Eigensolutions and Thermodynamic Properties of Kratzer plus generalized Morse Potential

In this study, we apply the parametric Nikiforov-Uvarov method to obtain the bound state solution of Schrodinger wave equation in the presence of Kratzer plus generalized Morse potential (KPGM). The energy eigen equation and the corresponding normalised wave function were obtained in closed form. The resulting energy eigen equation were use to study partition function and other thermodynamic properties such as vibrational mean energy, vibrational specific heat capacity, vibrational mean free energy and vibrational entropy for the proposed potential as applied to lithium hydride diatomic molecule. The thermodynamic plots obtained were in excellent agreement to work of existing literatures. The wave function and probability density plots for the diatomic molecules were obtained through a well designed and implemented maple programme.

The thermodynamic properties of a particular system is studied by finding the partition function which is a function of temperature. Other thermodynamic properties such as entropy, specific heat capacity, mean free energy and others are easily obtained using the partition function, which is widely applied in molecular physics and statistical physics [42,43,44].
In this research article, we solve the bound state solution of Schrödinger wave equation with Kratzer plus generalized Morse potential using the parametric Nikiforov-Uvarov method. We also extend our work to study the thermodynamic properties of the system we are studying. The Kratzer plus generalized Morse potential takes the form where D e is dissociation energy, r e is the equilibrium bond length, r represent the interatomic distance, α is the screening parameter. This article is organised as follows: Section 1 is the introduction of the article, The Parametric Nikiforov-Uvarov method and the non-relativistic solution is presented in section 2, Thermodynamic properties is expressed in section 3, while the numerical solution is shown in section 4.

The Parametric Nikiforov-Uvarov(NU) Method
In the parametric NU method, the second order linear differential equation is reduced to a generalised equation of hyper-geometric type which provides exact solutions interms of special orthogonal functions and the corresponding energy eigenvalues of the form. With the appropriate coordinate transformation S = S(x) the equation can be written as [17,45,47,48,49,50].
whereτ (s) is the polynomial of degree one, σ(s) andσ(s) are polynomials of at most degree two. Then the parametric NU differential equation is in the form [45] ψ (s) The parametric constants are obtained as follows The eigen energy equation is given as The corresponding total wave function is then given as

Non-relativistic solution with KPGMP
The Schrödinger wave equation for an arbitrary external potential V (r) in spherical coordinate is written as [4] d 2 ψ n (r) where E is the exact bound state energy eigenvalues, R n (r) is the eigenfunction, µ = m 1 m 2 m 1 +m 2 being the reduced mass, ( = µ = 1). n denotes the principal quantum number (n and are known as the vibration-rotation quantum numbers), r is the internuclear separation. Also, on substituting equation (1) into equation (7), the radial part of the Schrödinger equation for the KPGM is given as The Green-Aldrich approximation is given as [51] 1 On Substituting the transformation s = e αr and applying the Green-Aldrich approximation into equation (8) yields Comparing equation (10) with the standard parametric NU differential equation of (3), the parameters are obtained as follows: By substituting the appropriate parameters of equation (12) into equations (6) and equation (5), then simplify gives the respective wave function and energy eigenvalue equation for the KPGMP as and E n = − α 2 2 2µ where γ = 2µD The normalization constant in equation 13 can be obtain using the normalization condition the wavefunction is assumed to be in the bound at r ε (0, ∞) and s = e αr ε (1, 0) eqn 16 becomes Let z = 1 − 2s thus, the limit of integration of eqn17 changes from s ε(1, 0) to z ε(−1, 1). Then equation 17 reduces to Applying the standard integral [1, 4] Γ(x + n + 1)Γ(y + n + 1) n!Γ(x + y + n + 1)Γ(x + y + 2n + 1) .
Also, let z = w, x = 2γ − 1, y = 2δ. Then the normalization constant can be obtained as therefore, the total normalized wave function is given as The thermodynamic properties for KPGM will be studied by first obtaining the vibrational partition function defined as where λ is an upper bound of the vibrational quantum number obtain from the numerical solution of dEn dn = 0, β = 1 kT where K and T are Boltzmann constant and absolute temperature respectively. In the classical limit, the summation in 22 can be replaced with the integral : The energy equation of equation14 can be simplified to The energy equation 24 can then be express in the form Hence, the partition function equation 23 can be express in the classical limit as Equation23 is integrated using MAPLE package. Hence, the integral equation 23 which is the partition function is given as where Using the partition function 28, other thermodynamic properties are obtain as follows (a)Vibrational mean energy: where (b) Vibrational specific heat capacity: where (c) Vibrational entropy where (c) Vibrational free energy 4 Numerical Results

Numerical Results and Discussion
The thermodynamics properties of KPGM was studied, the plots of the wavefunction and thermodynamics properties as a function of the inverse temperature parameter β and λ for Lithium hydride (LiH) diatomic molecule are shown in figures 1-4 and figures 5-9 respectively. Figure 1(a) is the wave function plot for fixed l=0 which begins with a commom origin and proceed to a continous sinusoidal curve with various maximum and minimum turning points for Lithium hydride molecule. From the graph, it can be observed that the peaks of the turning points increases with increase in the principal quantum number (n), as such n=0 has the lowest peak. The probability density curve for fixed l=0 is displayed in figure 1(b). This plot follows normal distribution with several maximum points which also increases with an increase in the principal quantum number. The probability density curve completely describe the localization of electrons of LiH molecule, hence electron is more localized at n=3 which has the highest maximum point. The same description of figure 1(a) is applicable to figures 2(a) and 4(a) while figures 2(b) and 4(b) also have the same description as figure 1(b). The wavefunction graph in figure 3(a) has maximum and minimum turning points at higher quantum states except at the ground state (n=0) where there is a divergence in the curve. In figure 3(b), the probability density plot is sinusoidal in nature which shows the localization of electrons for higher quantum state while diverges at the ground state. Figures 5(a)and 5(b) show the variation of the vibrational partition function. It is observed in figure 5(a) that the partition function Z(β) increases exponentially from the origin with increase in the inverse temperature parameter (β) but the partition function Z(λ) decreases with increase in λ as presented in figure 5(b) for LiH diatomic molecule. The mean vibrational energy U (β) as displayed in figures 6(a) increases monotonically with increase in the values of β with slight maximum turning points. The plot of U (λ) against λ has a hyperbolic nature. From figures 7(a), The vibrational specific heat capacity C(β) first decreases with an increase in inverse temperature parameter to a minimum value and then increase monotonically. However, the graph of vibrational specific heat capacity (C(β)) as a function of β has various minimum turn points that touch the horizontal axis. C(λ) decreases exponentially with λ in figures 7(b).
Plots of the vibrational entropy with different values of β and λ is shown in figures 8(a) and 8(b) respectively. As seen in figure 8(a), the vibrational entropy C(β) increases linearly with increasing values of β while S(λ) decreases with increasing values of λ. Plots of the mean free energy F (β) as a function of β increases monotonically with an increase in β for various values of λ as presented in figure 9(a). In figure 9 (b), the vibrational free energy exhibited an hyperborlic nature which increases with an increase in λ

Conclusion
In this work, we have solved the Schrödinger wave equation in the presence of Kratzer plus generalized Morse potential (KPGM) using Parametric Nikiforov-Uvarov method. The energy eigenvalues and the corresponding wave function are obtained. However, we studied the thermodynamics properties of KPGM which are: vibrational partition Z, vibrational mean energy U, specific heat capacity C, entropy, and mean free energy. Also, we have plotted the variation of these thermodynamic functions as a function of β and λ for Lithium hydride diatomic molecule.
Conflict of interest-Not applicable Funding: The article processing charge is completely funded by SCOAP3 and licensed under CCBY 4.0