BOUND STATE AND RO-VIBRATIONAL ENERGIES EIGENVALUES OF SELECTED DIATOMIC MOLECULES WITH A CLASS OF INVERSELY QUADRATIC YUKAWA PLUS HULTHÉN POTENTIAL MODEL

The Nikiforov-Uvarov approach is used in this study to solve the Schrödinger equation utilizing a class of inversely quadratic Yukawa plus Hulthén potential model with an approximation to the centrifugal term. The normalized wave function and energy eigenvalue equation were obtained. The numerical bound state for a few diatomic molecules (N 2 , O 2 , NO, and CO) for various rotational and vibrational quantum numbers was calculated using the energy equation and the related spectroscopic data. Our results show that, with no divergence between the s-wave and l-wave, the energy eigenvalues are very sensitive to the potential and diatomic molecule properties, suggesting that the approximation approach is appropriate for this set of potentials. The results are consistent with earlier studies in the literature, and we also found four special cases of this potential.


INTRODUCTION
The time-independent Schrödinger wave equation can be used to study the dynamics and interactions of quantum mechanical processes and non-relativistic spinless particles [1][2][3][4].Because the eigenvalues and eigenfunctions associated with quantum problems include essential information regarding the quantum system.The analytical solutions to this equation with physical potentials are likely to play an important role in our understanding of the underlying principles of a quantum system [5,6].The bound state solutions of the Schrödinger equation for a few of these potentials, such as the Coulomb potential (CP) [7], Woods-Saxon [8], Hulthén [9], Manning-Rosen [10], and so on, are possible in some situations.A suitable approximation scheme can also be used to solve the Schrödinger equation approximately when the arbitrary rotational momentum quantum number is available [11].The approximation scheme proposed by Greene and Aldrich [11], the improved approximation scheme by Jia et al. [12], the approximation scheme by Hill [13], the Pekeris approximation [14], the approximation scheme by Yazarloo et al. [15], and the improved approximation scheme in Ref. [16] are a few examples of these approximations.
In order to investigate the interaction that exists between two particles, the Hulthén potential (HP) [36] is essential.It is utilized in the study of atomic, condensed matter, nuclear, and molecular physics as well as chemical physics [37,38].Another potential of interest is a newly proposed potential by Inyang et al. [39] called the class of inversely quadratic Yukawa potential.This work aims to use the class of inversely quadratic Yukawa plus Hulthén potential (CIQYHP) to obtain bound state approximate solutions to the Schrödinger equation.The obtain energy equation will be used to investigate the ro-vibrational energies of some selected diatomic molecules and the bound state energies.The potential model under study is of the form: where 01 11 21

31
, , , and V V V V are potential strength and 0 δ is the screening parameter.The shape of this potential as a function of the screening parameter is given in Fig. 1.The following is how the paper is set up: The energy eigenvalues and normalized eigenfunctions are obtained by solving the Schrödinger equation with the class of inversely quadratic Yukawa plus Hulthén potential using the Nikiforov-Uvarov method in Section 2. The resulting energy equation will be applied in Section 3 to compute numerically the energy eigenvalues at various states of the chosen diatomic molecules and the discussion.In Section 4, conclusions are provided.

APPROXIMATE SOLUTION OF SCHRÖDINGER EQUATION WITH THE CLASS OF INVERSELY QUADRATIC YUKAWA PLUS HULTHÉN POTENTIAL
In this study, the second-order differential equation of the hypergeometric type is solved using the Nikiforov-Uvarov method.The specifics are provided in Reference [17].The Schrödinger equation reads [40] ( ) , 2 where ( ) nl r ψ is the wavefunctions, nl E is the eigenvalues of the quantum system, r is the radial distance from the origin,  is the reduced Planck's constant and μ is the reduced mass.Replacing Eq. (1) into Eq.( 2 Equation ( 3) cannot be solved with the proposed potential because of the inverse square term.We then introduce the Greene-Aldrich approximation scheme [11] to deal with the inverse square term.This approximation scheme is a good approximation and is valid for 0 1, δ << and it becomes ( ) Applying Eq. ( 5), we have By using the change of variable from We put Eq. ( 6) into Eq.( 5) and simplify to get, where EEJP.
Linking Eq. ( 7) and Eq. ( 1) of Ref. [17], we obtain the polynomials: Inserting Eq. ( 9) into Eq.( 11) of Ref. [17], gives; ( ) ( ) ( ) where ( ) ( ) The NU approach states that the discriminant of this quadratic equation must be set to zero in order to solve the quadratic form of Eq. ( 10) under the square root sign.A new quadratic equation is generated by this discriminant, which can be solved for the constant K 0 to obtain the two roots: Replacing Eq. ( 12) into Eq.( 10 with Eq. ( 9) and Eq. ( 13).Therefore, we obtain ( ) ( ) Referring to Eq. ( 10) and Eq. ( 13) of Ref. [17], we have the following equations: With the aid of Eq. ( 8), we can compare Eqs. ( 16) and ( 17) and get the bound state energy eigenvalues of the Schrödinger equation with the class of inversely quadratic Yukawa plus Hulthén potential as follows: Therefore, the complete eigenfunction can be express as: Special cases 1. Setting 01 11 21 0 V V V = = = in Eq. ( 18), we obtain the energy equation for the Hulthén potential, Equation ( 20) is in agreement with Eq. ( 32) of [41] and Eq. ( 37) of [42].

Results and Discussion
We numerically computed the energy eigenvalues for the class of inversely quadratic Yukawa plus Hulthén potential in Table 1 through adjusting the principal quantum number at a fixed orbital angular momentum quantum number with the potential strength ( 01 For a fixed value of angular momentum quantum , the energy spectrum increases as the principal quantum number  increases for this range of potential strength as the screening parameter is not varied.In Table 2, we numerically show the energy eigenvalues of this potential at a fixed n by varying l for various screening parameters, 0 0.05,0.075,and 0.1 δ = .As the screening parameter and angular momentum quantum  increases for a fixed value of principal quantum number , the energy spectrum increases.In Table 3, we numerically present energy eigenvalues of Hulthén potential at 2p, 3p, 3d, 4p.As the screening parameter increases, the energy eigenvalues increase with increase in the quantum numbers.We compared our result for the Hulthén potential with the results from three other methods. Using the energy equation found in Eq. ( 18), we quantitatively presented the eigenvalues for four diatomic molecules in Tables 4. The model parameters for each molecule listed in Table 4 were entered to do this.These diatomic molecules were chosen due to their significance in chemical physics and chemistry.In addition, we have also used the following EEJP.

Figure 1 .
Figure 1.Variation of the potential ( ) V r against internuclear distance r for the N 2 , O 2 , NO and CO diatomic molecules eV for the class of inversely quadratic Yukawa plus Hulthén potential with 1 The Greene-Aldrich approximation scheme has been used to study the bound state solutions to the Schrödinger equation with the newly proposed potential of a class of inversely quadratic Yukawa plus Hulthén potential model.Using the NU technique, the eigenvalues and normalized eigenfunctions are determined.After that, we impute the experimental values for each molecular parameter and apply the solution for four diatomic molecules.The findings demonstrate that these diatomic molecules' bound state energy spectrum increases as various quantum numbers increase.These findings can be applied in Molecular and Chemical Physics.

Funding.
Not applicable.Conflicts of interest/Competing interests.The authors declare that they have no competing interests.

Table 1 .
Energy bound states ( ) eV for the class of inversely quadratic Yukawa plus Hulthén potential with

3 (2023)
The results show that the bound state energy spectra of these diatomic molecules increase as various quantum numbers n and l increases.Our newly developed potential models reduce to special cases by applying certain boundary conditions to validate the mathematical accuracy of our analytical calculations.