Eq. 27 (Jia et al., 2008). 4.3 Deformed quadratic exponential-type potential If 𝑉0 is set as 𝑉0 = 0, the potential model (1) reduces to deformed quadratic exponential-type potential: 𝑉𝑞(𝑟) = 𝑉1(𝑎+𝑏𝑒−𝛼𝑟+𝑐𝑒−2𝛼𝑟) (1−𝑞𝑒−𝛼𝑟)2 (27) and the energy equation as follows: 𝐸𝑛ℓ = 𝑉1𝑎 − ℏ2𝛼2 8𝜇 [ (𝑛+ 1 2 +√ 1 4 − 2𝜇𝑉1𝑏 ℏ2𝛼2𝑞 + ℓ(ℓ+1) 𝑞 + 2𝜇𝑉1𝑎 ℏ2𝛼2 − 2𝜇𝑉1𝑐 ℏ2𝛼2𝑞2) 2 + 2𝜇𝑉1𝑐 ℏ2𝛼2𝑞2+ 2𝜇𝑉1𝑎 ℏ2𝛼2 (𝑛+ 1 2 +√ 1 4 − 2𝜇𝑉1𝑏 ℏ2𝛼2𝑞 + ℓ(ℓ+1) 𝑞 + 2𝜇𝑉1𝑎 ℏ2𝛼2 − 2𝜇𝑉1𝑐 ℏ2𝛼2𝑞2) ] 2 (28) 4.4 Quadratic exponential-type potential If 𝑉0 is set as 𝑉0 = 0 and 𝑞 → 1 the potential model (1) reduces to quadratic exponential-type potential (Okorie et al., 2018): 𝑉(𝑟) = 𝑉1(𝑎+𝑏𝑒−𝛼𝑟+𝑐𝑒−2𝛼𝑟) (1−𝑒−𝛼𝑟)2 (29) and the energy equation as follows: 𝐸𝑛ℓ = 𝑉1𝑎 − ℏ2𝛼2 8𝜇 [ (𝑛+ 1 2 +√ 1 4 − 2𝜇𝑉1𝑏 ℏ2𝛼2 +ℓ(ℓ+1)+ 2𝜇𝑉1𝑎 ℏ2𝛼2 − 2𝜇𝑉1𝑐 ℏ2𝛼2 ) 2 + 2𝜇𝑉1𝑐 ℏ2𝛼2 + 2𝜇𝑉1𝑎 ℏ2𝛼2 (𝑛+ 1 2 +√ 1 4 − 2𝜇𝑉1𝑏 ℏ2𝛼2 +ℓ(ℓ+1)+ 2𝜇𝑉1𝑎 ℏ2𝛼2 − 2𝜇𝑉1𝑐 ℏ2𝛼2 ) ] 2 (30) Equation 30 is in agreement with the energy equation given in Eq. 31 (Okorie et al., 2018). http://revista.iq.unesp.br/ojs/index.php/ecletica/index https://doi.org/10.26850/1678-4618eqj.v46.4.2021.p60-73 Original article iq.unesp.br/ecletica 66 Eclética Química Journal, vol. 46, n. 4, 2021, 60-73 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v46.4.2021.p60-73 5. Results and discussions In our study, the energy (Eq. 18) and wave function (Eq. 22) of the deformed Hulthen potential plus the quadratic exponential-type potential obtained using the parametric NU. For validity purposes, were also obtained the energy eigenvalues and wave function of the deformed Hulthen potential, Hulthen potential, deformed quadratic exponential-type potential and quadratic exponential-type potential, shown in Eqs. 23–30 as special cases. In the present study, the energy spectrum was used to study the four selected diatomic molecules, H2, HCl, CO and LiH. The spectroscopic parameters of these molecules are given in Tab. 1 and taken from (Edet et al., 2020e; Rampho et al., 2021). The following conversions were used; ℏc = 1973.269 eVÅ and 1𝑎𝑚𝑢 = 931.5 × 106𝑒𝑉(Å) −1 for all computations (Edet et al., 2020e; Rampho et al., 2021). All non- spectroscopic parameters are kept in natural units. Table 1. Spectroscopic parameters of the molecules used in this work (Edet et al., 2020e; Rampho et al., 2021). Parameters H2 (Edet et al., 2020e) HCl (Rampho et al., 2021) CO (Edet et al., 2020e) LiH (Rampho et al., 2021)  (Å−1) 1.9426 1.8677 2.294 1.128 𝜇 (𝑎.𝑚. 𝑢) 0.50391 0.980105 6.860672 0.880122 In Tab. 2, the energy values for the q-deformed Hulthen-quadratic exponential-type potential was shown for H2, HCl, CO and LiH diatomic molecules for various values of the deformation parameter and of quantum states. It is seen clearly that when the deformation parameter is q = −0.5, the energy is low and for instance in LiH molecule it becomes more bounded. But for q = 0.5, the energy is raised significantly although drops slightly and remains positive when q = 1 (i.e., absence of deformation). Table 2. Energy values for the q-deformed Hulthen-quadratic exponential-type potential for H2, HCl, CO and LiH diatomic molecule for various values of the deformation parameter and of quantum states. State q H2 HCl CO LiH 1s –0.5 1.208540 1.104410 2.021420 –0.338560 0.5 3.865880 3.710050 4.565230 2.225200 1.0 3.547310 3.415690 4.299980 1.950540 2s –0.5 0.910264 0.901356 1.928360 –0.466642 0.5 3.885060 3.730360 4.578700 2.241860 1.0 3.423210 3.333520 4.263660 1.899950 2p –0.5 1.221470 1.110510 2.022730 –0.336094 0.5 3.867670 3.711040 4.565480 2.225650 1.0 3.542160 3.413300 4.299470 1.949580 3s –0.5 0.600211 0.692631 1.834060 –0.597053 0.5 3.872680 3.735150 4.588670 2.252040 1.0 3.284190 3.243990 4.225680 1.846270 3p –0.5 0.923715 0.907634 1.929690 –0.464130 0.5 3.885170 3.730790 4.578890 2.242160 1.0 3.417380 3.330900 4.263140 1.898930 3d –0.5 1.247280 1.122710 2.025350 –0.331163 0.5 3.870960 3.712950 4.565970 2.226530 1.0 3.531790 3.408500 4.298470 1.947680 4s –0.5 0.278707 0.478346 1.738530 –0.729764 0.5 3.830940 3.725190 4.595210 2.255940 1.0 3.131300 3.147490 4.186070 1.789610 4p –0.5 0.614176 0.699079 1.835410 –0.594496 0.5 3.871240 3.735050 4.588800 2.252200 1.0 3.277720 3.241150 4.225130 1.845200 Continue… http://revista.iq.unesp.br/ojs/index.php/ecletica/index https://doi.org/10.26850/1678-4618eqj.v46.4.2021.p60-73 Original article iq.unesp.br/ecletica 67 Eclética Química Journal, vol. 46, n. 4, 2021, 60-73 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v46.4.2021.p60-73 4d –0.5 0.950587 0.920182 1.932350 –0.459108 0.5 3.885150 3.731590 4.579270 2.242750 1.0 3.405660 3.325650 4.262080 1.896900 4f –0.5 1.285910 1.140990 2.029280 –0.323769 0.5 3.875230 3.715670 4.566710 2.227830 1.0 3.516090 3.401270 4.296950 1.944810 The variation of the energy eigenvalues with different parameters of the combined potential, such as V0, V1, a, b, c and q, is shown in Figs. 1–7, respectively, for H2, HCl, CO and LiH diatomic molecule in the ground state. In Fig. 1a and b, the variation of the energy spectrum was plotted for various values of n as a function of the parameter ℓ for q = 1 and q = −1, respectively. Figure 1a shows that the energy decreases as the angular momentum increases. In Fig. 1b, it is possible to observe that the energy increases as the ℓ increases up to a maximum and then drops again sporadically. Figure 2a and b shows the variation of the energy spectrum for various values of n as a function of the parameter V0 for q = 1 and q = 1, respectively, in the region 0 < V0 < 3. In Fig. 2a, the energy of the system increases as the parameter V0 increases. In Fig. 2b, the energy linearly decreases as V0 increases. Figure 3a and b shows the variation of the energy spectrum for various values of n as a function of the parameter V1 for q = 1 and q = −1, respectively, in the region 0 < V1 < 0.16. In both figures, the energy increases monotonically as the parameter V1 increases as well. Figure 4a and b shows the variation of the energy spectrum for various values of diatomic molecules as a function of the parameter a, respectively, in the interval 0 < a < 0.25. In both cases considered, the energy increases linearly with increasing a, but in the case q = −1 (Fig. 4b), there is a wider spacing between the energy levels. Figure 5a and b shows the variation of the energy spectrum for various values of n as a function of the parameter b for q = 1 and q = −1, respectively, in the region 0 < b < 1. In Fig. 5a, the energy increases linearly as the parameter b increases. In Fig. 5b, the energy decreases linearly as the parameter b increases. Figure 6a and b shows the variation of the energy spectrum for various values of n as a function of the parameter c for q = 1 and q = -1, respectively, in the region 0 < c < 0.20. In Fig. 6a, the energy decreases linearly as the parameter c increases. In Fig. 6b, the energy decreases linearly as the parameter c increases. Figure 7a and b shows the variation of the energy spectrum for various values of n as a function of the parameter −𝑞 and +𝑞 in the intervals −1 < q < 0 and 0 < q < 1, respectively. In Fig. 7a, the energy increases up to a maximum as q increases and then declines again. In Fig. 7b, the energy increases linearly as q upsurges. Figure 1. (a) The variation of the energy spectrum for various values of n as a function of the parameter ℓ for q = 1. (b) The variation of the energy spectrum for various values of n as a function of the parameter ℓ for q = −1. a = 2fm-1, b = 1fm−1, c = −3fm−1, V1 = 2fm−1 and V0 = 3fm−1. http://revista.iq.unesp.br/ojs/index.php/ecletica/index https://doi.org/10.26850/1678-4618eqj.v46.4.2021.p60-73 Original article iq.unesp.br/ecletica 68 Eclética Química Journal, vol. 46, n. 4, 2021, 60-73 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v46.4.2021.p60-73 Figure 2. (a) The variation of the energy spectrum for various values of 𝑛 as a function of the parameter 𝑉0 for q = 1. (b) The variation of the energy spectrum for various values of 𝑛 as a function of the parameter 𝑉0 for q = −1. h = 1, μ = 1, a = 2, b = 1, c = −3, V1 = 2 and α = 0.05. Figure 3. (a)The variation of the energy spectrum for various values of n as a function of the parameter V1 for q = 1. (b) The variation of the energy spectrum for various values of n as a function of the parameter V1 for q = −1. a = 2, b = 1, c = −3 and V0 = 3. http://revista.iq.unesp.br/ojs/index.php/ecletica/index https://doi.org/10.26850/1678-4618eqj.v46.4.2021.p60-73 Original article iq.unesp.br/ecletica 69 Eclética Química Journal, vol. 46, n. 4, 2021, 60-73 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v46.4.2021.p60-73 Figure 4. (a) The variation of the energy spectrum for various values of n as a function of the parameter a for q = 1. (b) The variation of the energy spectrum for various values of n as a function of the parameter a for q = −1. b = 1, c = −3, V0 = 3 and 𝑉1 = 2. Figure 5. (a) The variation of the energy spectrum for various values of n as a function of the parameter b for q = 1. (b) The variation of the energy spectrum for various values of n as a function of the parameter b for q = −1. a = 2, c = −3, V0 = 3 and 𝑉1 = 2. http://revista.iq.unesp.br/ojs/index.php/ecletica/index https://doi.org/10.26850/1678-4618eqj.v46.4.2021.p60-73 Original article iq.unesp.br/ecletica 70 Eclética Química Journal, vol. 46, n. 4, 2021, 60-73 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v46.4.2021.p60-73 Figure 6. (a) The variation of the energy spectrum for various values of n as a function of the parameter c for q = 1. (b) The variation of the energy spectrum for various values of n as a function of the parameter c for q = −1. a = 2, b = 1, V0 = 3 and 𝑉1 = 2. Figure 7. (a) The variation of the energy spectrum for various values of n as a function of the parameter –q. (b) The variation of the energy spectrum for various values of n as a function of the parameter +q. a = 2, b = 1, c = −3, V1 = 2 and V0 = 3. http://revista.iq.unesp.br/ojs/index.php/ecletica/index https://doi.org/10.26850/1678-4618eqj.v46.4.2021.p60-73 Original article iq.unesp.br/ecletica 71 Eclética Química Journal, vol. 46, n. 4, 2021, 60-73 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v46.4.2021.p60-73 6. Conclusion In this work, the bound state solutions of the Schrödinger equation were studied with the q- deformed Hulthen-quadratic exponential-type potential using parametric NU method. The Greene and Aldrich approximation scheme was used to deal with the centrifugal term; the energy eigenvalues and the corresponding eigenfunctions were obtained and some special cases of the potential were also discussed. The energy spectrum is applied to study four selected diatomic molecules, H2, HCl, CO and LiH. The effect of the deformation parameters and other potential parameters on the energy of the system were graphically and numerically analyzed. The results are in excellent agreement with literature. Finally, the results can find many applications in quantum mechanical systems, atomic and molecular physics. Authors’ contribution Conceptualization: Obogo, U. P.; Ubi, O. E. Data curation: Ikot, A. N.; Edet, C. O. Formal Analysis: Edet, C. O. Funding acquisition: Not applicable. Investigation: Obogo, U. P.; Ubi, O. E.; Edet, C. O.; Ikot, A. N. Methodology: Edet, C. O. Project administration: Obogo, U. P.; Ubi, O. E.; Edet, C. O.; Ikot, A. N. Resources: Edet, C. O.; Ikot, A. N. Software: Edet, C. O. Supervision: Obogo, U. P.; Edet, C. O.; Ikot, A. N. Validation: Edet, C. O.; Ikot, A. N. Visualization: Edet, C. O.; Ikot, A. N. Writing – original draft: Obogo, U. P.; Ubi, O. E. Writing – review & editing: Edet, C. O.; Ikot, A. N. Data availability statement All data sets were generated or analyzed in the current study. Funding Not applicable Acknowledgments We thank the anonymous referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. In addition, Collins Okon Edet acknowledges eJDS (ICTP). References Agboola, D. The Hulthén potential in D-dimensions. Phys. Scr. 2009, 80 (6), 065304. https://doi.org/10.1088/0031- 8949/80/06/065304 Bayrak, O.; Kocak, G.; Boztosun, I. Any l-state solutions of the Hulthén potential by the asymptotic iteration method. J. Phys. A: Math. Gen. 2006, 39 (37), 11521. https://doi.org/10.1088/0305-4470/39/37/012 Ciftci, H.; Hall, R. L.; Saad, N. Asymptotic iteration method for eigenvalue problems. J. Phys. A: Math. Gen. 2003, 36 (47), 11807. https://doi.org/10.1088/0305-4470/36/47/008 Ciftci, H.; Hall, R. L.; Saad, N. 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