Bound state solutions of the Schrödinger equation with energy- dependent molecular Kratzer potential via asymptotic iteration method

The exact or approximate solutions of the Schrödinger equations play a vital role in many branches of modern physics and chemistry. The solution of this equation is used in the description of particle dynamics in the non-relativistic regime. Even though the Schrödinger equation was developed many decades ago, it is still very challenging to solve it analytically. The solution of the Schrödinger equation contains all the necessary information needed for the full description of a quantum state such as the probability density and entropy of the system. The Schrödinger equation with many physical potentials model have been investigated in recent times with different advance mathematical technique such as Nikiforov-Uvarov (NU) method, asymptotic iteration method (AIM), functional analysis approach, supersymmetric quantum mechanics (SUSYQM) among others. One of such potential models is the Kratzer potential,


Introduction
The exact or approximate solutions of the Schrödinger equations play a vital role in many branches of modern physics and chemistry 1,2 . The solution of this equation is used in the description of particle dynamics in the non-relativistic regime 3,4 . Even though the Schrödinger equation was developed many decades ago, it is still very challenging to solve it analytically 5,6 . The solution of the Schrödinger equation contains all the necessary information needed for the full description of a quantum state such as the probability density and entropy of the system 7,8 . The Schrödinger equation with many physical potentials model have been investigated in recent times with different advance mathematical technique such as Nikiforov-Uvarov (NU) method [9][10][11] , asymptotic iteration method (AIM) [12][13][14][15][16] , functional analysis approach 16 , supersymmetric quantum mechanics (SUSYQM) 17-20 among others 21 . One of such potential models is the Kratzer potential 22 , (1) where D is the dissociation energy and a is the equilibrium internuclear length.
The Kratzer potential has been used as a potential model to describe internuclear vibration of diatomic molecules 23,24 . Many authors have investigated the bound state solutions of the Kratzer potential within relativistic and nonrelativistic quantum mechanics 25,26 . Recently, Budaca 27 studied an energy-dependent Coulomb-ABSTRACT: In this paper, we obtained the exact bound state energy spectrum of the Schrödinger equation with energy dependent molecular Kratzer potential using asymptotic iteration method (AIM). The corresponding wave function expressed in terms of the confluent hypergeometric function was also obtained. As a special case, when the energy slope parameter in the energydependent molecular Kratzer potential is set to zero, then the well-known molecular Kratzer potential is recovered. Numerical results for the energy eigenvlaues are also obtained for different quantum states, in the presence and absence of the energy slope parameter. These results are discussed extensively using graphical representation. Our results are seen to agree with the results in literature. like potential within the framework of Bohr Hamiltonian. Furthermore, Budaca 27 had 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. The energy-dependent potentials have been studied in nuclear physics with applications to quark confinement 28,29 . Several researchers have also given great attention to investigate the energy dependent potentials 30,31 . Boumali and Labidi 32 solved the Klein-Gordon equation with an energydependent potential, the Shannon and Fisher information theory was also considered. Also, Lombard et.al. 33 studied the wave equation energydependent potential for confined systems. Therefore, the energy dependent potential in the Schrödinger equation or other wave equation in physics has many applications such as features in spectrum of confined systems and heavy quark systems in nuclear and molecular physics 34 .
In this paper, we shall study the influence of the energy-dependent Kratzer potential on some diatomic molecules defined as: where the energy slope parameter  must be positive definite in order to describe a physical system 27 .
The shape of the energy dependent Kratzer molecular potential with different energy slope parameters, as it varies with equilibrium internuclear length are illustrated in Fig. 1A-1C, for four selected diatomic molecules (CO, NO, O2 and I2).

Asymptotic Iteration Method
The AIM has been proposed and used to solve the homogenous linear second-order differential equation of the form 12-16 : where 0 0   and the prime denote the derivative with respect to x .
The functions, 0 () sx and 0 () x  must be sufficiently differentiable. Differentiating Eq. 3 with respect to x , we get Taking the second derivative of Eq. 3 yields Again by taking the ( ) , we obtain the following differential equations: where, Solving Eq. 8, we obtain the following relation: For sufficiently large values of k , the () x  values are obtained as (11) This method consists of converting the Schrödinger-like equation into the form of Eq. 3 for a given potential model. The corresponding energy eigenvalues are calculated by means of the quantization condition [12][13][14][15][16] .
The general solutions of Eq. 3 is obtain from Eq. 10 as: where 1 C and 2 C are integration constant. Also, the eigenfunction can be obtained by transforming the Schrödinger-like equation of the form: The exact solutions for Eq. 14 are given by where,

Bound state solution of the Schrödinger equation with energy-dependent Kratzer potential
The radial part of the Schrödinger equation with energy dependent potential ( , ) where  is the reduced mass of the molecules, nl E is the energy of the system, denotes the reduced Planck constant, n and represent the principal and orbital angular momentum quantum numbers respectively.
Substituting Eq. 1 into Eq. 17 yields, Now using a new variable transformation, y ar = , we obtain a second order differential equation of the form: with the following definitions for the used parameters: In order to transform Eq. 19 into form suitable for the AIM, we write the wave function in the form: Substituting Eq. 21 into Eq. 19, we obtain: Using Eq. 20 with Eq. 25, the energy eigenvalues of the Schrödinger equation with energy-dependent Kratzer potential is obtained as: This result is consistent with those reported by Bayrak, Boztosun and Ciftci 29 .

Results and Discussion
We compute the energy eigenvalues (in eV) of energy dependent Kratzer molecular potential for CO, NO, O2 and I2 diatomic molecules. This was done using the spectroscopic parameters given in The energy eigenvalues of energy dependent molecular Kratzer potential for the ground state diatomic molecules selected are shown in Tables 2 and 3, in the absence of the energy slope parameter  . Our results are very consistent with the results obtained by Bayrak, Boztosun and Ciftci 29 . Also, it is observed that the energy eigenvalues become more bounded as the quantum states of these molecules increases. Moreover, with the introduction of the energy slope parameter, the energy eigenvalues for the different diatomic molecules tends to increase drastically (See Tables   4-6), as compared to the absence of energy slope parameter in Tables 2 and 3.

Conclusions
We applied the asymptotic iteration method (AIM) to solve the Schrödinger equation with energy dependent molecular Kratzer potential. Its energy eigenvalues and corresponding wave functions in terms of confluent hypergeometric function have been obtained. The numerical results of the energy eigenvalues have been presented in the presence and absence of the energy slope parameter, respectively, for four different diatomic molecules (CO, NO, O2 and I2). Our results agree with the results in available literature, especially when the energy slope parameter is set to zero. We have also shown graphically the variation of the energy eigenvalues with some of the potential parameters like energy slope parameter and the potential strength. The behavior of the energy eigenvalues with these parameters is similar in all the diatomic molecules studied. The result obtained in this study finds application in quantum chemistry, molecular physics amongst others.

Acknowledgment
The authors thank the kind reviewers for the positive comments and suggestions that lead to an improvement of our manuscript