Eigensolutions and Thermodynamic Properties of Kratzer plus generalized Morse
Potential

Cecilia N. Isonguyo a,111E-mail<EMAIL_ADDRESS>Ituen B. Okon
a,222E-mail<EMAIL_ADDRESS>Akaninyene D. Antia a,333E-mail:
<EMAIL_ADDRESS>Kayode J. Oyewumi b,444E-mail:
kjoyewumi66@unilorin.edu.ng,Ekwevugbe Omugbe c,555E-mail:
omugbeekwevugbe@gmail.com,Clement A. Onate d,666E-mail:
<EMAIL_ADDRESS>Roseline U. Joshua e,777E-mail:
<EMAIL_ADDRESS>
Monday E. Udoh a,888E-mail<EMAIL_ADDRESS>and Eno E. Ituen
a,999E-mail<EMAIL_ADDRESS>
${}^{\textbf{a}}$Theoretical Physics Group, Department of Physics, University
of Uyo, Akwa Ibom State, Nigeria.
${}^{\textbf{b}}$Department of Physics, University of Ilorin, Kwara State,
Nigeria.

${}^{\textbf{c}}$ Department of Physics, Federal University of Petroleum
Resources, Effurun, Delta State, Nigeria.

${}^{\textbf{d}}$ Department of Physical Sciences, Landmark University, Omu-
Aran, Nigeria

${}^{\textbf{e}}$ Physics Programme, Department of Physics, University of Uyo,
Akwa Ibom State, Nigeria.

Corresponding Author’s email<EMAIL_ADDRESS>
###### Abstract

In this study, we apply the parametric Nikiforov-Uvarov method to obtain the
bound state solution of Schrödinger 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.

Keywords: Nikiforov-Uvarov Method, Kratzer plus generalized Morse Potential,
thermodynamic properties.

PACS Nos: 03.65Ge; 03.65-w; 03.65Ca

##  1 Introduction

The exact solutions of Schrödinger wave equation is one of the essential part
in quantum mechanics, this is because Schrödinger wave equation is used to
describe non-relativistic spinless particles and also has many applications in
atomic, nuclear and high energy Physics [1, 2, 3, 4, 5, 6, 7, 8, 9]. This has
prompted many researchers over the years to search for the solution of
Schcrödinger wave equation with different potentials [10, 11, 12, 13, 14, 15].
However, different methods have been used to obtain approximate solution of
Schcrödinger wave equation, they include Nikiforov-Uvarov method (NU) [16, 17,
18, 19, 20, 21, 22], Supersymmetry quantum mechanics (SUSY) [23, 24, 25, 26,
27, 28], Asymtotic Iteration method (AIM)[29, 30], Factorization method [31,
32], Exact and proper quantization method [33, 34, 35, 36]. In solving the
wave equation, the results obtained for various potential models are vastly
applied [4, 31, 37, 38, 39, 38, 39, 40, 41].

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

$\displaystyle
V(r)=-2D_{e}\left(\frac{r_{e}}{r}-\frac{r^{2}_{e}}{2r^{2}}\right)+D\left(1-\frac{be^{\alpha
r}}{1-e^{\alpha r}}\right)^{2},$ (1)

where $D_{e}$ is dissociation energy, $r_{e}$is the equilibrium bond length,
$r$ represent the interatomic distance, $\alpha$ 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.

##  2 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].

$\displaystyle\psi^{\prime\prime}(s)+\frac{\tilde{\tau}(s)}{\sigma(s)}\psi^{\prime}(s)+\frac{\tilde{\sigma}(s)}{\sigma^{2}(s)}\psi(s)=0$
(2)

where $\tilde{\tau}(s)$ is the polynomial of degree one, $\sigma(s)$ and
$\tilde{\sigma}(s)$ are polynomials of at most degree two. Then the parametric
NU differential equation is in the form [45]

$\displaystyle\psi^{\prime\prime}(s)+\frac{(c_{1}-c_{2}s)}{s(1-c_{3}s)}\psi^{\prime}(s)+\frac{1}{s^{2}(1-c_{3}s)^{2}}\left[-\Omega_{1}s^{2}+\Omega_{2}s-\Omega_{3}\right]\psi(s)=0$
(3)

The parametric constants are obtained as follows

$\displaystyle\left.\begin{array}[]{l}c_{1}=c_{2}=c_{3}=1;\,\,{c_{4}=\frac{1}{2}\left(1-c_{1}\right);\,\,\,c_{5}=\frac{1}{2}\left(c_{2}-c_{3}\right);\,\,c_{6}=c_{5}^{2}+\epsilon_{1}}\\\
{c_{7}=2c_{4}c_{5}-\Omega_{2};\,\,c_{8}=c_{4}^{2}+\Omega_{3};\,\,\,c_{9}=c_{3}c_{7}+c_{3}^{2}c_{8}+c_{6}}\\\
{c_{10}=c_{1}+2c_{4}+2\sqrt{c_{8}};\,\,\,c_{11}=c_{2}-2c_{5}+2\left(\sqrt{c_{9}}+c_{3}\sqrt{c_{8}}\right)}\\\
{c_{12}=c_{4}+\sqrt{c_{8}};\,\,c_{13}=c_{5}-\left(\sqrt{c_{9}}+c_{3}\sqrt{c_{8}}\right)}\end{array}\right\\}.$
(8)

The eigen energy equation is given as

$\displaystyle
c_{2}n-\left(2n+1\right)c_{5}\left(2n+1\right)\left(\sqrt{c_{9}}+c_{3}\sqrt{c_{8}}\right)+n\left(n-1\right)c_{3}+c_{7}2c_{3}c_{8}+2\sqrt{c_{8}c_{9}}=0$
(9)

The corresponding total wave function is then given as

$\displaystyle\Psi(s)=N_{nl}s^{c_{12}}\left(1-c_{3}s\right)^{-c_{12}-\frac{c_{11}}{c_{3}}}P_{n}^{\left(c_{10}-1,\,\,\frac{c_{11}}{c_{3}}-c_{10}-1\right)}\left(1-2c_{3}s\right)$
(10)

###  2.1 Non-relativistic solution with KPGMP

The Schrödinger wave equation for an arbitrary external potential $V(r)$ in
spherical coordinate is written as [4]

$\displaystyle\frac{d^{2}\psi_{n\ell}(r)}{dr^{2}}+\frac{2\mu}{\hbar^{2}}\left[E-V(r)-\frac{\hbar^{2}\ell(\ell+1)}{2\hbar
r^{2}}\right]\psi_{n\ell}=0$ (11)

where $E$ is the exact bound state energy eigenvalues, $R_{n\ell}(r)$ is the
eigenfunction, $\mu=\frac{m_{1}m_{2}}{m_{1}+m_{2}}$ being the reduced mass,
$(\hbar=\mu=1)$. $n$ denotes the principal quantum number ($n$ and $\ell$ are
known as the vibration-rotation quantum numbers), $r$ is the internuclear
separation.

Also, on substituting equation (1) into equation (11), the radial part of the
Schrödinger equation for the KPGM is given as

$\displaystyle\frac{d^{2}\psi_{n\ell}(r)}{dr^{2}}+\frac{2\mu}{\hbar^{2}}\left[E+\left(-2D_{e}\left(\frac{r_{e}}{r}-\frac{r^{2}_{e}}{2r^{2}}\right)+D\left(1-\frac{be^{\alpha
r}}{1-e^{\alpha r}}\right)^{2}\right)-\frac{\hbar^{2}\ell(\ell+1)}{2\mu
r^{2}}\right]\psi_{n\ell}(r)=0.$ (12)

The Green-Aldrich approximation is given as [51]

$\displaystyle\frac{1}{r^{2}}=\frac{4\alpha^{2}e^{-2\alpha r}}{(1-e^{-\alpha
r})^{2}}~{}~{},~{}~{}\frac{1}{r}=\frac{2\alpha e^{-\alpha r}}{1-e^{-\alpha
r}}.$ (13)

On Substituting the transformation $s=e^{\alpha r}$ and applying the Green-
Aldrich approximation into equation (12) yields

$\displaystyle\frac{d^{2}\psi(s)}{dr^{2}}+\frac{(1-s)}{s(1-s)}\frac{d\psi(s)}{dr}+\frac{1}{s^{2}(1-2)^{2}}\left[\begin{array}[]{l}(-\xi^{2}-A-B-
C-F-G-4\lambda)s^{2}\\\
+(2\xi^{2}+A+2C+F)s-(-\xi^{2}-C)\end{array}\right]\psi(s)=0$ (16)

where

$\displaystyle-\xi^{2}=\frac{2\mu E}{\alpha^{2}\hbar^{2}}~{},A=\frac{8\mu
D_{e}r_{e}}{\alpha\hbar^{2}}~{},B=\frac{8\mu
D_{e}r^{2}_{e}}{\alpha^{2}\hbar^{2}}~{},C=\frac{2\mu
D}{\alpha^{2}\hbar^{2}}~{},F\frac{4\mu
Db}{\alpha^{2}\hbar^{2}}~{},G=\frac{2\mu Db^{2}}{\alpha^{2}\hbar^{2}}.$ (17)

Comparing equation (16) with the standard parametric NU differential equation
of (3), the parameters are obtained as follows:

$\displaystyle\left.\begin{array}[]{lll}\Omega_{1}&=&\xi^{2}-A-B-C-
F-G-4\lambda,\Omega_{2}=2\xi^{2}+A+2C+F,~{}~{}\Omega_{3}=\xi^{2}-C\\\
c_{1}&=&c_{2}=c_{3}=1,~{}c_{4}=0,~{}c_{5}=-\frac{1}{2},~{}c_{6}=\frac{1}{4}+\xi+A+B+C+F+G+4\lambda,\\\
c_{7}&=&-2\xi^{2}-A-2C-F,~{}c_{8}=\xi^{2},~{}c_{9}=\frac{1}{4}+B+G+4\lambda,~{}c_{1}0=1+2\sqrt{\xi^{2}+C}\\\
c_{11}&=&2+2\left(\sqrt{\frac{1}{4}+B+G+C+4\lambda+\xi^{2}}\right),~{}c_{12}=\sqrt{\xi^{2}+C}~{},\\\
c_{13}&=&-\frac{1}{2}-\left(\sqrt{\frac{1}{4}+B+G+C+4\lambda+\xi^{2}}\right)\end{array}\right\\}.$
(23)

By substituting the appropriate parameters of equation (23) into equations
(10) and equation (9), then simplify gives the respective wave function and
energy eigenvalue equation for the KPGMP as

$\displaystyle\psi_{n\ell}=N_{n\ell}s^{\gamma}\left(1-s\right)^{\delta}P_{n}^{\left[2\gamma,2\delta-1\right]}\left(1-2s\right),\
\ \ s=e^{\alpha r}$ (24)

and

$\displaystyle
E_{n\ell}=-\frac{\alpha^{2}\hbar^{2}}{2\mu}\left[\frac{\left(n^{2}+n+\frac{1}{2}\right)+(2n+1)\eta-\frac{8\mu
D_{e}r_{e}}{\alpha\hbar^{2}}-\frac{8\mu
Db}{\alpha^{2}\hbar^{2}}}{(2n+1)+2\eta}\right]^{2}-\frac{2\mu
Db^{2}}{\alpha^{2}\hbar^{2}}$ (25)

where

$\displaystyle\gamma=\sqrt{\frac{2\mu D}{\alpha^{2}\hbar^{2}}-\frac{2\mu
E}{\alpha^{2}\hbar^{2}}},\ \ \delta=\frac{1}{2}+\sqrt{\frac{1}{4}+\frac{8\mu
D_{e}r_{e}^{2}}{\alpha^{2}\hbar^{2}}+\frac{2\mu
Db^{2}}{\alpha^{2}\hbar^{2}}+4\ell(\ell+1)},$
$\displaystyle\eta=\sqrt{\frac{1}{4}+\frac{8\mu
D_{e}r_{e}^{2}}{\alpha^{2}\hbar^{2}}+\frac{2\mu
Db^{2}}{\alpha^{2}\hbar^{2}}+4\ell(\ell+1)}$ (26)

The normalization constant in equation 24 can be obtain using the
normalization condition [4, 1]

$\displaystyle\int_{0}^{\infty}|\psi_{n\ell}|^{2}dr=\int_{0}^{\infty}|N_{n\ell}s^{\gamma}\left(1-s\right)^{\delta}P_{n}^{\left[2\gamma,2\delta-1\right]}\left(1-2s\right)|^{2}ds=1,$
(27)

the wavefunction is assumed to be in the bound at $r\ \varepsilon\ (0,\infty)$
and $s=e^{\alpha r}\ \varepsilon\ (1,0)$ eqn 27 becomes

$\displaystyle-\frac{N^{2}_{n\ell}}{\alpha}\int_{0}^{1}s^{2\gamma}\left(1-s\right)^{2\delta}P_{n}^{\left[2\gamma,2\delta-1\right]}\left(1-2s\right)|^{2}\frac{ds}{s}=1.$
(28)

Let $z=1-2s$ thus, the limit of integration of eqn28 changes from $s\
\varepsilon(1,0)$ to $z\ \varepsilon(-1,1)$. Then equation 28 reduces to

$\displaystyle\frac{N^{2}_{n\ell}}{2\alpha}\int_{-1}^{1}\left(\frac{1-z}{2}\right)^{2\gamma-1}\left(\frac{1+z}{2}\right)^{2\delta}\left[P_{n}^{\left[2\gamma,2\delta-1\right]}(z)\right]^{2}dz=1.$
(29)

Applying the standard integral [1, 4]

$\displaystyle\int_{-1}^{1}\left(\frac{1-w}{2}\right)^{x}\left(\frac{1+w}{2}\right)^{y}\left[P_{n}^{(x,y-1)}(w)\right]^{2}dw=\frac{2^{x+y+1}\Gamma(x+n+1)\Gamma(y+n+1)}{n!\Gamma(x+y+n+1)\Gamma(x+y+2n+1)}.$
(30)

Also, let $z=w,\ x=2\gamma-1,\ y=2\delta$. Then the normalization constant can
be obtained as

$\displaystyle
N_{n\ell}=\sqrt{\frac{2\alpha(n!)\Gamma(2\gamma+2\delta+n)\Gamma(2\gamma+2\delta+2n)}{2^{(2\gamma+2\delta)}\Gamma(2\gamma+n)\Gamma(2\delta+n+1)}},$
(31)

therefore, the total normalized wave function is given as

$\displaystyle\psi(s)=\sqrt{\frac{2\alpha(n!)\Gamma(2\gamma+2\delta+n)\Gamma(2\gamma+2\delta+2n)}{2^{(2\gamma+2\delta)}\Gamma(2\gamma+n)\Gamma(2\delta+n+1)}}\
\ s^{\gamma}(1-s)^{\delta}P_{n}^{\left[2\gamma,2\delta-1\right]}(1-2s).$ (32)

Figure 1(a): Wavefunction plot for fixed $l=0$ for LiH molecule

Figure 1(b): Probabilty density plot for fixed $l=0$ for LiH molecule

Figure 2(a): Wavefunction plot for fixed $l=1$ for LiH molecule

Figure 2(b): Probabilty density plot for fixed $l=1$ for LiH molecule

Figure 3(a): Wavefunction plot for fixed $l=2$
for LiH molecule

Figure 3(b): Probabilty density plot for fixed $l=2$ for LiH molecule

Figure 4(a): Wavefunction plot for fixed $l=3$
for LiH molecule

Figure 4(b): Probabilty density plot for fixed $l=3$ for LiH molecule

##  3 Thermodynamic properties for the Potential model

The thermodynamic properties for KPGM will be studied by first obtaining the
vibrational partition function defined as

$\displaystyle Z(\beta)=\sum_{n=0}^{\lambda}e^{-\beta E_{n}}$ (33)

where $\lambda$ is an upper bound of the vibrational quantum number obtain
from the numerical solution of $\frac{dE_{n}}{dn}=0$, $\beta=\frac{1}{kT}$
where $K$ and $T$are Boltzmann constant and absolute temperature respectively.
In the classical limit, the summation in 33 can be replaced with the integral
:

$\displaystyle Z(\beta)=\int_{0}^{\lambda}e^{-\beta E_{n}}dn$ (34)

The energy equation of equation25 can be simplified to

$\displaystyle
E_{nl}=Q_{1}-Q_{2}\left[(n+\Delta)+\frac{Q_{3}}{(n+\Delta)}\right]^{2}$ (35)

where

$\displaystyle Q_{1}=\frac{-2\mu
Db^{2}}{\alpha^{2}\hbar^{2}},Q_{2}=\frac{\alpha^{2}\hbar^{2}}{8\mu},Q_{3}=-\frac{\mu
D_{e}r_{e}^{2}}{2\alpha^{2}\hbar^{2}}-\frac{\mu
Db^{2}}{8\alpha^{2}\hbar^{2}}-\frac{\ell(\ell+1)}{4}-\frac{4\mu
D_{e}}{\alpha^{2}\hbar^{2}},\,\ $
$\displaystyle\Delta=\frac{1}{2}+\frac{1}{2}\sqrt{1+\frac{2\mu
D_{e}r_{e}^{2}}{\alpha^{2}\hbar^{2}}+\frac{\mu
Db^{2}}{2\alpha^{2}\hbar^{2}}+\ell(\ell+1)}$ (36)

The energy equation 35 can then be express in the form

$\displaystyle
E_{nl}=-\left[Q_{2}\rho^{2}+\frac{Q_{2}Q_{3}^{2}}{\rho^{2}}\right]-[2Q_{2}Q_{3}-Q_{1}],\,\,\rho=n+\Delta$
(37)

Hence, the partition function equation 34 can be express in the classical
limit as

$\displaystyle
Z(\beta)=e^{\beta(2Q_{2}Q_{3}-Q_{1})}\int_{0}^{\lambda}e^{(Q_{2}\rho^{2}+\frac{Q_{2}Q_{3}^{2}}{\rho^{2}})}d\rho$
(38)

Equation34 is integrated using MAPLE package. Hence, the integral equation 34
which is the partition function is given as

$\displaystyle Z(\beta)=\frac{\zeta_{1}}{4\sqrt{-\beta
Q_{2}}}\left[\begin{array}[]{l}1+\text{erf}\left(\lambda\sqrt{-\beta
Q_{2}}-\frac{\sqrt{-\beta Q_{2}Q_{3}^{2}}}{\lambda}\right)-e^{4\sqrt{-\beta
Q_{2}}\sqrt{-\beta Q_{2}Q_{3}^{2}}}\\\ \text{erfc}\left(\lambda\sqrt{-\beta
Q_{2}}+\frac{\sqrt{-\beta Q_{2}Q_{3}^{2}}}{\lambda}\right)\end{array}\right]$
(41)

where

$\displaystyle\zeta_{1}=\sqrt{\pi}\exp\left(-\beta Q_{1}+2\beta
Q_{2}Q_{3}-2\sqrt{-\beta Q_{2}}\sqrt{-\beta Q_{2}Q_{3}^{2}}\right)$ (42)

Using the partition function 41, other thermodynamic properties are obtain as
follows
(a)Vibrational mean energy:

$\displaystyle U(\beta)=-\frac{\partial\ln
Z(\beta)}{\partial\beta}=\left[\frac{\begin{array}[]{l}\sqrt{-\beta
Q_{2}Q_{3}^{2}}\left(\sqrt{\pi}\zeta_{2}\left(2\beta
Q_{1}+1\right)+\zeta_{4}\right)\\\ +\zeta_{2}\sqrt{-\beta
Q_{2}Q_{3}^{2}}-4\sqrt{\pi}\beta\zeta_{3}Q_{2}Q_{3}^{2}\sqrt{-\beta
Q_{2}}\end{array}}{2\sqrt{\pi}\beta\zeta_{2}\sqrt{-\beta
Q_{2}Q_{3}^{2}}}\right]$ (45)

where

$\displaystyle\left.\begin{array}[]{l}\zeta_{2}=\text{erfc}\left(\lambda\sqrt{-\beta
Q_{2}}-\frac{\sqrt{-\beta Q_{2}Q_{3}^{2}}}{\lambda}\right)+e^{4\sqrt{-\beta
Q_{2}}\sqrt{-\beta Q_{2}Q_{3}^{2}}}\text{erfc}\left(\lambda\sqrt{-\beta
Q_{2}}+\frac{\sqrt{-\beta Q_{2}Q_{3}^{2}}}{\lambda}\right)-2\\\
\zeta_{3}=\text{erfc}\left(\lambda\sqrt{-\beta Q_{2}}-\frac{\sqrt{-\beta
Q_{2}Q_{3}^{2}}}{\lambda}\right)-e^{4\sqrt{-\beta Q_{2}}\sqrt{-\beta
Q_{2}Q_{3}^{2}}}\text{erfc}\left(\lambda\sqrt{-\beta Q_{2}}+\frac{\sqrt{-\beta
Q_{2}Q_{3}^{2}}}{\lambda}\right)-2\\\ \zeta_{4}=4\lambda\sqrt{-\beta
Q_{2}}\exp\left(\frac{\beta
Q_{2}\left(\lambda^{4}+Q_{3}^{2}\right)}{\lambda^{2}}+2\sqrt{-\beta
Q_{2}}\sqrt{-\beta Q_{2}Q_{3}^{2}}\right)\\\ \end{array}\right\\}.$ (49)

(b) Vibrational specific heat capacity:

$\displaystyle C(\beta)=k\beta^{2}(\frac{\partial^{2}\ln
Z(\beta)}{\partial\beta^{2}})=\frac{\text{k$\beta$}^{2}\left(-\pi\lambda\zeta_{2}^{2}\sqrt{-\beta
Q_{2}}\sqrt{-\beta
Q_{2}Q_{3}^{2}}+\zeta_{8}-\zeta_{9}\right)}{2\pi\beta^{2}\lambda\zeta_{2}^{2}\sqrt{-\beta
Q_{2}}\sqrt{-\beta Q_{2}Q_{3}^{2}}}$ (50)

where

$\displaystyle\left.\begin{array}[]{lll}\zeta_{5}&=&e^{\frac{\beta
Q_{2}\left(\lambda^{4}+Q_{3}^{2}\right)}{\lambda^{2}}}\left(4\lambda^{2}\sqrt{-\beta
Q_{2}}+\sqrt{-\beta Q_{2}Q_{3}^{2}}\right)-8\sqrt{\pi}\lambda e^{2\sqrt{-\beta
Q_{2}}\sqrt{-\beta Q_{2}Q_{3}^{2}}}\\\ &&\sqrt{-\beta Q_{2}}\sqrt{-\beta
Q_{2}Q_{3}^{2}}\text{erfc}\left(\lambda\sqrt{-\beta Q_{2}}+\frac{\sqrt{-\beta
Q_{2}Q_{3}^{2}}}{\lambda}\right)\\\ \zeta_{6}&=&\left(\sqrt{-\beta
Q_{2}Q_{3}^{2}}-4\lambda^{2}\sqrt{-\beta Q_{2}}\right)\exp\left(\frac{\beta
Q_{2}\left(\lambda^{4}+Q_{3}^{2}\right)}{\lambda^{2}}+2\sqrt{-\beta
Q_{2}}\sqrt{-\beta Q_{2}Q_{3}^{2}}\right)\\\
&&+16\sqrt{\pi}\lambda\sqrt{-\beta Q_{2}}\sqrt{-\beta Q_{2}Q_{3}^{2}}\\\
\zeta_{7}&=&\zeta_{5}\text{erfc}\left(\lambda\sqrt{-\beta
Q_{2}}-\frac{\sqrt{-\beta
Q_{2}Q_{3}^{2}}}{\lambda}\right)+\zeta_{6}e^{2\sqrt{-\beta Q_{2}}\sqrt{-\beta
Q_{2}Q_{3}^{2}}}\\\ &&\text{erfc}\left(\lambda\sqrt{-\beta
Q_{2}}+\frac{\sqrt{-\beta Q_{2}Q_{3}^{2}}}{\lambda}\right)-2e^{\frac{\beta
Q_{2}\left(\lambda^{4}+Q_{3}^{2}\right)}{\lambda^{2}}}\left(4\lambda^{2}\sqrt{-\beta
Q_{2}}+\sqrt{-\beta Q_{2}Q_{3}^{2}}\right)\\\
\zeta_{8}&=&4\sqrt{\pi}\beta^{2}Q_{2}^{2}e^{2\sqrt{-\beta Q_{2}}\sqrt{-\beta
Q_{2}Q_{3}^{2}}}\left(\zeta_{7}Q_{3}^{2}-\lambda^{4}\zeta_{2}\sqrt{-\beta
Q_{2}Q_{3}^{2}}e^{\frac{\beta
Q_{2}\left(\lambda^{4}+Q_{3}^{2}\right)}{\lambda^{2}}}\right)\\\
\zeta_{9}&=&2\beta\lambda^{2}Q_{2}\sqrt{-\beta
Q_{2}Q_{3}^{2}}\exp\left(\frac{\beta
Q_{2}\left(\lambda^{4}+Q_{3}^{2}\right)}{\lambda^{2}}+2\sqrt{-\beta
Q_{2}}\sqrt{-\beta Q_{2}Q_{3}^{2}}\right)\\\ &&\left(4\lambda\sqrt{-\beta
Q_{2}}\exp\left(\frac{\beta
Q_{2}\left(\lambda^{4}+Q_{3}^{2}\right)}{\lambda^{2}}+2\sqrt{-\beta
Q_{2}}\sqrt{-\beta
Q_{2}Q_{3}^{2}}\right)-\sqrt{\pi}\zeta_{2}\right)\end{array}\right\\}.$ (60)

(c) Vibrational entropy

$\displaystyle S(\beta)$ $\displaystyle=$ $\displaystyle k\ln
Z(\beta)-k\beta\frac{\partial\ln Z(\beta)}{\partial\beta}$ (61)
$\displaystyle=$
$\displaystyle\text{kLog}\left(-\frac{\sqrt{\pi}\zeta_{265}\exp\left(-\beta
Q_{1}+2\beta Q_{2}Q_{3}-2\sqrt{-\beta Q_{2}}\sqrt{-\beta
Q_{2}Q_{3}^{2}}\right)}{4\sqrt{-\beta Q_{2}}}\right)$
$\displaystyle+\frac{k\left(\zeta_{10}-\zeta_{11}+\sqrt{\pi}\zeta_{2}\left(2\beta
Q_{1}-4\beta Q_{2}Q_{3}+1\right)\right)}{2\sqrt{\pi}\zeta_{3}}$

where

$\displaystyle\left.\begin{array}[]{lll}\zeta_{10}&=&4\sqrt{\pi}\sqrt{-\beta
Q_{2}}\sqrt{-\beta Q_{2}Q_{3}^{2}}\text{erfc}\left(\lambda\sqrt{-\beta
Q_{2}}-\frac{\sqrt{-\beta
Q_{2}Q_{3}^{2}}}{\lambda}\right)+4\lambda\sqrt{-\beta Q_{2}}\\\
&&\exp\left(\frac{\beta
Q_{2}\left(\lambda^{4}+Q_{3}^{2}\right)}{\lambda^{2}}+2\sqrt{-\beta
Q_{2}}\sqrt{-\beta Q_{2}Q_{3}^{2}}\right)-8\sqrt{\pi}\sqrt{-\beta
Q_{2}}\sqrt{-\beta Q_{2}Q_{3}^{2}}\\\ \zeta_{11}&=&4\sqrt{\pi}e^{4\sqrt{-\beta
Q_{2}}\sqrt{-\beta Q_{2}Q_{3}^{2}}}\sqrt{-\beta Q_{2}}\sqrt{-\beta
Q_{2}Q_{3}^{2}}\text{erfc}\left(\lambda\sqrt{-\beta Q_{2}}+\frac{\sqrt{-\beta
Q_{2}Q_{3}^{2}}}{\lambda}\right)\end{array}\right\\}.$ (65)

(c) Vibrational free energy

$\displaystyle F(\beta)$ $\displaystyle=$ $\displaystyle-kT\ln
Z(\beta)=\frac{\sqrt{\pi}\zeta_{2}\ln\exp\left(-\beta Q_{1}+2\beta
Q_{2}Q_{3}-2\sqrt{-\beta Q_{2}}\sqrt{-\beta
Q_{2}Q_{3}^{2}}\right)}{4\beta\sqrt{-\beta Q_{2}}}$ (66)

##  4 Numerical Results

Figure 5(a): Variation of Partition Function With Respect to $\beta$

Figure 5(b): Variation of Partition Function With Respect to $\lambda$

Figure 6(a): Variation of Vibrational Mean Energy With Respect to $\beta$

Figure 6(b): Variation of Vibrational Mean Energy With Respect to $\lambda$

Figure 7(a): Variation of Specific Heat Capacity With Respect to $\lambda$

Figure 7(b): Variation of Specific Heat Capacity With Respect to $\beta$

Figure 8(a): Variation of Vibrational Entropy
With Respect to $\beta$

Figure 8(b): Variation of Vibrational Entropy With Respect to $\lambda$

Figure 9(a): Variation of Vibrational Free Energy With Respect to $\beta$

Figure 9(b): Variation of Vibrational Free Energy With Respect to $\lambda$

##  5 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 $\beta$ and $\lambda$ 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 $Z(\beta)$ increases
exponentially from the origin with increase in the inverse temperature
parameter ($\beta$) but the partition function $Z(\lambda)$ decreases with
increase in $\lambda$ as presented in figure 5(b) for LiH diatomic molecule.
The mean vibrational energy $U(\beta)$ as displayed in figures 6(a) increases
monotonically with increase in the values of $\beta$ with slight maximum
turning points. The plot of $U(\lambda)$ against $\lambda$ has a hyperbolic
nature. From figures 7(a), The vibrational specific heat capacity $C(\beta)$
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 ($C(\beta)$) as a function of $\beta$ has various
minimum turn points that touch the horizontal axis. $C(\lambda)$ decreases
exponentially with $\lambda$ in figures 7(b). Plots of the vibrational entropy
with different values of $\beta$ and $\lambda$ is shown in figures 8(a) and
8(b) respectively. As seen in figure 8(a), the vibrational entropy $C(\beta)$
increases linearly with increasing values of $\beta$ while $S(\lambda)$
decreases with increasing values of $\lambda$. Plots of the mean free energy
$F(\beta)$ as a function of $\beta$ increases monotonically with an increase
in $\beta$ for various values of $\lambda$ as presented in figure 9(a). In
figure 9 (b), the vibrational free energy exhibited an hyperborlic nature
which increases with an increase in $\lambda$

##  6 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 $\beta$ and $\lambda$ 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

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