Welcome to visit Communications in Theoretical Physics,
Mathematical Physics

Generalised tanh-shaped hyperbolic potential: Klein–Gordon equation's bound state solution

  • V H Badalov , 1 ,
  • S V Badalov , 2, 3,
Expand
  • 1Institute for Physical Problems, Baku State University, 1148 Baku, Azerbaijan
  • 2Theoretical Materials Physics, Paderborn University, D-33098 Paderborn, Germany
  • 3Theoretical Physics VII, University of Bayreuth, D-95440 Bayreuth, Germany

Author to whom any correspondence should be addressed.

Received date: 2022-12-09

  Revised date: 2023-05-11

  Accepted date: 2023-05-11

  Online published: 2023-06-26

Copyright

© 2023 Institute of Theoretical Physics CAS, Chinese Physical Society and IOP Publishing

Abstract

The development of potential theory heightens the understanding of fundamental interactions in quantum systems. In this paper, the bound state solution of the modified radial Klein–Gordon equation is presented for generalised tanh-shaped hyperbolic potential from the Nikiforov–Uvarov method. The resulting energy eigenvalues and corresponding radial wave functions are expressed in terms of the Jacobi polynomials for arbitrary l states. It is also demonstrated that energy eigenvalues strongly correlate with potential parameters for quantum states. Considering particular cases, the generalised tanh-shaped hyperbolic potential and its derived energy eigenvalues exhibit good agreement with the reported findings. Furthermore, the rovibrational energies are calculated for three representative diatomic molecules, namely H2, HCl and O2. The lowest excitation energies are in perfect agreement with experimental results. Overall, the potential model is displayed to be a viable candidate for concurrently prescribing numerous quantum systems.

Cite this article

V H Badalov , S V Badalov . Generalised tanh-shaped hyperbolic potential: Klein–Gordon equation's bound state solution[J]. Communications in Theoretical Physics, 2023 , 75(7) : 075003 . DOI: 10.1088/1572-9494/acd441

1. Introduction

Following quantum mechanics’ genesis, the study of precisely solvable problems plays a critical role in comprehending the underlying quantum-mechanical systems [13]. Analytical solutions of the Schrödinger, Klein–Gordon (KG), and Dirac equations are of particular importance in quantum mechanics because the wave function contains all the information necessary for a complete description of a particle’s behaviour in a force field [38].
In any n radial and l orbital quantum states, a limited number of physical potentials can be solved exactly for the Schrödinger, KG, and Dirac equations [2, 3]. Generally, many quantum systems can only be solved numerically or through approximation techniques [79]. Therefore, several methods including the supersymmetry method [10], the factorisation method [3], the Laplace transform approach [11], the path integral method [12], the Nikiforov–Uvarov (NU) method [13], asymptotic iteration method [14, 15] and the quantisation rule method [1618]  have been developed so far and they have been applied for the solution of the quantum wave equation. The NU method yields more practicality to solve second-order differential equations by transforming them into hypergeometric-type equations. Furthermore, the various exponential and hyperbolic potentials are analytically solved by using different approximation schemes with the NU method.
In principle, the exponential potential models always draw considerable attention and are widely used in various physical systems, including quantum cosmology, nuclear physics, molecular physics, elementary particle physics, and condensed matter physics [1929]. Up to now, many exponential-type potentials, including the Morse [30, 31], Hulthén [3238], Woods–Saxon [27, 3943], Rosen–Morse [4448], Eckart-type [4951], Manning–Rosen [5254], Deng–Fan [55, 56], Pöschl–Teller like [57], Mathieu [58], sine-type hyperbolic [59] and Schiöberg [6063] potentials have been investigated, and some analytical bound state solutions were obtained using an approximation for these models in l ≠ 0 state. Some known exponential potentials can also be transformed into a hyperbolic potential model, which can help understand quantum systems’ natural dynamics [6476]. For instance, the thermodynamic properties of some molecules have been successfully predicted using the improved Rosen–Morse potential and the Fu-Wang-Jia potential to describe the internal vibrations of molecules [7782].
Motivated by the simplicity and applicability of the generalisations of the hyperbolic potentials, the generalised tanh-shaped hyperbolic potential (GTHP) [83] was recently proposed as follows:
$\begin{eqnarray}V(r)={V}_{1}+{V}_{2}\tanh (\alpha r)+{V}_{3}{\tanh }^{2}(\alpha r),\end{eqnarray}$
where V1, V2, V3 are the depths of potential well and α is the adjustable parameter representing the properties of the interaction potential. For clarity about potential, see also the S2 section4(4See supplementary material, which includes [29, 56, and 63], for additional details of potential information and theoretical derivations.). GTHP is the general case of the significant physical potential such as the standard and generalised Woods–Saxon [39, 42], Rosen–Morse [44], Manning–Rosen type, generalised and standard Morse [30], Schiöberg [60], four-parametric exponential-type [6769], Williams–Poulios potential [84, 85], and the sum of the linear and harmonic oscillator potentials, see S2 section in4. As it seems, GTHP’s characteristics can be used to explain the interactions of molecular, atomic, and nuclear particles.
In this study, we extend our study of GTHP by considering it in the KG equation. We apply the NU method to analytically solve KG and obtain the bound state for this potential, and we compare the results with the previously reported ones in particular cases. Then, the potential is modelled for several diatomic molecules, and the obtained results are in good agreement with experimental ones. This study allows us to correctly explain a broad variety of quantum systems’ characteristics and behaviour, including retardation effects, without needing a great deal of complex derivation or massive computing resources. The remainder of this study covers the following sections: the bound-state solution of the radial KG equation is presented in section 2. In section 3, we explore the results for energy levels and the corresponding normalised eigenfunctions in some special cases and diatomic molecules. Finally, some concluding remarks are stated in section 4.

2. Bound state solutions

The time-independent KG equation with scalar and vector potentials S(r) and V(r) a spin-zero particles takes the general form: [2]
$\begin{eqnarray}\begin{array}{l}{{\rm{\nabla }}}^{2}\psi (r,\theta ,\varphi )+\displaystyle \frac{1}{{\hslash }^{2}{c}^{2}}\{{\left(E-V(r)\right)}^{2}\\ \,-{\left({{Mc}}^{2}+S(r)\right)}^{2}\}\psi (r,\theta ,\varphi )=0,\end{array}\end{eqnarray}$
where E is the relativistic energy system an M denotes the rest mass of the particle. To denote the radial and angular components of the wave function ψnlm(r, θ, φ), the concept of variable separation is defined as:
$\begin{eqnarray}{\psi }_{{nlm}}(r,\theta ,\varphi )=\displaystyle \frac{{\chi }_{{nl}}(r)}{r}{{\rm{Y}}}_{{lm}}(\theta ,\varphi ),\end{eqnarray}$
and substituting it into equation (2), the modified radial KG equation is obtained as follows:
$\begin{eqnarray}\left[\frac{{{\rm{d}}}^{2}}{{\rm{d}}{r}^{2}}+\frac{(E-{{Mc}}^{2}-V(r)-S(r))(E+{{Mc}}^{2}-V(r)+S(r))}{{\hslash }^{2}{c}^{2}}-\frac{l(l+1)}{{r}^{2}}\right]\chi (r)=0.\end{eqnarray}$
After choosing of equal scalar and vector potentials, that is, S(r) = V(r), equation (4) becomes in the following form:
$\begin{eqnarray}\begin{array}{l}\frac{{{\rm{d}}}^{2}\chi (r)}{{\rm{d}}{r}^{2}}+\frac{1}{{\hslash }^{2}{c}^{2}}\\ \times \,\left[{E}^{2}-{M}^{2}{c}^{4}-2(E+{{Mc}}^{2})V(r)-\frac{{\hslash }^{2}{c}^{2}l(l+1)}{{r}^{2}}\right]\chi (r)=0.\end{array}\end{eqnarray}$
Considering the expression of 2V(r) equal as equation (1), that is
$\begin{eqnarray}2V(r)={V}_{\mathrm{GTHP}}(r)={V}_{1}+{V}_{2}\tanh (\alpha r)+{V}_{3}{\tanh }^{2}(\alpha r).\end{eqnarray}$
It should be noted that this choice the potential enables us to reduce the resulting relativistic states to their non-relativistic limit under appropriate transformations. Therefore, we obtain it as:
$\begin{eqnarray}\begin{array}{l}\frac{{{\rm{d}}}^{2}\chi (r)}{{\rm{d}}{r}^{2}}+\frac{1}{{\hslash }^{2}{c}^{2}}\{{E}^{2}-{M}^{2}{c}^{4}-(E+{{Mc}}^{2})\\ \ \ \times \,[{V}_{1}+{V}_{2}\tanh (\alpha r)+{V}_{3}{\tanh }^{2}(\alpha r)]\\ \,-\frac{{\hslash }^{2}{c}^{2}l(l+1)}{{r}^{2}}\}\chi (r)=0.\end{array}\end{eqnarray}$
This equation cannot be solved analytically for l ≠ 0 due to the centrifugal term. Therefore, the Pekeris approximation [4042, 83, 86] which is the most widely used and convenient for our purposes can be taken to solve this equation. According to the Pekeris approximation scheme to deal with the centrifugal term is used [83]
$\begin{eqnarray}\displaystyle \frac{1}{{r}^{2}}\approx \displaystyle \frac{1}{{r}_{e}^{2}}[{A}_{0}+{A}_{1}\tanh (\alpha r)+{A}_{2}{\tanh }^{2}(\alpha r),\end{eqnarray}$
where the parameters A0, A1 and A2 were found as: [83]
$\begin{eqnarray}\left\{\begin{array}{l}{A}_{0}=1+\tfrac{{\cosh }^{4}(\alpha {r}_{e})}{{\alpha }^{2}{r}_{e}^{2}}[3{\tanh }^{2}(\alpha {r}_{e})+2\alpha {r}_{e}\tanh (\alpha {r}_{e})(1-2{\tanh }^{2}(\alpha {r}_{e}))]\\ {A}_{1}=-\tfrac{2{\cosh }^{4}(\alpha {r}_{e})}{{\alpha }^{2}{r}_{e}^{2}}[3\tanh (\alpha {r}_{e})+\alpha {r}_{e}(1-3{\tanh }^{2}(\alpha {r}_{e}))]\,\,\,\,\,\,.\\ {A}_{2}=\tfrac{{\cosh }^{4}(\alpha {r}_{e})}{{\alpha }^{2}{r}_{e}^{2}}(3-2\alpha {r}_{e}\tanh (\alpha {r}_{e})).\end{array}\right.\end{eqnarray}$
After inserting the equation (8) into equation (7), and further using a new variable $\tanh (\alpha r)=s$, s ∈ [0, 1], we obtain it as:
$\begin{eqnarray}\chi ^{\prime\prime} (s)-\displaystyle \frac{2s}{1-{s}^{2}}\chi ^{\prime} (s)-\displaystyle \frac{\varepsilon +\beta s+\gamma {s}^{2}}{{\left(1-{s}^{2}\right)}^{2}}\chi (s)=0,\end{eqnarray}$
where
$\begin{eqnarray}\left\{\begin{array}{l}\tfrac{{E}^{2}-{M}^{2}{c}^{4}-(E+{{Mc}}^{2}){V}_{1}}{{\alpha }^{2}{\hslash }^{2}{c}^{2}}-\tfrac{l(l+1){A}_{0}}{{\alpha }^{2}{r}_{e}^{2}}=-\varepsilon \\ \tfrac{(E+{{Mc}}^{2}){V}_{2}}{{\alpha }^{2}{\hslash }^{2}{c}^{2}}+\tfrac{l(l+1){A}_{1}}{{\alpha }^{2}{r}_{e}^{2}}=\beta \\ \tfrac{(E+{{Mc}}^{2}){V}_{3}}{{\alpha }^{2}{\hslash }^{2}{c}^{2}}+\tfrac{l(l+1){A}_{2}}{{\alpha }^{2}{r}_{e}^{2}}=\gamma \end{array}\right.\end{eqnarray}$
with the boundary conditions χ(0) = 0 and χ(1) = 0. Now for implementing the NU method, equation (10) should be rewritten as the hypergeometric type equation form as:
$\begin{eqnarray}\chi ^{\prime\prime} (s)+\displaystyle \frac{\tilde{\tau }(s)}{\sigma (s)}\chi ^{\prime} (s)+\displaystyle \frac{\tilde{\sigma }(s)}{{\sigma }^{2}(s)}\chi (s)=0.\end{eqnarray}$
After comparing equation (10) and equation (12), we obtain:
$\begin{eqnarray}\tilde{\tau }(s)=-2s;\,\sigma (s)=1-{s}^{2};\,\tilde{\sigma }(s)=-\varepsilon -\beta s-\gamma {s}^{2}.\end{eqnarray}$
The new function π(s) as given in [13] can be obtained by substituting equation (13) and taking $\sigma ^{\prime} (s)=-2s$. Hence, the function π(s) is defined as:
$\begin{eqnarray}\pi (s)=\pm \sqrt{(\gamma -k){s}^{2}+\beta s+\varepsilon +k}.\end{eqnarray}$
The value of the constant parameter k can be calculated by performing the condition that the discriminant of the expression equation (14) under the square root is equal to zero. Hence, we obtain it as:
$\begin{eqnarray}\left\{\begin{array}{l}{k}_{1}={D}^{2}-\varepsilon ,\\ {k}_{2}={P}^{2}-\varepsilon ,\end{array}\right.\end{eqnarray}$
where
$\begin{eqnarray}\left\{\begin{array}{l}D=\sqrt{\tfrac{(\varepsilon +\gamma )+\sqrt{{\left(\varepsilon +\gamma \right)}^{2}-{\beta }^{2}}}{2}},\\ P=\sqrt{\tfrac{(\varepsilon +\gamma )-\sqrt{{\left(\varepsilon +\gamma \right)}^{2}-{\beta }^{2}}}{2}}\end{array}\right.\end{eqnarray}$
with D > P, 2DP = ∣β∣, D2 + P2 = ϵ + γ. When the individual values of k are given in equation (15) are substituted into equation (14), the eight possible forms of π(s) are written in the following forms:
$\begin{eqnarray}\pi (s)=\pm \left\{\begin{array}{c}{Ps}\pm D,\,{for}\,\,k\,=\,{D}^{2}-\varepsilon \\ {Ds}\pm P,\,{for}\,\,k\,=\,{P}^{2}-\varepsilon \,\end{array}\right..\end{eqnarray}$
Even π(s) have eight different values, but according to NU method, we select only one of them such that the function $\tau (s)=\tilde{\tau }(s)+2\pi (s)$ has the negative derivative and a root on the interval (0, 1), that is, $\tau ^{\prime} (s)\lt 0$ and τ(s) = 0, s ∈ (0, 1). Noticing that the other forms have no physical meaning, we will take:
$\begin{eqnarray}\left\{\begin{array}{l}\pi (s)=P-{Ds}\\ \tau (s)=2P-2(D+1)s\\ k\,=\,{P}^{2}-\varepsilon \end{array}\right..\end{eqnarray}$
After using the following relations: $\lambda =k+\pi ^{\prime} (s)$ and ${\lambda }_{n}=-n\tau ^{\prime} (s)-\tfrac{n(n-1)}{2}\sigma $, (n = 0, 1, 2,….) [13], we obtain λ as:
$\begin{eqnarray}\lambda =k+\pi ^{\prime} (s)={P}^{2}-\varepsilon -D\end{eqnarray}$
and
$\begin{eqnarray}\lambda ={\lambda }_{n}=2{Dn}+n(n+1),\end{eqnarray}$
where n is the radial quantum number (n = 0, 1, 2,….). After comparing equation (19) with equation (20), we obtain the following relation:
$\begin{eqnarray}D=n^{\prime} \gt 0,\end{eqnarray}$
where
$\begin{eqnarray}n^{\prime} =\sqrt{\displaystyle \frac{1}{4}+\gamma }-n-\displaystyle \frac{1}{2}.\end{eqnarray}$
By inserting the expression D into equation (21) we obtain
$\begin{eqnarray}\varepsilon +\beta +\gamma ={\left(n^{\prime} +\displaystyle \frac{\beta }{2n^{\prime} }\right)}^{2}.\end{eqnarray}$
After inserting the equations (11) and (22) into equation (23) for energy level equation, we obtain it as:
$\begin{eqnarray}\begin{array}{l}{M}^{2}{c}^{4}-{E}^{2}+(E+{{Mc}}^{2})({V}_{1}+{V}_{2}+{V}_{3})\\ \ \ +\,\displaystyle \frac{{\hslash }^{2}{c}^{2}l(l+1)}{{r}_{e}^{2}}({A}_{0}+{A}_{1}+{A}_{2})={\alpha }^{2}{\hslash }^{2}{c}^{2}\\ \ \ \times \,\left[\Space{0ex}{5.25ex}{0ex}\sqrt{\displaystyle \frac{(E+{{Mc}}^{2}){V}_{3}}{{\alpha }^{2}{\hslash }^{2}{c}^{2}}+\displaystyle \frac{l(l+1)}{{\alpha }^{2}{r}_{e}^{2}}{A}_{2}+\displaystyle \frac{1}{4}}\right.\\ \ \ {\left.-\,n-\displaystyle \frac{1}{2}+\displaystyle \frac{\tfrac{(E+{{Mc}}^{2}){V}_{2}}{2{\alpha }^{2}{\hslash }^{2}{c}^{2}}+\tfrac{l(l+1)}{2{\alpha }^{2}{r}_{e}^{2}}{A}_{1}}{\sqrt{\tfrac{(E+{{Mc}}^{2}){V}_{3}}{{\alpha }^{2}{\hslash }^{2}{c}^{2}}+\tfrac{l(l+1)}{{\alpha }^{2}{r}_{e}^{2}}{A}_{2}+\tfrac{1}{4}}-n-\tfrac{1}{2}}\right]}^{2}.\end{array}\end{eqnarray}$
with n = 0, 1, 2,... $\unicode{x0230A}\sqrt{\tfrac{(E+{{Mc}}^{2}){V}_{3}}{{\alpha }^{2}{\hslash }^{2}{c}^{2}}+\tfrac{l(l+1)}{{\alpha }^{2}{r}_{e}^{2}}{A}_{2}+\tfrac{1}{4}}-\tfrac{1}{2}-\sqrt{-\tfrac{(E+{{Mc}}^{2}){V}_{2}}{2{\alpha }^{2}{\hslash }^{2}{c}^{2}}-\tfrac{l(l+1)}{2{\alpha }^{2}{r}_{e}^{2}}{A}_{1}}\unicode{x0230B}$.
By applying the NU method, we can obtain the radial eigenfunctions. After substituting π(s) and Σ(s) into $\tfrac{{\rm{\Phi }}^{\prime} (s)}{{\rm{\Phi }}(s)}=\tfrac{\pi (s)}{\sigma (s)}$ and $\tfrac{\rho ^{\prime} (s)}{\rho (s)}+\tfrac{\sigma ^{\prime} (s)}{\sigma (s)}=\tfrac{\tau (s)}{\sigma (s)}$ solving the order differential equation, one can find the finite function Φ(s) and ρ(s) in the interval (0, 1) it is easily obtained:
$\begin{eqnarray}\begin{array}{rcl}{\rm{\Phi }}(s) & = & {\left(1-s\right)}^{\tfrac{\eta }{2}}{\left(1+s\right)}^{\tfrac{\nu }{2}},\\ \rho (s) & = & {\left(1-s\right)}^{\eta }{\left(1+s\right)}^{\nu },\end{array}\end{eqnarray}$
where η = DP = $\sqrt{\varepsilon -| \beta | +\gamma }\gt 0$, ν = D + P = $\sqrt{\varepsilon +| \beta | +\gamma }\gt 0$. Beyond that, the other part of the wave function yn(s) is the hypergeometric-type function whose polynomial solutions are given by Rodrigues relation ${y}_{n}(s)=\tfrac{{B}_{n}}{\rho (s)}\tfrac{{{\rm{d}}}^{n}}{{\rm{d}}{s}^{n}}\left[{\sigma }^{n}(s)\rho (s)\right]$, we obtain it as:
$\begin{eqnarray}{y}_{n}(s)={{\rm{P}}}_{n}^{(\eta ,\,\nu )}(s)=\frac{{\left(-1\right)}^{n}}{{2}^{n}n!}{\left(1-s\right)}^{-\eta }{\left(1+s\right)}^{-\nu }\frac{{{\rm{d}}}^{n}}{{\rm{d}}{s}^{n}}\left[{\left(1-s\right)}^{n+\eta }{\left(1+s\right)}^{n+\nu }\right],\end{eqnarray}$
where ${{\rm{P}}}_{n}^{(\eta ,\,\nu )}(s)$ is the Jacobi polinomial [87].
According to the relation χ(s) = Φ(s)y(s) [13], we obtain the radial wave functions as:
$\begin{eqnarray}{\chi }_{{nl}}(s)={C}_{{nl}}{\left(1-s\right)}^{\tfrac{\eta }{2}}{\left(1+s\right)}^{\tfrac{\nu }{2}}{{\rm{P}}}_{n}^{(\eta ,\,\nu )}(s),\end{eqnarray}$
where Cnl is the normalisation constant. By using the normalisation condition, we obtain Cnl as:
$\begin{eqnarray}{C}_{{nl}}=\frac{{2}^{n}}{(n+\eta )!(n+\nu )!}\sqrt{\frac{\alpha }{\displaystyle \sum _{k,m=0}^{n}\frac{{\left(-1\right)}^{k+m}{\rm{F}}(1,1-\nu -k-m;\eta +2n-k-m+1;-1)}{k!(\eta +n-k)!m!(\eta +n-m)!(n-k)!(n-m)!(\nu +k)!(\nu +m)!}}}.\end{eqnarray}$
where F(a, b; c; z) is the hypergeometric function.

3. Results and discussion

3.1. Particular cases

In this part, we discuss the results by investigating the expression of analytically obtained energy level equation (24) for this potential based on some special cases:
(i) By choosing the parameters of GTHP as ${V}_{1}=-\tfrac{{V}_{0}}{2}-\tfrac{W}{4}$, ${V}_{2}=\tfrac{{V}_{0}}{2}$, ${V}_{3}=\tfrac{W}{4}$ and $\alpha =\tfrac{1}{2a}$, we obtain the energy spectrum equation of the generalised Woods–Saxon potential:
$\begin{eqnarray}\begin{array}{l}{M}^{2}{c}^{4}-{E}^{2}+\displaystyle \frac{{\hslash }^{2}{c}^{2}l(l+1){C}_{0}}{{R}_{0}^{2}}=\displaystyle \frac{{\hslash }^{2}{c}^{2}}{4{a}^{2}}\\ \ \ \times \,\left[\sqrt{\Space{0ex}{5.25ex}{0ex}\displaystyle \frac{(E+{{Mc}}^{2}){a}^{2}W}{{\hslash }^{2}{c}^{2}}+\displaystyle \frac{l(l+1){a}^{2}{C}_{2}}{{R}_{0}^{2}}+\displaystyle \frac{1}{4}}\right.\\ \ \ {\left.-\,n-\displaystyle \frac{1}{2}+\displaystyle \frac{\tfrac{(E+{{Mc}}^{2}){a}^{2}{V}_{0}}{{\hslash }^{2}{c}^{2}}-\tfrac{l(l+1){a}^{2}({C}_{1}+{C}_{2})}{{R}_{0}^{2}}}{\sqrt{\tfrac{(E+{{Mc}}^{2}){a}^{2}W}{{\hslash }^{2}{c}^{2}}+\tfrac{l(l+1){a}^{2}{C}_{2}}{{R}_{0}^{2}}+\tfrac{1}{4}}-n-\tfrac{1}{2}}\right]}^{2},\end{array}\end{eqnarray}$
where C0, C1 and C2 are defined as:
$\begin{eqnarray}\left\{\begin{array}{l}{C}_{0}=\tfrac{{A}_{0}+{A}_{1}+{A}_{2}}{{\left(1+{x}_{e}\right)}^{2}}=\tfrac{1}{{\left(1+{x}_{e}\right)}^{2}}+\tfrac{{\left(1+{{\rm{e}}}^{\alpha {x}_{e}}\right)}^{2}}{\alpha {{\rm{e}}}^{\alpha {x}_{e}}{\left(1+{x}_{e}\right)}^{3}}\left[\tfrac{{{\rm{e}}}^{-\alpha {x}_{e}}-3}{1+{{\rm{e}}}^{\alpha {x}_{e}}}+\tfrac{3{{\rm{e}}}^{-\alpha {x}_{e}}}{\alpha (1+{x}_{e})}\right]\\ {C}_{1}=-\tfrac{2({A}_{1}+2{A}_{2})}{{\left(1+{x}_{e}\right)}^{2}}=\tfrac{2{\left(1+{{\rm{e}}}^{\alpha {x}_{e}}\right)}^{2}}{\alpha {{\rm{e}}}^{\alpha {x}_{e}}{\left(1+{x}_{e}\right)}^{3}}\left[2-{{\rm{e}}}^{-\alpha {x}_{e}}-\tfrac{3(1+{{\rm{e}}}^{-\alpha {x}_{e}})}{\alpha (1+{x}_{e})}\right]\,\,\,\,\,\,.\\ {C}_{2}=\tfrac{4{A}_{2}}{{\left(1+{x}_{e}\right)}^{2}}=\tfrac{{\left(1+{{\rm{e}}}^{\alpha {x}_{e}}\right)}^{3}}{\alpha {{\rm{e}}}^{\alpha {x}_{e}}{\left(1+{x}_{e}\right)}^{3}}\left[{{\rm{e}}}^{-\alpha {x}_{e}}-1+\tfrac{3(1+{{\rm{e}}}^{-\alpha {x}_{e}})}{\alpha (1+{x}_{e})}\right]\end{array}\right.\end{eqnarray}$
(ii) By considering W = 0 and xe = 0 in equation (29), we obtain the energy spectrum equation for the standard Woods–Saxon potential, as follows:
$\begin{eqnarray}\begin{array}{l}{M}^{2}{c}^{4}-{E}^{2}+\displaystyle \frac{{\hslash }^{2}{c}^{2}l(l+1){C}_{0}}{{R}_{0}^{2}}=\displaystyle \frac{{\hslash }^{2}{c}^{2}}{4{a}^{2}}\\ \ \ \times \,{\left[\sqrt{\displaystyle \frac{l(l+1){a}^{2}{C}_{2}}{{R}_{0}^{2}}+\displaystyle \frac{1}{4}}-n-\displaystyle \frac{1}{2}+\displaystyle \frac{\tfrac{(E+{{Mc}}^{2}){a}^{2}{V}_{0}}{{\hslash }^{2}{c}^{2}}-\tfrac{l(l+1){a}^{2}({C}_{1}+{C}_{2})}{{R}_{0}^{2}}}{\sqrt{\tfrac{l(l+1){a}^{2}{C}_{2}}{{R}_{0}^{2}}+\tfrac{1}{4}}-n-\tfrac{1}{2}}\right]}^{2}.\end{array}\end{eqnarray}$
Here, ${C}_{0}={A}_{0}+{A}_{1}+{A}_{2}=1-\tfrac{4}{\alpha }+\tfrac{12}{{\alpha }^{2}}$, ${C}_{1}=-2({A}_{1}+2{A}_{2})=\tfrac{8}{\alpha }-\tfrac{48}{{\alpha }^{2}}$, ${C}_{2}=4{A}_{2}=\tfrac{48}{{\alpha }^{2}}$ [40, 41], where $\alpha =\tfrac{{R}_{0}}{a}$.
(iii) By taking the parameters of GTHP as V3 = − V1 = C and V2 = B, the energy spectrum equation of the Rosen–Morse potential is obtained as:
$\begin{eqnarray}\begin{array}{l}{M}^{2}{c}^{4}-{E}^{2}+(E+{{Mc}}^{2})B+\displaystyle \frac{{\hslash }^{2}{c}^{2}l(l+1)}{{r}_{e}^{2}}({A}_{0}+{A}_{1}+{A}_{2})={\alpha }^{2}{\hslash }^{2}{c}^{2}\\ \ \ \times \,\left[\sqrt{\Space{0ex}{5.25ex}{0ex}\displaystyle \frac{(E+{{Mc}}^{2})C}{{\alpha }^{2}{\hslash }^{2}{c}^{2}}+\displaystyle \frac{l(l+1)}{{\alpha }^{2}{r}_{e}^{2}}{A}_{2}+\displaystyle \frac{1}{4}}\right.\\ \ \ {\left.-\,n-\displaystyle \frac{1}{2}+\displaystyle \frac{\tfrac{(E+{{Mc}}^{2})B}{2{\alpha }^{2}{\hslash }^{2}{c}^{2}}+\tfrac{l(l+1)}{2{\alpha }^{2}{r}_{e}^{2}}{A}_{1}}{\sqrt{\tfrac{(E+{{Mc}}^{2})C}{{\alpha }^{2}{\hslash }^{2}{c}^{2}}+\tfrac{l(l+1)}{{\alpha }^{2}{r}_{e}^{2}}{A}_{2}+\tfrac{1}{4}}-n-\tfrac{1}{2}}\right]}^{2}.\end{array}\end{eqnarray}$
From the above expression, when l = 0, the energy spectrum is in good agreement with the result in [46, 47]
$\begin{eqnarray}\begin{array}{l}{M}^{2}{c}^{4}-{E}^{2}+(E+{{Mc}}^{2})B={\alpha }^{2}{\hslash }^{2}{c}^{2}\\ \ \ \times \,{\left[\sqrt{\displaystyle \frac{(E+{{Mc}}^{2})C}{{\alpha }^{2}{\hslash }^{2}{c}^{2}}+\displaystyle \frac{1}{4}}-n-\displaystyle \frac{1}{2}+\displaystyle \frac{\tfrac{(E+{{Mc}}^{2})B}{2{\alpha }^{2}{\hslash }^{2}{c}^{2}}}{\sqrt{\tfrac{(E+{{Mc}}^{2})C}{{\alpha }^{2}{\hslash }^{2}{c}^{2}}+\tfrac{1}{4}}-n-\tfrac{1}{2}}\right]}^{2}.\end{array}\end{eqnarray}$
(iv) By taking the parameters of GTHP as ${V}_{1}=\tfrac{{\hslash }^{2}}{8{{Mb}}^{2}}[\beta (\beta -1)-2A]$, ${V}_{2}=-\tfrac{{\hslash }^{2}}{4{{Mb}}^{2}}[\beta (\beta -1)-A]$, ${V}_{3}=\tfrac{{\hslash }^{2}}{8{{Mb}}^{2}}\beta (\beta -1)$ and $2\alpha =\tfrac{1}{b}$, we obtain the energy spectrum equation of the Manning–Rosen-type potential as follows:
$\begin{eqnarray}\begin{array}{l}{M}^{2}{c}^{4}-{E}^{2}+\displaystyle \frac{{\hslash }^{2}{c}^{2}l(l+1)}{{r}_{e}^{2}}({A}_{0}+{A}_{1}+{A}_{2})=\displaystyle \frac{{\hslash }^{2}{c}^{2}}{4{b}^{2}}\\ \ \ \times \,\left[\sqrt{\Space{0ex}{5.25ex}{0ex}\displaystyle \frac{(E+{{Mc}}^{2})\beta (\beta -1)}{2{{Mc}}^{2}}+\displaystyle \frac{4l(l+1){b}^{2}}{{r}_{e}^{2}}{A}_{2}+\displaystyle \frac{1}{4}}\right.\\ \ \ {\left.-\,n-\displaystyle \frac{1}{2}+\displaystyle \frac{-\tfrac{(E+{{Mc}}^{2})[\beta (\beta -1)-A]}{2{{Mc}}^{2}}+\tfrac{2l(l+1){b}^{2}}{{r}_{e}^{2}}{A}_{1}}{\sqrt{\tfrac{(E+{{Mc}}^{2})\beta (\beta -1)}{2{{Mc}}^{2}}+\tfrac{4l(l+1){b}^{2}}{{r}_{e}^{2}}{A}_{2}+\tfrac{1}{4}}-n-\tfrac{1}{2}}\right]}^{2}.\end{array}\end{eqnarray}$
(v) By choosing the parameters of GTHP as ${V}_{1}=\tfrac{1}{4}{D}_{e}{(b-2)}^{2},{V}_{2}=-\tfrac{1}{2}{D}_{e}b(b-2),{V}_{3}=\tfrac{1}{4}{D}_{e}{b}^{2}$, 2α = δ and $b={{\rm{e}}}^{\delta {r}_{e}}+1$, we obtain the following energy spectrum equation for the improved Rosen–Morse potential (the generalised Morse-type potential):
$\begin{eqnarray}\begin{array}{l}{M}^{2}{c}^{4}-{E}^{2}+(E+{{Mc}}^{2}){D}_{e}+\displaystyle \frac{{\hslash }^{2}{c}^{2}l(l+1)}{{r}_{e}^{2}}({A}_{0}+{A}_{1}+{A}_{2})=\displaystyle \frac{{\delta }^{2}{\hslash }^{2}{c}^{2}}{4}\\ \ \ \times \,\left[\sqrt{\Space{0ex}{5.25ex}{0ex}\displaystyle \frac{(E+{{Mc}}^{2}){D}_{e}{b}^{2}}{{\delta }^{2}{\hslash }^{2}{c}^{2}}+\displaystyle \frac{4l(l+1)}{{\delta }^{2}{r}_{e}^{2}}{A}_{2}+\displaystyle \frac{1}{4}}\right.\\ \ \ {\left.-\,n-\displaystyle \frac{1}{2}+\displaystyle \frac{-\tfrac{(E+{{Mc}}^{2}){D}_{e}b(b-2)}{{\delta }^{2}{\hslash }^{2}{c}^{2}}+\tfrac{2l(l+1)}{{\delta }^{2}{r}_{e}^{2}}{A}_{1}}{\sqrt{\tfrac{(E+{{Mc}}^{2}){D}_{e}{b}^{2}}{{\delta }^{2}{\hslash }^{2}{c}^{2}}+\tfrac{4l(l+1)}{{\delta }^{2}{r}_{e}^{2}}{A}_{2}+\tfrac{1}{4}}-n-\tfrac{1}{2}}\right]}^{2}.\end{array}\end{eqnarray}$
From the above expression, we obtain the same expression equation (42) of [70] with a0 = A0 + A1 + A2, a1 = − 2(A1 + 2A2), a2 = 4A2 and ${{\rm{e}}}^{\alpha {r}_{e}}+1=b$.
And finally, from the expression equation (35), when considering l = 0, we get the same expression as in [56]:
$\begin{eqnarray}\begin{array}{l}{M}^{2}{c}^{4}-{E}^{2}+(E+{{Mc}}^{2}){D}_{e}=\displaystyle \frac{{\delta }^{2}{\hslash }^{2}{c}^{2}}{4}\\ \ \ \times \,{\left[\sqrt{\displaystyle \frac{(E+{{Mc}}^{2}){D}_{e}{b}^{2}}{{\delta }^{2}{\hslash }^{2}{c}^{2}}+\displaystyle \frac{1}{4}}-n-\displaystyle \frac{1}{2}-\displaystyle \frac{\tfrac{(E+{{Mc}}^{2}){D}_{e}b(b-2)}{{\delta }^{2}{\hslash }^{2}{c}^{2}}}{\sqrt{\tfrac{(E+{{Mc}}^{2}){D}_{e}{b}^{2}}{{\delta }^{2}{\hslash }^{2}{c}^{2}}+\tfrac{1}{4}}-n-\tfrac{1}{2}}\right]}^{2}.\end{array}\end{eqnarray}$
(vi) By choosing the parameters of GTHP V1 = δ2D, V2 = −2δΣD and V3 = Σ2D, we obtain the energy spectrum equation for the Schiöberg potential, as follows:
$\begin{eqnarray}\begin{array}{l}{M}^{2}{c}^{4}-{E}^{2}+(E+{{Mc}}^{2})D{\left(\delta -\sigma \right)}^{2}+\displaystyle \frac{{\hslash }^{2}{c}^{2}l(l+1){C}_{0}}{{r}_{e}^{2}}={\alpha }^{2}{\hslash }^{2}{c}^{2}\\ \ \ \times \,\left[\sqrt{\Space{0ex}{5.25ex}{0ex}\displaystyle \frac{(E+{{Mc}}^{2}){\sigma }^{2}D}{{\alpha }^{2}{\hslash }^{2}{c}^{2}}+\displaystyle \frac{l(l+1){C}_{2}}{4{\alpha }^{2}{r}_{e}^{2}}+\displaystyle \frac{1}{4}}\right.\\ \ \ {\left.-\,n-\displaystyle \frac{1}{2}+\displaystyle \frac{-\tfrac{(E+{{Mc}}^{2})\delta \sigma D}{{\alpha }^{2}{\hslash }^{2}{c}^{2}}+\tfrac{l(l+1)}{4{\alpha }^{2}{r}_{e}^{2}}({C}_{1}-{C}_{2})}{\sqrt{\tfrac{(E+{{Mc}}^{2}){\sigma }^{2}D}{{\alpha }^{2}{\hslash }^{2}{c}^{2}}+\tfrac{l(l+1){C}_{2}}{4{\alpha }^{2}{r}_{e}^{2}}+\tfrac{1}{4}}-n-\tfrac{1}{2}}\right]}^{2}.\end{array}\end{eqnarray}$
Here
$\begin{eqnarray}\left\{\begin{array}{l}{C}_{0}={A}_{0}+{A}_{1}+{A}_{2}=1-{\left(\tfrac{1+{{\rm{e}}}^{-2\alpha {r}_{e}}}{2\alpha {r}_{e}}\right)}^{2}\left[\tfrac{8\alpha {r}_{e}}{1+{{\rm{e}}}^{-2\alpha {r}_{e}}}-3-2\alpha {r}_{e}\right]\\ {C}_{1}=2({A}_{1}+2{A}_{2})=-(1+{{\rm{e}}}^{2\alpha {r}_{e}})\cdot \tfrac{1\,+\,{{\rm{e}}}^{-2\alpha {r}_{e}}}{\alpha {r}_{e}}\left[3-(3+2\alpha {r}_{e})\cdot \tfrac{1+{{\rm{e}}}^{-2\alpha {r}_{e}}}{2\alpha {r}_{e}}\right]\\ {C}_{2}=4{A}_{2}={\left(1+{{\rm{e}}}^{2\alpha {r}_{e}}\right)}^{2}{\left(\tfrac{1+{{\rm{e}}}^{-2\alpha {r}_{e}}}{2\alpha {r}_{e}}\right)}^{2}\left[3+2\alpha {r}_{e}-\tfrac{4\alpha {r}_{e}}{1+{{\rm{e}}}^{-2\alpha {r}_{e}}}\right]\end{array}\right.\end{eqnarray}$
From the expression equation (37), we obtain the same expression equation (42) of [70] with a0 = C0, a1 = − C1, a2 = C2, De = D(δΣ)2 and ${{\rm{e}}}^{\alpha {r}_{e}}+1=\tfrac{2}{1-\tfrac{\delta }{\sigma }}$.
(vii) By taking the parameters of GTHP ${V}_{1}={P}_{1}+\tfrac{{P}_{2}}{2}+\tfrac{{P}_{3}}{4},{V}_{2}=-\tfrac{{P}_{2}}{2}-\tfrac{{P}_{3}}{2}$ and ${V}_{3}=\tfrac{{P}_{3}}{4}$ for energy spectrum equation of the four-parameter exponential-type potential, we have
$\begin{eqnarray}\begin{array}{l}{M}^{2}{c}^{4}-{E}^{2}+(E+{{Mc}}^{2}){P}_{1}+\displaystyle \frac{{\hslash }^{2}{c}^{2}l(l+1)}{{r}_{e}^{2}}({A}_{0}+{A}_{1}+{A}_{2})={\alpha }^{2}{\hslash }^{2}{c}^{2}\\ \ \ \times \,\left[\sqrt{\Space{0ex}{5.25ex}{0ex}\displaystyle \frac{(E+{{Mc}}^{2}){P}_{3}}{4{\alpha }^{2}{\hslash }^{2}{c}^{2}}+\displaystyle \frac{l(l+1)}{{\alpha }^{2}{r}_{e}^{2}}{A}_{2}+\displaystyle \frac{1}{4}}\right.\\ \ \ {\left.-\,n-\displaystyle \frac{1}{2}+\displaystyle \frac{-\tfrac{(E+{{Mc}}^{2})({P}_{2}+{P}_{3})}{4{\alpha }^{2}{\hslash }^{2}{c}^{2}}+\tfrac{l(l+1)}{2{\alpha }^{2}{r}_{e}^{2}}{A}_{1}}{\sqrt{\tfrac{(E+{{Mc}}^{2}){P}_{3}}{4{\alpha }^{2}{\hslash }^{2}{c}^{2}}+\tfrac{l(l+1)}{{\alpha }^{2}{r}_{e}^{2}}{A}_{2}+\tfrac{1}{4}}-n-\tfrac{1}{2}}\right]}^{2}.\end{array}\end{eqnarray}$
(viii) By choosing the parameters of GTHP ${V}_{1}=\tfrac{{W}_{1}+{W}_{2}+{W}_{3}}{4};{V}_{2}=\tfrac{{W}_{3}-{W}_{1}}{2}$ and ${V}_{3}=\tfrac{{W}_{1}-{W}_{2}+{W}_{3}}{4}$, we obtain the energy spectrum equation for the Williams–Poulios-type potential as follows:
$\begin{eqnarray}\begin{array}{l}{M}^{2}{c}^{4}-{E}^{2}+(E+{{Mc}}^{2}){W}_{3}+\displaystyle \frac{{\hslash }^{2}{c}^{2}l(l+1)}{{r}_{e}^{2}}({A}_{0}+{A}_{1}+{A}_{2})=\ {\alpha }^{2}{\hslash }^{2}{c}^{2}\\ \ \times \,\left[\sqrt{\Space{0ex}{5.25ex}{0ex}\displaystyle \frac{(E+{{Mc}}^{2})({W}_{1}-{W}_{2}+{W}_{3})}{4{\alpha }^{2}{\hslash }^{2}{c}^{2}}+\displaystyle \frac{l(l+1)}{{\alpha }^{2}{r}_{e}^{2}}{A}_{2}+\displaystyle \frac{1}{4}}\right.\\ \ \ {\left.-\,n-\displaystyle \frac{1}{2}+\displaystyle \frac{\tfrac{(E+{{Mc}}^{2})({W}_{3}-{W}_{1})}{4{\alpha }^{2}{\hslash }^{2}{c}^{2}}+\tfrac{l(l+1)}{2{\alpha }^{2}{r}_{e}^{2}}{A}_{1}}{\sqrt{\tfrac{(E+{{Mc}}^{2})({W}_{1}-{W}_{2}+{W}_{3})}{4{\alpha }^{2}{\hslash }^{2}{c}^{2}}+\tfrac{l(l+1)}{{\alpha }^{2}{r}_{e}^{2}}{A}_{2}+\tfrac{1}{4}}-n-\tfrac{1}{2}}\right]}^{2}.\end{array}\end{eqnarray}$
(ix) When α is chosen as much smaller than one, the allowed values of the energy level equation of the equation (24) are defined as follows:
$\begin{eqnarray}\begin{array}{l}{M}^{2}{c}^{4}-{E}^{2}+\left({V}_{1}-\displaystyle \frac{{V}_{2}^{2}}{4{V}_{3}}\right)(E+{{Mc}}^{2})+\displaystyle \frac{{\hslash }^{2}{c}^{2}l(l+1)}{{r}_{e}^{2}}\\ \ \ +\,2\alpha \hslash c\left(1-\displaystyle \frac{{V}_{2}^{2}}{4{V}_{3}^{2}}\right)\sqrt{(E+{{Mc}}^{2}){V}_{3}}\left(n+\displaystyle \frac{1}{2}\right)\\ \ \ -\,{\alpha }^{2}{\hslash }^{2}{c}^{2}\left[\left(1+\displaystyle \frac{3{V}_{2}^{2}}{4{V}_{3}^{2}}\right){\left(n+\displaystyle \frac{1}{2}\right)}^{2}+\displaystyle \frac{1}{4}\left(1-\displaystyle \frac{{V}_{2}^{2}}{4{V}_{3}^{2}}\right)\right]=o({\alpha }^{2})\end{array}\end{eqnarray}$
for small values of n and l.
Finally, using the following transformations EMc2Enl and E + Mc2 → 2μc2, we obtain the energy level equation of equation (24) for the non-relativistic case, as follows:
$\begin{eqnarray}\begin{array}{l}{E}_{{nl}}={V}_{1}+{V}_{2}+{V}_{3}+\displaystyle \frac{{\hslash }^{2}l(l+1)}{2\mu {r}_{e}^{2}}({A}_{0}+{A}_{1}+{A}_{2})-\displaystyle \frac{{\alpha }^{2}{\hslash }^{2}}{2\mu }\\ \ \ \times \,\left[\sqrt{\displaystyle \frac{2\mu {V}_{3}}{{\alpha }^{2}{\hslash }^{2}}+\displaystyle \frac{l(l+1)}{{\alpha }^{2}{r}_{e}^{2}}{A}_{2}+\displaystyle \frac{1}{4}}\right.\\ \ \ {\left.-\,n-\displaystyle \frac{1}{2}+\displaystyle \frac{\tfrac{\mu {V}_{2}}{{\alpha }^{2}{\hslash }^{2}}+\tfrac{l(l+1)}{2{\alpha }^{2}{r}_{e}^{2}}{A}_{1}}{\sqrt{\tfrac{2\mu {V}_{3}}{{\alpha }^{2}{\hslash }^{2}}+\tfrac{l(l+1)}{{\alpha }^{2}{r}_{e}^{2}}{A}_{2}+\tfrac{1}{4}}-n-\tfrac{1}{2}}\right]}^{2}\,,\end{array}\end{eqnarray}$
and this result are same as the result obtained in [83]. When α is much smaller than unity then the allowed values of the energy spectrum become
$\begin{eqnarray}\begin{array}{l}{E}_{{nl}}=V({r}_{e})+\hslash {\omega }_{e}\left(n+\displaystyle \frac{1}{2}\right)+\displaystyle \frac{{\hslash }^{2}l(l+1)}{2\mu {r}_{e}^{2}}-\displaystyle \frac{{\alpha }^{2}{\hslash }^{2}}{2\mu }\\ \ \ \times \,\left[\left(1+\displaystyle \frac{3{V}_{2}^{2}}{4{V}_{3}^{2}}\right){\left(n+\displaystyle \frac{1}{2}\right)}^{2}+\displaystyle \frac{1}{4}\left(1-\displaystyle \frac{{V}_{2}^{2}}{4{V}_{3}^{2}}\right)\right]+o({\alpha }^{2})\end{array}\end{eqnarray}$
for small values of n and l, where ${\omega }_{e}=\alpha \left(1-\tfrac{{V}_{2}^{2}}{4{V}_{3}^{2}}\right)\sqrt{\tfrac{2{V}_{3}}{\mu }}$ is the classical frequency of oscillation about the minimum point, r = re. Here, the first term $V({r}_{e})={V}_{1}-\tfrac{{V}_{2}^{2}}{4{V}_{3}}$ is the minimum value of GTHP; the second term is the energy levels of the harmonic oscillator; the third term is the energy levels of rotational energy corresponding to a fixed distance between atoms and the fourth term is the energy levels of anharmonic correction.

3.2. The bound state energy eigenvalues and lowest excitation in diatomic molecules

Spectroscopic parameters of the diatomic molecules H2, HCl and O2 are given in table 1, which are taken from [88, 89]. Based on the experimental values such as the dissociation energy De, the equilibrium bond length re, and the equilibrium vibrational frequency νe, the potential parameters V1, V2, V3 and the screening parameter α can be defined by using the expressions (S19), (S20), (S24), and (S28) in4.
Table 1. Spectroscopic molecular parameters for H2, HCl, O2 diatomic molecules.
Molecule μ re De νe
(a. m. u. ) (Å) (cm−1) (cm−1)
H2 0.5041 0.7417 36 118.062 4395.2
HCl 0.9799 1.274 563 03 37 243 2990.875
O2 8.000 1.207 5358 42 047 1580.194
By using these parameters, we present the potential energy curves calculated for H2, HCl, O2 diatomic molecules, see figure 1. Further, we can easily calculate the bound state energy eigenvalues for the diatomic molecules H2, HCl, O2 at n and l quantum states by using the ${E}_{{nl}}^{{NR}}$ expressions of the Schrödinger molecule and ${E}_{{nl}}^{b}={E}_{{nl}}^{R}-{{Mc}}^{2}$ expression of the binding energies KG molecule [2], see table 2. The obtained eigenvalues of the HCl diatomic molecule are in good agreement with experimentally reported values in [90].
Figure 1. Potential energy curves of the diatomic molecules H2, HCl and O2 as a function of the interatomic distance.
Table 2. The bound state energy eigenvalues ${E}_{{nl}}^{{NR}}$ of the Schrödinger molecule and ${E}_{{nl}}^{b}={E}_{{nl}}^{R}-{{Mc}}^{2}$ of the Klein–Gordon molecule in the GTHP calculated using equation (42) and equation (24) for several n and l, respectively.
n l ${E}_{{nl}}^{{NR}},({{\rm{H}}}_{2})$ ${E}_{{nl}}^{b},({{\rm{H}}}_{2})$ ${E}_{{nl}}^{{NR}},(\mathrm{HCl})$ ${E}_{{nl}}^{b},(\mathrm{HCl})$ ${E}_{{nl}}^{{NR}},({{\rm{O}}}_{2})$ ${E}_{{nl}}^{b},({{\rm{O}}}_{2})$
(cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (cm−1)
0 0 2163.332 479 2163.332 477 1480.444 771 1480.444 675 786.437 8165 786.437 6894
1 0 6327.103 462 6327.103 410 4359.482 381 4359.482 341 2338.054 436 2338.054 024
1 1 6444.845 686 6444.845 634 4379.847 347 4379.847 337 2340.894 357 2340.894 427
2 0 10 252.927 23 10 252.927 26 7125.499 716 7125.499 649 3860.922 397 3860.922 186
2 1 10 367.885 83 10 367.885 90 7145.313 357 7145.313 215 3863.729 451 3863.729 320
2 2 10 596.757 12 10 596.757 28 7184.927 896 7184.927 714 3869.343 429 3869.344 512
3 0 13 934.324 20 13 934.324 17 9777.517 902 9777.517 934 5354.985 484 5354.985 616
3 1 14 046.229 51 14 046.229 52 9796.773 872 9796.773 864 5357.759 581 5357.759 412
3 2 14 269.024 31 14 269.024 47 9835.273 336 9835.273 166 5363.306 888 5363.307 911
3 3 14 600.720 22 14 600.720 31 9892.991 022 9892.990 941 5371.629 453 5371.630 377
4 0 17 364.449 80 17 364.449 77 12 314.543 22 12 314.543 18 6820.187 243 6820.187 560
4 1 17 473.029 85 17 473.029 83 12 333.235 31 12 333.235 23 6822.927 545 6822.927 953
4 2 17 689.208 23 17 689.208 38 12 370.607 07 12 370.606 85 6828.408 793 6828.409 623
4 3 18 011.064 17 18 011.064 30 12 426.633 34 12 426.633 29 6836.630 730 6836.631 845
4 4 18 435.819 49 18 435.819 57 12 501.277 34 12 501.277 24 6847.592 259 6847.593 247
5 0 20 536.070 41 20 536.070 39 14 735.565 70 14 735.565 70 8256.471 021 8256.471 069
5 1 20 641.051 30 20 641.051 28 14 753.687 76 14 753.687 60 8259.177 484 8259.177 992
5 2 20 850.069 73 20 850.069 82 14 789.919 18 14 789.918 98 8264.591 767 8264.592 703
5 3 21 161.280 64 21 161.280 76 14 844.235 14 14 844.235 24 8272.713 618 8272.714 473
5 4 21 572.017 81 21 572.017 97 14 916.599 24 14 916.599 34 8283.541 215 8283.541 974
5 5 22 078.906 37 22 078.906 53 15 006.962 14 15 006.962 07 8297.076 277 8297.076 196
6 0 23 441.536 12 23 441.536 08 17 039.560 05 17 039.559 91 9663.779 212 9663.779 010
6 1 23 542.642 61 23 542.642 54 17 057.105 39 17 057.105 32 9666.452 501 9666.452 396
6 2 23 743.954 74 23 743.954 88 17 092.184 01 17 092.183 82 9671.798 959 9671.800 000
6 3 24 043.712 12 24 043.712 23 17 144.770 88 17 144.770 97 9679.820 497 9679.821 126
6 4 24 439.372 71 24 439.372 81 17 214.829 91 17 214.829 95 9690.513 889 9690.514 445
6 5 24 927.723 10 24 927.723 19 17 302.311 92 17 302.311 89 9703.880 795 9703.880 933
6 6 25 505.011 24 25 505.011 31 17 407.155 87 17 407.155 76 9719.918 459 9719.919 524
While we compare the lowest excitation energies for diatomic molecules, the obtained results are in perfect agreement with the sophisticated high-resolution measurements, see table 3. Although the six parameters Lennard–Jones potential model is a little bit more accurate than GTHP, the fact that GTHP has four parameters is a great advantage for easier modelling of physical systems. Generally, the obtained results allow one to tune and optime the potential concerning its desired properties in atomic, molecular, chemical, condensed matter and high energy physics applications.
Table 3. The lowest rotational △E(l) and vibrational △E(n) excitation energies, all values in cm−1.
Molecule H2 H2 HCl HCl O2 O2
Exc. type l = 0 → 1 n = 0 → 1 l = 0 → 1 n = 0 → 1 l = 0 → 1 n = 0 → 1
Theorya 117.742 224 4163.770 983 20.364 966 2879.037 610 2.839 921 1551.616 620
Theoryb 117.742 224 4163.770 933 20.364 996 2879.037 666 2.840 403 1551.616 335
Theoryc 2885.86
Theoryd 2872.98
Theorye 118.486 812(9) 4161.1661(9)
$\mathrm{Exp}.$ 118.486 84(10)f 4161.1660(03)g,
4161.16632(18)h 2885.98i

aThis work for non-relativistic case.

bThis work for relativistic case.

cAnalytical calculation by extended Lennard-Jones potential, see [89].

dCalculation by ab initio multi-reference configuration interaction calculation with the digit standing for diatomics (EHFACE2), see [90].

eCalculation by nonadiabatic perturbation theory with considering relativistic quantum electrodynamic, see [91]; the experimental value:

fSee [92].

gSee [93].

hSee [94].

iSee [90].

4. Concluding remarks

In this study, we proposed a new potential model, which holds numerous important physical potentials. Next, the bound state solution of the radial KG equation with this potential is examined analytically within the framework of the NU method. It is also presented that the energy eigenvalues are sensitively associated with potential parameters for quantum states. GTHP and its obtained energy eigenvalues are in remarkable overlap with the reported results in some cases, so this potential model is a desirable candidate for displaying multiple quantum systems concurrently. For more specific cases, GTHP was used to study for modelling several diatomic molecules, and the study showed that the good agreement between the lowest rotational △E(l) and vibrational △E(n) excitation energies and the experimental of H2, HCl and O2 diatomic molecules. In view of the simplicity and accuracy, our work provides additional physical insights about the systems and sheds some light on this potential’s representative power.

We thank Wolf Gero Schmidt for his valuable intellectual comments and discussion.

1
Bagrov V G Gitman D M 1990 Exact Solutions of Relativistic Wave Equations Dordrecht Kluwer (http://openlibrary.org/books/OL2204930M)

2
Greiner W 1987 Relativistic Quantum Mechanics. Wave Equations Berlin Springer 3rd edn.

DOI

3
Dong S-H 2007 Factorization Method in Quantum Mechanics The Netherlands Springer

DOI

4
Boivin L Kärtner F X Haus H A 1994 Analytical solution to the quantum field theory of self-phase modulation with a finite response time Phys. Rev. Lett. 73 240 243

DOI

5
Bialynicki-Birula I 2004 Particle beams guided by electromagnetic vortices: new solutions of the Lorentz, Schrödinger, Klein–Gordon, and Dirac equations Phys. Rev. Lett. 93 020402

DOI

6
Belić M Petrović N Zhong W P Xie R H Chen G 2008 Analytical light bullet solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation Phys. Rev. Lett. 101 123904

DOI

7
Garavelli S L Oliveira F A 1991 Analytical solution for a Yukawa-type potential Phys. Rev. Lett. 66 1310 1313

DOI

8
Flügge S 1999 Practical Quantum Mechanics Berlin Springer

DOI

9
Schneider B Gharibnejad H 2020 Numerical methods every atomic and molecular theorist should know Nat. Rev. Phys. 2 89 102

DOI

10
Witten E 1981 Dynamical breaking of supersymmetry Nucl. Phys. B 188 513 554

DOI

11
Kreyszig E 1993 Advanced Engineering Mathematics New York Wiley

12
Grosche C 1995 Conditionally solvable path integral problems J. Phys. A: Math. Gen. 28 5889 5902

DOI

13
Nikiforov A F Uvarov V B 1988 Special Functions of Mathematical Physics Basel Birkhäuser

DOI

14
Ciftci H Hall R L Saad N 2003 Asymptotic iteration method for eigenvalue problems J. Phys. A: Math. Gen. 36 11807 11816

DOI

15
Ciftci H Hall R L Saad N 2005 Construction of exact solutions to eigenvalue problems by the asymptotic iteration method J. Phys. A: Math. Gen. 38 1147 1155

DOI

16
Ma Z Q Xu B W 2005 Quantum correction in exact quantization rules EPL 69 685 691

DOI

17
Qiang W-C Dong S-H 2010 Proper quantization rule EPL 89 10003

DOI

18
Serrano F A Cruz-Irisson M Dong S-H 2011 Proper quantization rule as a good candidate to semiclassical quantization rules Ann. Phys. 523 771 782

DOI

19
Polyakov A 1981 Quantum geometry of bosonic strings Phys. Lett. B 103 207 210

DOI

20
Vilenkin A 1994 Approaches to quantum cosmology Phys. Rev. D 50 2581 2594

DOI

21
Socorro J D’Oleire M 2010 Inflation from supersymmetric quantum cosmology Phys. Rev. D 82 044008

DOI

22
Rebesh A P Lev B I 2019 Analytical solutions of the classical and quantum cosmological models with an exponential potential Phys. Rev. D 100 123533

DOI

23
Song X Q Wang C W Jia C S 2017 Thermodynamic properties for the sodium dimer Chem. Phys. Lett. 673 50 55

DOI

24
Cole J T Makris K G Musslimani Z H Christodoulides D N Rotter S 2016 Twofold ${ \mathcal P }{ \mathcal T }$ symmetry in doubly exponential optical lattices Phys. Rev. 93 013803 A

DOI

25
Pivano A Dolocan V O 2020 Analytical description of the topological interaction between magnetic domain walls in nanowires Phys. Rev. 101 014438 B

DOI

26
Dechant A Kindermann F Widera A Lutz E 2019 Continuous-time random walk for a particle in a periodic potential Phys. Rev. Lett. 123 070602

DOI

27
Hoff D E M 2020 Mirror-symmetry violation in bound nuclear ground states Nature 580 52 55

DOI

28
Ruiz R F G 2020 Spectroscopy of short-lived radioactive molecules Nature 581 396 400

DOI

29
Jia C-S Diao Y-F Liu X-J Wang P-Q Liu J-Y Zhang G-D 2012 Equivalence of the Wei potential model and Tietz potential model for diatomic molecules J. Chem. Phys. 137 014101

DOI

30
Morse P M 1929 Diatomic molecules according to the wave mechanics. II. Vibrational levels Phys. Rev. 34 57 64

DOI

31
Berkdemir C Han J 2005 Any l-state solutions of the Morse potential through the Pekeris approximation and Nikiforov–Uvarov method Chem. Phys. Lett. 409 203 207

DOI

32
Hulthén L 1942 On the virtual state of the deuteron Ark. Mat. Astron. Fys. 29B 1

33
Bayrak O Kocak G Boztosun I 2006 Any l-state solutions of the Hulthén potential by the asymptotic iteration method J. Phys. A: Math. Gen. 39 11521 11529

DOI

34
Domínguez-Adame F 1989 Bound states of the Klein–Gordon equation with vector and scalar Hulthén-type potentials Phys. Lett. A 136 175 177

DOI

35
Ahmadov A I Aslanova S M Sh M Orujova S V Badalov Dong S-H 2019 Approximate bound state solutions of the Klein–Gordon equation with the linear combination of Hulthén and Yukawa potentials Phys. Lett. A 383 3010 3017

DOI

36
Ahmadov A I Aslanova S M Orujova M S Badalov S V 2021 Analytical bound state solutions of the Klein–Fock–Gordon equation for the sum of Hulthén and Yukawa potential within SUSY quantum mechanics Adv. High Energy Phys. 2021 8830063

DOI

37
Okon I B Popoola O Isonguyo C N 2017 Approximate solution of Schrodinger equation with some diatomic molecular interactions using Nikiforov–Uvarov method Adv. High Energy Phys. 2017 9671816

DOI

38
Okon I B Omugbe E Antia A D Onate C A Akpabio L E Osafile O E 2021 Spin and pseudospin solutions to Dirac equation and its thermodynamic properties using hyperbolic Hulthen plus hyperbolic exponential inversely quadratic potential Sci. Rep. 11 892

DOI

39
Woods R D Saxon D S 1954 Diffuse surface optical model for nucleon-nuclei scattering Phys. Rev. 95 577 578

DOI

40
Badalov V H Ahmadov H I Ahmadov A I 2009 Analytical solutions of the Schrödinger equation with the Woods–Saxon potential for arbitrary l state Int. J. Mod. Phys. E 18 631 641

DOI

41
Badalov V H Ahmadov H I Badalov S V 2010 Any l-state analytical solutions of the Klein–Gordon equation for the Woods–Saxon potential Int. J. Mod. Phys. E 19 1463 1475

DOI

42
Badalov V H Baris B Uzun K 2019 Bound states of the D-dimensional Schrödinger equation for the generalized Woods–Saxon potential Mod. Phys. Lett. A 34 1950107

DOI

43
Otsuka T Gade A Sorlin O Suzuki T Utsuno Y 2020 Evolution of shell structure in exotic nuclei Rev. Mod. Phys. 92 015002

DOI

44
Rosen N Morse P M 1932 On the vibrations of polyatomic molecules Phys. Rev. 42 210 217

DOI

45
Taşkın F 2009 Approximate solutions of the Schrödinger equation for the Rosen–Morse potential including centrifugal term Int. J. Theor. Phys. 48 2692 2697

DOI

46
Yi L Z Diao Y F Liu J Y Jia C S 2004 Bound states of the Klein–Gordon equation with vector and scalar Rosen–Morse-type potentials Phys. Lett. A 333 212 217

DOI

47
Soylu A Bayrak O Boztosun I 2008 Exact solutions of Klein–Gordon equation with scalar and vector Rosen–Morse-type potentials Chin. Phys. Lett. 25 2754 2757

DOI

48
Gu X-Y Dong S-H Ma Z-Q 2009 Energy spectra for modified Rosen–Morse potential solved by the exact quantization rule J. Phys. A: Math. Theor. 42 035303

DOI

49
Eckart C 1930 The penetration of a potential barrier by electrons Phys. Rev. 35 1303 1309

DOI

50
Dong S-H Qiang W-C Sun G-H Bezerra V B 2007 Analytical approximations to the l-wave solutions of the Schrödinger equation with the Eckart potential J. Phys. A: Math. Theor. 40 10535 10540

DOI

51
Liu X-Y Wei G-F Long C-Y 2009 Arbitrary wave relativistic bound state solutions for the Eckart potential Int. J. Theor. Phys. 48 463 470

DOI

52
Manning M F Rosen N 1933 A potential function for the vibrations of diatomic molecules Phys. Rev. 44 951 (Minutes of the Middletown Meeting, October 14, 1933) 10

DOI

53
Ahmadov A I Aydin C Uzun O 2014 Analytical solutions of the Klein–Fock–Gordon equation with the Manning–Rosen potential plus a ring-shaped-like potential Int. J. Mod. Phys. A 29 1450002

DOI

54
Ahmadov A I Demirci M Aslanova S M Mustamin M F 2020 Arbitrary l-state solutions of the Klein–Gordon equation with the Manning–Rosen plus a class of Yukawa potentials Phys. Lett. A 384 126372

DOI

55
Hamzavi M Ikhdair S M Thylwe K E 2013 Equivalence of the empirical shifted Deng-Fan oscillator potential for diatomic molecules J. Math. Chem. 51 227 238

DOI

56
Chen T Lin S-R Jia C-S 2013 Solutions of the Klein–Gordon equation with the improved Rosen–Morse potential energy model Eur. Phys. J. Plus 128 69

DOI

57
Sun G-H Aoki M A Dong S-H 2013 Quantum information entropies of the eigenstates for the Pöschl–Teller-like potential Chin. Phys. B 22 050302

DOI

58
Sun G-H Chen C-Y Taud H Yáñez-Márquez C Dong S-H 2020 Exact solutions of the 1D Schrödinger equation with the Mathieu potential Phys. Lett. A 384 126480

DOI

59
Dong Q Sun G-H Jing J Dong New S-H 2019 Findings for two new type sine hyperbolic potentials Phys. Lett. A 383 270 275

DOI

60
Schiöberg D 1986 The energy eigenvalues of hyperbolical potential functions Mol. Phys. 59 1123 1137

DOI

61
Ikhdair S M Sever R 2009 Improved analytical approximation to arbitrary l-state solutions of the Schrödinger equation for the hyperbolical potential Ann Phys. 521 189 197

DOI

62
Hu X-T Liu J-Y Jia C-S 2013 The ${3}^{3}{\sum }_{g}^{+}$ state of Cs2 molecule Comput. Theor. Chem. 1019 137 140

DOI

63
Wang P-Q Liu J-Y Zhang L-H Cao S-Y Jia C-S 2012 Improved expressions for the Schiöberg potential energy models for diatomic molecules J. Mol. Spectrosc. 278 23 26

DOI

64
Jia C S Zhang L H Wang C W 2017 Thermodynamic properties for the lithium dimer Chem. Phys. Lett. 667 211

DOI

65
Hua W 1990 Four-parameter exactly solvable potential for diatomic molecules Phys. Rev. A 42 2524 2529

DOI

66
Tietz T 1963 Potential-energy function for diatomic molecules J. Chem. Phys. 38 3036 3037

DOI

67
Jia C-S Zhang Y Zeng X-L Liang L-T 2001 Identity for the exponential-type molecule potentials and the supersymmetry shape invariance Commun. Theor. Phys. 36 641 646

DOI

68
Ikot A N Lütfüog˜lu B C Ngwueke M I Udoh M E Zare S Hassanabadi H 2016 Klein–Gordon equation particles in exponential-type molecule potentials and their thermodynamic properties in D dimensions Eur. Phys. J. Plus 131 419

DOI

69
Fu K-X Wang M Jia C-S 2019 Improved five-parameter exponential-type potential energy model for diatomic molecules Commun. Theor. Phys. 71 103 106

DOI

70
Tan M-S He S Jia C-S 2014 Molecular spinless energies of the improved Rosen–Morse potential energy model in D dimensions Eur. Phys. J. Plus 129 264

DOI

71
Liu J-Y Hu X-T Jia C-S 2014 Molecular energies of the improved Rosen–Morse potential energy model Can. J. Chem. 92 40 44

DOI

72
Hu X-T Zhang L-H Jia C-S 2014 D-dimensional energies for cesium and sodium dimers Can. J. Chem. 92 386 391

DOI

73
Araújo J P Ballester M Y 2021 A comparative review of 50 analytical representation of potential energy interaction for diatomic systems: 100 years of history Int. J. Quantum Chem. 121 e26808

DOI

74
Wang C-W Peng X-L Liu J-Y Jiang R Li X-P Liu Y-S Liu S-Y Wei L-S Zhang L-H Jia C-S 2022 A novel formulation representation of the equilibrium constant for water gas shift reaction Int. J. Hydrogen Energy 47 27821 27838

DOI

75
Okon I B Popoola O O Omugbe E Antia A D Isonguyo C N Ituen E E 2021 Thermodynamic properties and bound state solutions of Schrodinger equation with Mobius square plus screened-Kratzer potential using Nikiforov–Uvarov method Comput. Theor. Chem. 1196 113132

DOI

76
Okon I B Isonguyo C N Antia A D Ikot A N Popoola O O 2020 Fisher and Shannon information entropies for a noncentral inversely quadratic plus exponential Mie-type potential Commun. Theor. Phys. 72 065104

DOI

77
Jia C-S Wang C-W Zhang L-H Peng X-L Tang H-M Zeng R 2018 Enthalpy of gaseous phosphorus dimer Chem. Eng. Sci. 183 26 29

DOI

78
Jia C-S Zeng R Peng X-L Zhang L-H Zhao Y-L 2018 Entropy of gaseous phosphorus dimer Chem. Eng. Sci. 190 1 4

DOI

79
Peng X-L Jiang R Jia C-S Zhang L-H Zhao Y-L 2018 Gibbs free energy of gaseous phosphorus dimer Chem. Eng. Sci. 190 122 125

DOI

80
Ding Q-C Jia C-S Wang C-W Peng X-L Liu J-Y Zhang L-H Jiang R Zhu S-Y Yuan H Tang H-X 2023 Unified non-fitting formulation representation of thermodynamic properties for diatomic substances J. Mol. Liq. 371 121088

DOI

81
Liang D-C Zeng R Wang C-W Ding Q-C Wei L-S Peng X-L Liu J-Y Yu J Jia C-S 2022 Prediction of thermodynamic properties for sulfur dioxide J. Mol. Liq. 352 118722

DOI

82
Ding Q-C Jia C-S Liu J-Z Li J Du R-F Liu J-Y Peng X-L Wang C-W Tang H-X 2022 Prediction of thermodynamic properties for sulfur dimer Chem. Phys. Lett. 803 139844

DOI

83
Ahmadov H I Dadashov E A Huseynova N S Badalov V H 2021 Generalized tanh-shaped hyperbolic potential: bound state solution of Schrödinger equation Eur. Phys. J. Plus 136 244

DOI

84
Williams B W Poulios D P 1993 A simple method for generating exactly solvable quantum mechanical potentials Eur. J. Phys. 14 222 226

DOI

85
Peña J J Morales J García-Ravelo J 2017 Bound state solutions of Dirac equation with radial exponential-type potentials J. Math. Phys. 58 043501

DOI

86
Pekeris C L 1934 The rotation-vibration coupling in diatomic molecules Phys. Rev. 45 98 103

DOI

87
Abramowitz M Stegun I A 1965 Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables New York Dover (https://bibsonomy.org/bibtex/2bfc117729f3b97b4f2c1c9227e60ed1a/drmatusek)

88
Zhang Y P Cheng C H Kim J T Stanojevic J Eyler E E 2004 Dissociation energies of molecular hydrogen and the hydrogen molecular ion Phys. Rev. Lett. 92 203003

DOI

89
Hajigeorgiou P G 2010 An extended Lennard-Jones potential energy function for diatomic molecules: application to ground electronicstates J. Mol. Spectrosc. 263 101 110

DOI

90
Peña-Gallego A Abreu P E Varandas A J C 2000 MRCI calculation, scaling of the external correlation, and modeling of potential energy curves for HCl and OCl J. Phys. Chem. A 104 6241 6246

DOI

91
Komasa J Piszczatowski K Yach G Przybytek M Jeziorski B Pachucki K 2011 Quantum electrodynamics effects in rovibrational spectra of molecular hydrogen J. Chem. Theory Comput. 7 3105 3115

DOI

92
Salumbides E J Dickenson G D Ivanov T I Ubachs W 2011 QED effects in molecules: test on rotational quantum states of H2 Phys. Rev. Lett. 107 043005

DOI

93
Stanke M Kedziera D Bubin S Molski M Adamowicz L 2008 Orbit–orbit relativistic corrections to the pure vibrational non-Born-Oppenheimer energies of H2 J. Chem. Phys. 128 114313

DOI

94
Dickenson G D Niu M L Salumbides E J Komasa J Eikema K S E Pachucki K Ubachs W 2013 Fundamental vibration of molecular hydrogen Phys. Rev. Lett. 110 193601

DOI

Outlines

/