The Generalized Fractional NU Method for the Diatomic Molecules in the Deng-
Fan Model

M. Abu-Shady<EMAIL_ADDRESS>T. A. Abdel-Karim,
<EMAIL_ADDRESS>and E. M. Khokha<EMAIL_ADDRESS>

1,2 Department of Mathematics and Computer Science, Faculty of Science,
Menoufia University, Egypt

3 Department of Basic Science, Modern Academy for Engineering and Technology
in Maadi, Cairo, Egypt

###### Abstract

A solution of the fractional ${N}$-dimensional radial Schrödinger equation
(SE) with the Deng-Fan potential (DFP) is investigated by the generalized
fractional NU method. The analytical formulas of energy eigenvalues and
corresponding eigenfunctions for the DFP are generated. Furthermore, the
current results are applied to several diatomic molecules (DMs) for the DFP as
well as the shifted Deng-Fan potential (SDFP). For both the DFP and its
shifted potential, the effect of the fractional parameter (${\delta}$) on the
energy levels of various DMs is examined numerically and graphically. We found
that the energy eigenvalues are gradually improved when the fractional
parameter increases. The energy spectra of various DMs are also evaluated in
three-dimensional space and higher dimensions. It’s worthy to note that the
energy spectrum raises as the number of dimensions increases. In addition, the
dependence of the energy spectra of the DFP and its shifted potential on the
reduced mass, screening parameter, equilibrium bond length, rotational and
vibrational quantum numbers is illustrated. To validate our findings, we
estimate the energy levels of the DFP and SDFP at the classical case
(${\delta=1}$) for various DMs and found that they are entirely compatible
with earlier studies.

Keywords: ${N}$-dimensional Radial Schrödinger Equation; Generalized
Fractional Derivative; Diatomic Molecules; Nikiforov–Uvarov Method; Deng-Fan
Potential.

##  1 Introduction

Fractional calculus (FC) has attracted a huge number of researchers throughout
the last decades in the last and present century. The prominence of FC in
diverse disciplines of science and engineering has grown considerably [1]\-
[3]. The foundations of FC are non-integer order differentiation and
integration. In the literature, there are numerous definitions have been
suggested for the fractional differential equations. The definitions of
Jumarie [4], Riemann-Liouville [5] and Caputo [6] are the most appropriate for
physical circumstances and have received a lot of attention.

By using Jumarie-type derivative rules, Das et al. [7] -[8] investigated the
approximate solutions of the ${N}$-dimensional fractional SE for generalized
Mie-type potentials [7] and pseudoharmonic potential [8] for a typical DM, and
they obtained the mass spectra of quarkonia with the Cornell potential
corresponds to the fractional parameter ${\alpha=0.5}$. In addition they
employed the power series approach to investigate the solution of the
fractional Klein–Gordon (KG) equation with fractional scalar and vector and
potentials [9].

Al-Raeei and El-Daher relied on the definition of Riemann–Liouville fractional
derivative with a numerical technique to solve the space-dependent fractional
SE for the Coulomb potential [10], Van Der Walls potential [11], Lennard-Jones
potential [12] and Morse potential [13].

Recently, Khalil [14] has proposed a new definition of fractional derivative
termed conformable fractional derivative (CFD) which upholds essential
classical characteristics. Abdeljawad [15] extended the definition and
established the fundamental notions of the CFD. Depending on the CFD, Karayer
et al. [16] presented conformable fractional (CF) NU method to investigate the
solutions of the SE with different potentials.

Chung et al.[17] used the Heun function to investigate the CF SE for the
Killingbeck and hyperbolic potentials. Within the context of CF quantum
mechanics, the three-dimensional fractional harmonic oscillator was explored
and some expectation values were obtained in Ref. [18]. The CF NU approach was
used to obtain the one-dimensional KG equation for the generalized Hulthén
potential [19]. In Ref. [20], the fractional SE for a particle with position-
dependent mass in an infinite potential well was studied using the CFD. In the
context of the CFD, the ${N}$-dimensional radial SE was used to investigate
the properties of heavy quarkonia for the dependent temperature potential
[21], Trigonometric Rosen–Morse potential [22], hot-magnetized interaction
potential [23], and generalized Cornell potential [24]. The impact of
fraction-order and dimensional number on heavy-quarkonium masses was also
examined. Hammad et al. [25] used the CF NU method to develop the solutions of
the CF Bohr Hamiltonian with the Kratzer potential for triaxial nuclei. Abu-
Shady [26] explored the mathematical model for the Coronavirus Disease 2019
using the concept of the CFD. The fractional SE with the screened Kratzer
potential [27] and Morse potential [28] was solved using the CF NU technique
and the energies of several DMs were estimated for various fractional values.

The investigation of DMs is a prominent branch of chemistry and molecular
physics. Multiple authors have recently examined the solution of relativistic
and nonrelativistic wave equations in order to understand the characteristics
of some DMs using different molecular potentials. There is a considerable
number of empirical potentials in the literature that provide a proper
description of the interactions between atoms in the DM as the Kratzer [29],
Morse [30], Pöschl-Teller [31], Manning-Rosen [32], Hulthén [33], Schlöberg
[34], and Tietz-Hua potentials [35].

The DFP is generally renowned as one of the most empirical potentials for
accurately describing the DM energy spectrum and electromagnetic transitions.
In 1957, Deng and Fan [36] postulated the DFP to explain the ro-vibrational
spectrum of the DM, also called the general Morse potential. The DFP is
regarded an empirical potential because it has the proper physical boundary
conditions at the origin and infinity. The DFP takes the form [37]\- [38]

$V(r)=D_{e}\Big{(}1-\frac{be^{-\alpha r}}{1-e^{-\alpha r}}\Big{)}^{2},\qquad
b=e^{\alpha r_{e}}-1,\qquad r\in(0,\infty)$ (1)

The SDFP is also defined as [39]

$V(r)=D_{e}\Big{(}1-\frac{be^{-\alpha r}}{1-e^{-\alpha r}}\Big{)}^{2}-D_{e},$
(2)

where ${D_{e}}$, ${r_{e}}$ and ${\alpha}$ are the dissociation energy, the
equilibrium bond length and the screening parameter which indicates the radius
of the potential, respectively.

Copious theoretical studies were reported to derive the solutions of the DFP
and SDFP in both relativistic and non-relativistic regimes . For example, the
approximate solutions of the DFP were investigated using the hypergeometric
functions with the SE [37] and with Klein–Gordon equation [38]. Zhang et al.
[39] solved the SE with the DFP using the supersymmetric shape invariance
formalism and also computed the rotational transition frequencies for HF
molecule. By using the exact quantization rule method, Falaye et al. [40]
obtained the solutions of the SE with the DFP and calculated the ro-
vibrational energy of some DMs. Ikhdair [41] studied the solutions of the
Dirac equation for the DFP with spin and pseudospin symmetries by the NU
method. Oluwadare et al. [42] solved the KG and Dirac equations using the NU
method with the DFP. Mustafa [43] calculated the vibrational energies of Li2
DM with the SE for the DFP. The NU technique was used to solve the SE with the
DFP [44] and SDFP [45], as well as the energy levels for a set of DMs were
also reported.

The solutions of the SE with the SDFP were found and the ro-vibrational
energies for various DMs are also computed using the asymptotic iteration
method [46], and the generalized pseudospectral method [47]. Oluwadare and
Oyewumi [48] used the proper quantization rule to explore the approximate
solutions of the SE with the SDFP, as well as the energy levels and the
expectation values of DMs were investigated. The Feynman path integral method
[49] was employed to explore the approximate solutions of the SDFP.
Furthermore, the energy eigenvalues and the eigenfunctions were investigated
for some DMs. Omugbe [50] used the WKB method to estimate the energy spectrum
of the SE with the DFP for HCl, LiH, and ScH molecules. Nath and Roy [51]
utilized the NU method to evaluate the ro-vibrational energy and thermodynamic
characteristics of the DFP for H2, LiH, HCl, and CO molecules. The numerical
values of energy and thermodynamic quantities of K2 molecule with the DFP were
reported by Onyenegecha et al. [52] using the formula technique. The KG
equation with the DFP was examined by the SWKB and WKB methods [53].
Furthermore, the vibrational energies and scattering phase shift of HCl and
LiH molecules were discusseed in the higher dimensions using the NU functional
analysis [54].

The study of diverse problems of quantum mechanics in the higher dimensional
space has recently piqued the interest of theoretical physicists. Since the
higher dimension studies aid in a broad approach to the problem, allowing the
desired outcomes in the lower dimensions to be acquired directly by dialing
appropriate with ${N}$. Therefore, a wide number of works [55]-[66] have been
reported to derive the energy spectra of SE solutions in ${N}$-dimensional
space.

Abu-Shady and Kaabar [67] recently proposed a new definition for the
fractional derivative known as the generalized fractional derivative (GFD).
The GFD definition is regarded as a comprehensive type for the fractional
derivative because it provides more features than the other definitions [4, 5,
6, 14]. Where the CFD can be simply obtained as a special case from the GFD.

As a novel investigation, we employ the perception of the GFD to derive the
bound state solutions of the ${N}$-dimensional SE with the DFP via the GF NU
method motivated by the recent work [67]. Moreover, we explore the impact of
the fractional order and the dimensional number on the ro-vibrational energy
spectra of multiple DMs.

This investigation is organized as follows: In Sec. 2, the theoretical tools
required for this investigation are displayed. In Sec. 3, we get the bound
state solutions of the ${N}$-dimensional SE for the DFP via the GFD NU method.
In Sec. 4, the numerical estimates of the ro-vibrational energy levels of
several DMs are reported. Finally, in Sec.5, we provide the conclusion of our
investigation.

##  2 Mathematical Tools

###  2.1 Overview of the fractional derivative definitions

Here, we introduce some of the basic definitions of the fractional derivative.
For ${\delta\in(n-1,n)}$, the Riemann–Liouville definition is defined as
[68]\- [69]

$\mathcal{D}^{RL}{f(r)}=\frac{1}{\Gamma(n-\delta)}\frac{d^{n}}{dr^{n}}\int_{r_{0}}^{r}\frac{f(s)}{(x-s)^{\delta-n+1}}\,ds.$
(3)

and the Caputo definition is also defined as [68]\- [69]

$\mathcal{D}^{C}{f(r)}=\frac{1}{\Gamma(n-\delta)}\int_{r_{0}}^{r}\frac{f^{n}(s)}{(r-s)^{\delta-n+1}}\,ds.$
(4)

Due to the fact that the Riemann–Liouville and Caputo definitions do not
satisfy some classical features such as chain, product, and quotient rules,
Khalil et al suggested a new fractional derivative definition Known as the
conformable fractional derivative (CFD) to overcome this problem. The CFD has
the following form [14]

$\mathcal{D}^{CF}{f(r)}=\lim_{\varphi\to 0}\frac{f(r+\varphi
r^{1-\delta})-f(r)}{\varphi};\qquad 0<\varphi\leqslant
1,\qquad\mathcal{D}^{CF}{f(0)}=\lim_{r\to 0^{+}}f^{CF}(r).$ (5)

where ${\mathcal{D}^{CF}}$ is the local fractional derivative operator which
satisfies the interesting properties that traditional fractional derivatives
do not, such as the formula of the derivative of the product or quotient of
two functions and the chain rule.

More recently, Abu-Shady and Kaabar [67] proposed a new definition for the
fractional derivative known as the GFD that has the form:

$\mathcal{D}^{GF}{f(r)}=\lim_{\varphi\to
0}\frac{f\Big{(}r+\frac{\Gamma(\gamma)}{\Gamma(\gamma-\delta+1)}\varphi
r^{1-\delta}\Big{)}-f(r)}{\varphi};\qquad\gamma>-1,\gamma\in
R^{+},\qquad\mathcal{D}^{GF}{f(0)}=\lim_{r\to 0^{+}}f^{GF}(r).$ (6)

The main advantage of the GFD is that it fulfills the property:
${\mathcal{D}^{\delta}{\mathcal{D}^{\gamma}f(r)}=\mathcal{D}^{\delta+\gamma}f(r)}$,
for a differentiable function ${f(r)}$, which is not satisfied by the CFD.

###  2.2 Generalized fractional NU method

In this subsection, our aim is to generalize the NU method into the framework
of GFD. The NU method [70] is utilized to solve the second-order differential
equation which has the following form:

$\mathcal{H}^{\prime\prime}(\rho)+\frac{\tilde{\tau}(\rho)}{\sigma(\rho)}\mathcal{H}^{\prime}(\rho)+\frac{\tilde{\sigma}(\rho)}{\sigma^{2}(\rho)}\mathcal{H}(\rho),$
(7)

where ${\sigma(\rho)}$ and ${\tilde{\sigma}(\rho)}$ are functions of maximum
second degree and ${\tilde{\tau}(\rho)}$ is a function of maximum first
degree.

Now we consider the generalized fractional differential equation of the
following form:

$\mathcal{D}^{\delta}[\mathcal{D}^{\delta}\mathcal{H}(\rho)]+\frac{\tilde{\tau}(\rho)}{\sigma(\rho)}\mathcal{D}^{\delta}\mathcal{H}(\rho)+\frac{\tilde{\sigma}(\rho)}{\sigma^{2}(\rho)}\mathcal{H}(\rho),$
(8)

Using the key property of the GFD,

$\mathcal{D}^{\delta}\mathcal{H}(\rho)=Q\rho^{1-\delta}\mathcal{H}^{\prime}(\rho),$
(9)
$\mathcal{D}^{\delta}[\mathcal{D}^{\delta}\mathcal{H}(\rho)]=Q^{2}\Big{[}\rho^{2(1-\delta)}\mathcal{H}^{\prime\prime}(\rho)+(1-\delta)\rho^{1-2\delta}\mathcal{H}^{\prime}(\rho)\Big{]},$
(10)

where

$Q=\frac{\Gamma(\gamma)}{\Gamma(\gamma-\delta+1)}.$ (11)

and substituting Eqs. (9) and (10) into Eq. (8) we obtain

$\mathcal{H}^{\prime\prime}(\rho)+\frac{Q(1-\delta)\rho^{-\delta}\sigma(\rho)+\tilde{\tau}(\rho)}{Q\rho^{1-\delta}\sigma(\rho)}\mathcal{H}^{\prime}(\rho)+\frac{\tilde{\sigma}(\rho)}{Q^{2}\rho^{2-2\delta}\sigma^{2}(\rho)}\mathcal{H}(\rho),$
(12)

Eq. (11) can be reduced to a hypergeometric type equation as follows:

$\mathcal{H}^{\prime\prime}(\rho)+\frac{\tilde{\tau}_{GF}(\rho)}{\sigma_{GF}(\rho)}\mathcal{H}^{\prime}(\rho)+\frac{\tilde{\sigma}(\rho)}{\sigma_{GF}^{2}(\rho)}\mathcal{H}(\rho),$
(13)

where

$\tilde{\tau}_{GF}(\rho)=Q(1-\delta)\rho^{-\delta}\sigma(\rho)+\tilde{\tau}(\rho),\qquad\sigma_{GF}(\rho)=Q\rho^{1-\delta}\sigma(\rho).$
(14)

where the subscript GF stands for the generalized fractional, and
${\sigma(\rho)}$ and ${\tilde{\sigma}(\rho)}$ are polynomials of maximum
${2\delta}$-th degree and ${\tilde{\tau}(\rho)}$ is a function at most
${\delta}$-th degree.

Setting

$\mathcal{H}(\rho)=\chi(\rho)\mathcal{V}(\rho).$ (15)

Substituting Eq. (15) into Eq. (13), we obtain

$\sigma_{GF}(\rho)\mathcal{V}^{\prime\prime}(\rho)+\tau_{GF}(\rho)\mathcal{V}^{\prime}(\rho)+\lambda(\rho)\mathcal{V}(\rho)=0.$
(16)

where ${\chi(\rho)}$ satisfies the following relation

$\frac{\chi^{\prime}(\rho)}{\chi(\rho)}=\frac{\pi_{GF}(\rho)}{\sigma_{GF}(\rho)}.$
(17)

and

$\lambda(\rho)=k(\rho)+\pi_{GF}^{\prime}(\rho).$ (18)

Here, ${\mathcal{V}(\rho)}=\mathcal{V}_{n}(\rho)$ is a hypergeometric-type
function, whose polynomial solutions are given by the Rodrigues formula

$\mathcal{V}_{n}(\rho)=\frac{G_{n}}{\omega(\rho)}\frac{d^{n}}{d\rho^{n}}[\sigma_{GF}^{n}(\rho)\omega(\rho)],$
(19)

where ${G_{n}}$ is the normalization constant, and ${\omega(\rho)}$ is a
weight function that defined as follows [51]:

$\omega(\rho)=\Big{[}\sigma_{GF}(\rho)\Big{]}^{-1}\text{exp}\Big{(}\int\frac{\tau_{GF}(\rho)}{\sigma_{GF}(\rho)}\,d\rho\Big{)}$
(20)

The function ${\pi_{GF}(\rho)}$ has the following form:

$\pi_{GF}(\rho)=\frac{\sigma_{GF}^{\prime}(\rho)-\tilde{\tau}_{GF}(\rho)}{2}\pm\sqrt{\Bigg{[}\frac{\sigma_{GF}^{\prime}(\rho)-\tilde{\tau}_{GF}(\rho)}{2}\Bigg{]}^{2}-\tilde{\sigma}(\rho)+k(\rho)\sigma_{GF}(\rho)},$
(21)

The value of ${k(\rho)}$ can be determined if the function under the square
root is the square of a polynomial. Thus, the equation of the eigenvalues can
be obtained from the following relation:

$\lambda(\rho)=\lambda_{n}(\rho)=-n\Big{[}\tau_{GF}^{\prime}(\rho)+\frac{(n-1)}{2}\sigma_{GF}^{\prime\prime}(\rho)\Big{]},$
(22)

where

$\tau_{GF}(\rho)=\tilde{\tau}_{GF}(\rho)+2\pi_{GF}(\rho).$ (23)

Finally, the eigenfunctions ${\mathcal{H}(\rho)}$ can be determined from Eq.
(15) using Eq. (17) and Eq. (19).

##  3 Solution of the generalized fractional SE for the DFP

The ${N}$-dimensional radial SE for a DM can be written as [55]\- [66]

$\Biggl{\\{}\frac{d^{2}}{dr^{2}}+\frac{N-1}{r}\frac{d}{dr}-\frac{l(l+N-2)}{r^{2}}+\frac{2\mu}{\hbar^{2}}\Big{(}E-V(r)\Big{)}\Biggl{\\}}\phi(r)=0,$
(24)

where ${\mu,\hbar,E}$ and ${N}$ are the reduced mass, the reduced Planck’s
constant, the energy spectrum and the dimensional number respectively. Let us
set,

$\phi(r)=\frac{1}{r^{\frac{N-1}{2}}}F(r).$ (25)

Eq. (24) becomes

$\frac{d^{2}F(r)}{dr^{2}}+\Bigg{[}\frac{2\mu}{\hbar^{2}}\Big{(}E-V(r)\Big{)}-\frac{(\eta^{2}-\frac{1}{4})}{r^{2}}\Bigg{]}F(r)=0,$
(26)

with

$\eta=l+\frac{N-2}{2},$ (27)

Here we consider a general form of the DFP as follows:

$V(r)=D_{e}\Big{(}1-\frac{be^{-\alpha r}}{1-e^{-\alpha
r}}\Big{)}^{2}+V_{0},\qquad b=e^{\alpha r_{e}}-1,\qquad r\in(0,\infty)$ (28)

The DFP can be found by setting ${V_{0}=0}$. In the case if ${V_{0}=-D_{e}}$,
Eq. (28) turns to the SDFP. By inserting Eq. (28) into Eq. (26), we obtain:

$\frac{d^{2}F(r)}{dr^{2}}+\Biggl{\\{}\frac{2\mu}{\hbar^{2}}\Bigg{[}E-D_{e}\Big{(}1-\frac{be^{-\alpha
r}}{(1-e^{-\alpha
r})}\Big{)}^{2}-V_{0}\Bigg{]}-\frac{(\eta^{2}-\frac{1}{4})}{r^{2}}\Bigg{)}\Biggl{\\}}F(r)=0,$
(29)

Now we utilize a proper approximation for the term ${1/r^{2}}$ as [51]

$\frac{1}{r^{2}}\approx\alpha^{2}\Bigg{[}c_{0}+\frac{e^{-\alpha
r}}{(1-e^{-\alpha r})}+\frac{e^{-2\alpha r}}{(1-e^{-\alpha
r})^{2}}\Bigg{]};\qquad c_{0}=\frac{1}{12}.$ (30)

Inserting Eq. (30) into Eq. (29), we obtain:

$\frac{d^{2}F(r)}{dr^{2}}+\Biggl{\\{}\frac{2\mu}{\hbar^{2}}\Bigg{[}E-D_{e}\Big{(}1-\frac{be^{-\alpha
r}}{(1-e^{-\alpha
r})}\Big{)}^{2}-V_{0}\Bigg{]}-\alpha^{2}\Big{(}\eta^{2}-\frac{1}{4}\Big{)}\Bigg{[}c_{0}+\frac{e^{-\alpha
r}}{(1-e^{-\alpha r})}+\frac{e^{-2\alpha r}}{(1-e^{-\alpha
r})^{2}}\Bigg{]}\Biggl{\\}}F(r)=0,$ (31)

By defining the variable ${\rho=e^{-\alpha r}}$, Eq. (31) turns to

$F^{\prime\prime}(\rho)+\frac{(1-\rho)}{\rho(1-\rho)}F^{\prime}(\rho)+\frac{1}{\rho^{2}(1-\rho)^{2}}\Big{[}-\mathcal{A}\rho^{2}+\mathcal{B}\rho-\mathcal{C}\Big{]}F(\rho)=0,$
(32)

where

$\mathcal{A}=c_{0}\Big{(}\eta^{2}-\frac{1}{4}\Big{)}+\frac{2\mu}{\alpha^{2}\hbar^{2}}\Big{[}V_{0}+D_{e}(b+1)^{2}\Big{]}-\epsilon,$
(33)
$\mathcal{B}=(2c_{0}-1)\Big{(}\eta^{2}-\frac{1}{4}\Big{)}+\frac{4\mu}{\alpha^{2}\hbar^{2}}\Big{[}V_{0}+D_{e}(b+1)\Big{]}-2\epsilon,$
(34)
$\mathcal{C}=c_{0}\Big{(}\eta^{2}-\frac{1}{4}\Big{)}+\frac{2\mu}{\alpha^{2}\hbar^{2}}\Big{(}V_{0}+D_{e}\Big{)}-\epsilon,$
(35)

with

$\epsilon=\frac{2\mu E}{\alpha^{2}\hbar^{2}}$ (36)

The generalized fractional form of the SE for the DFP can be stated by
changing integer orders with fractional orders in Eq. (32) as follows:

$\mathcal{D}^{\delta}\mathcal{D}^{\delta}F(\rho)+\frac{(1-\rho^{\delta})}{\rho^{\delta}(1-\rho^{\delta})}\mathcal{D}^{\delta}F(\rho)+\frac{1}{\rho^{2\delta}(1-\rho^{\delta})^{2}}\Big{[}-\mathcal{A}\rho^{2\delta}+\mathcal{B}\rho^{\delta}-\mathcal{C}\Big{]}F(\rho)=0,$
(37)

Substituting Eqs. (9) and (10) into Eq. (37), we obtain

$F^{\prime\prime}(\rho)+\frac{\Big{[}1+Q(1-\delta)\Big{]}(1-\rho^{\delta})}{Q\rho(1-\rho^{\delta})}F^{\prime}(\rho)+\frac{1}{Q^{2}\rho^{2}(1-\rho^{\delta})^{2}}\Big{[}-\mathcal{A}\rho^{2\delta}+\mathcal{B}\rho^{\delta}-\mathcal{C}\Big{]}F(\rho)=0,$
(38)

By comparing Eq. (38) with Eq. (13), we get the following functions:

$\tilde{\tau}_{GF}(\rho)=\Big{(}1+Q(1-\delta)\Big{)}(1-\rho^{\delta}),\qquad\sigma_{GF}(\rho)=Q\rho(1-\rho^{\delta}),\qquad\tilde{\sigma}_{GF}(\rho)=-\mathcal{A}\rho^{2\delta}+\mathcal{B}\rho^{\delta}-\mathcal{C}.$
(39)

The function ${\pi_{GF}(\rho)}$ can be defined after substituting Eq. (39)
into Eq. (21) as follows:

$\begin{split}&\pi_{GF}(\rho)=\frac{(Q\delta-1)+(1-2Q\delta)\rho^{\delta}}{2}\pm\\\
&\sqrt{\Big{[}\frac{(1-2Q\delta)^{2}}{4}+\mathcal{A}-Qk\rho^{1-\delta}\Big{]}\rho^{2\delta}+\Big{[}\frac{(Q\delta-1)(1-2Q\delta)}{2}-\mathcal{B}+Qk\rho^{1-\delta}\Big{]}\rho^{\delta}+\Big{[}\frac{(Q\delta-1)^{2}}{4}+\mathcal{C}\Big{]}}.\end{split}$
(40)

Eq. (40) can be written in the following form:

$\pi_{GF}(\rho)=\frac{(Q\delta-1)+(1-2Q\delta)\rho^{\delta}}{2}\pm\sqrt{P\rho^{2\delta}+W\rho^{\delta}+R},$
(41)

where

$P=T_{1}-Qk\rho^{1-\delta},\qquad W=T_{2}+Qk\rho^{1-\delta},\qquad R=T_{3},$
(42)

with

$T_{1}=\frac{(1-2Q\delta)^{2}}{4}+\mathcal{A},\qquad
T_{2}=\frac{(Q\delta-1)(1-2Q\delta)}{2}-\mathcal{B},\qquad
T_{3}=\frac{(Q\delta-1)^{2}}{4}+\mathcal{C}.$ (43)

We can get the two possible roots of ${k}$ by employing the condition that the
discriminant of the expression under the square root of Eq. (41) must be zero.
Hence,

$k_{\pm}=\Lambda\Big{[}-(T_{2}+2T_{3})\pm
2\sqrt{T_{3}(T_{1}+T_{2}+T_{3})}\Big{]}\rho^{\delta-1};\qquad\Lambda=\frac{1}{Q}.$
(44)

Inserting Eq. (44) into Eq. (41), we have

$\pi_{GF}(\rho)=\frac{(Q\delta-1)+(1-2Q\delta)\rho^{\delta}}{2}\pm\begin{cases}\big{(}\sqrt{T_{3}}-\sqrt{T_{1}+T_{2}+T_{3}}\big{)}\rho^{\delta}-\sqrt{T_{3}},\qquad
k=k_{+}\\\
\big{(}\sqrt{T_{3}}+\sqrt{T_{1}+T_{2}+T_{3}}\big{)}\rho^{\delta}-\sqrt{T_{3}},\qquad
k=k_{-}\\\ \end{cases}.$ (45)

We select the negative sign in Eq. (45) in order to obtain a physically
acceptable solution. Hence, the function ${\pi_{GF}(\rho)}$ becomes

$\pi^{-}_{GF}(\rho)=\frac{(Q\delta-1)+(1-2Q\delta)\rho^{\delta}}{2}-\Big{[}\big{(}\sqrt{T_{3}}+\sqrt{T_{1}+T_{2}+T_{3}}\big{)}\rho^{\delta}-\sqrt{T_{3}}\Big{]},$
(46)

and

$k_{-}=-\Lambda\Big{[}T_{2}+2T_{3}+2\sqrt{T_{3}(T_{1}+T_{2}+T_{3})}\Big{]}\rho^{\delta-1}.$
(47)

Therefore, the functions ${\lambda(\rho),\tau_{GF}(\rho)}$ and
${\lambda_{n}(\rho)}$ can be found as follows:

$\lambda(\rho)=\Bigg{[}-\Lambda(T_{2}+2T_{3})+\delta\Bigg{(}\frac{1}{2}\Big{(}1-2Q\delta\Big{)}-\sqrt{T_{3}}\Bigg{)}-\sqrt{T_{1}+T_{2}+T_{3}}\Big{(}\delta+2\Lambda\sqrt{T_{3}}\Big{)}\Bigg{]}\rho^{\delta-1},$
(48)
$\tau_{GF}(\rho)=\Big{(}2\sqrt{T_{3}}+Q\Big{)}-\Big{[}Q(\delta+1)+2\Big{(}\sqrt{T_{3}}+\sqrt{T_{1}+T_{2}+T_{3}}\Big{)}\Big{]}\rho^{\delta},$
(49)
$\lambda_{n}(\rho)=n\delta\Big{[}\frac{Q(n+1)(\delta+1)}{2}+2\Big{(}\sqrt{T_{3}}+\sqrt{T_{1}+T_{2}+T_{3}}\Big{)}\Big{]}\rho^{\delta-1}.$
(50)

By equating Eqs. (48) and (50), the energy spectrum formula for a DM in the
${N}$ dimensional space can be written as follows:

$E^{N}_{nl}=\frac{\alpha^{2}\hbar^{2}}{2\mu}\Biggl{\\{}\xi_{3}-\Bigg{(}\frac{\delta\Big{[}\frac{1}{2}\Big{(}1-2Q\delta-
Qn(n+1)(\delta+1)\Big{)}-(2n+1)\sqrt{\xi_{1}+\xi_{2}+\xi_{3}}-\Lambda(\xi_{2}+2\xi_{3})\Big{]}}{\Big{(}\delta(2n+1)+2\Lambda\sqrt{\xi_{1}+\xi_{2}+\xi_{3}}\Big{)}}\Bigg{)}^{2}\Biggl{\\}},$
(51)

where

$\xi_{1}=\frac{(1-2Q\delta)^{2}}{4}+c_{0}\Big{(}\eta^{2}-\frac{1}{4}\Big{)}+\frac{2\mu}{\alpha^{2}\hbar^{2}}\Big{[}V_{0}+D_{e}(b+1)^{2}\Big{]},$
(52)
$\xi_{2}=\frac{(Q\delta-1)(1-2Q\delta)}{2}-(2c_{0}-1)\Big{(}\eta^{2}-\frac{1}{4}\Big{)}-\frac{4\mu}{\alpha^{2}\hbar^{2}}\Big{[}D_{e}(b+1)+V_{0}\Big{]},$
(53)
$\xi_{2}=\frac{(Q\delta-1)^{2}}{4}+c_{0}\Big{(}\eta^{2}-\frac{1}{4}\Big{)}+\frac{2\mu}{\alpha^{2}\hbar^{2}}\Big{(}D_{e}+V_{0}\Big{)}.$
(54)

Now we determine the corresponding eigenfunctions. Using Eq. (17) the function
${\chi(\rho)}$ is

$\chi(\rho)=\rho^{\Big{(}\frac{(Q\delta-1)}{2}+\sqrt{\xi_{3}-\epsilon}\Big{)}}\Big{(}1-\rho^{\delta}\Big{)}^{\Big{(}{\frac{1}{2}+\frac{\Lambda}{\delta}\sqrt{\xi_{1}+\xi_{2}+\xi_{3}}}\Big{)}}.$
(55)

The weight function ${\omega(\rho)}$ is found using Eq. (20) as

$\omega(\rho)=\Lambda\rho^{\Big{(}2\Lambda\sqrt{\xi_{3}-\epsilon}\Big{)}}\Big{(}1-\rho^{\delta}\Big{)}^{\Big{(}{\frac{2\Lambda}{\delta}\sqrt{\xi_{1}+\xi_{2}+\xi_{3}}}\Big{)}}.$
(56)

Using Eq. (19), the expression for the function ${\mathcal{V}_{n}(\rho)}$
takes the form

$\mathcal{V}_{n}(\rho)=G_{n}\rho^{\Big{(}-2\Lambda\sqrt{\xi_{3}-\epsilon}\Big{)}}\Big{(}1-\rho^{\delta}\Big{)}^{\Big{(}{\frac{-2\Lambda}{\delta}\sqrt{\xi_{1}+\xi_{2}+\xi_{3}}}\Big{)}}\frac{d^{n}}{d\rho^{n}}\Bigg{[}Q^{n}\rho^{\Big{(}n+2\Lambda\sqrt{\xi_{3}-\epsilon}\Big{)}}\Big{(}1-\rho^{\delta}\Big{)}^{\Big{(}{n+\frac{2\Lambda}{\delta}\sqrt{\xi_{1}+\xi_{2}+\xi_{3}}}\Big{)}}\Bigg{]}.$
(57)

Finally, using Eq. (15), we get the complete solution of Eq. (32) as

$F(\rho)=G_{n}\rho^{\Lambda\Big{(}\frac{(Q\delta-1)}{2}-\sqrt{\xi_{3}-\epsilon}\Big{)}}\Big{(}1-\rho^{\delta}\Big{)}^{\Big{(}{\frac{1}{2}-\frac{\Lambda}{\delta}\sqrt{\xi_{1}+\xi_{2}+\xi_{3}}}\Big{)}}\frac{d^{n}}{d\rho^{n}}\Bigg{[}Q^{n}\rho^{\Big{(}n+2\Lambda\sqrt{\xi_{3}-\epsilon}\Big{)}}\Big{(}1-\rho^{\delta}\Big{)}^{\Big{(}{n+\frac{2\Lambda}{\delta}\sqrt{\xi_{1}+\xi_{2}+\xi_{3}}}\Big{)}}\Bigg{]}.$
(58)

(a)

(b)

Figure 1: Shape of DFP and SDFP for some DMs. Table 1: Spectroscopic parameters for various DMs [48].  Molecule |  ${r_{e}}$ (${\overset{o}{A}}$) |  ${\alpha}$ (${\overset{o}{A}^{-1}}$) |  ${\mu}$ (a.m.u.) |  ${D_{e}}$ ${(cm^{-1})}$
---|---|---|---|---
$\begin{array}[]{c}\text{NO}\\\ \text{CO}\\\ \text{I}_{2}\\\ \text{N}_{2}\\\ \text{O}_{2}\\\ \text{H}_{2}\\\ \text{HF}\\\ \text{LiH}\\\ \text{ScH}\\\ \text{HCl}\\\ \end{array}$ | $\begin{array}[]{c}1.1508\\\ 1.1282\\\ 2.6620\\\ 1.0940\\\ 1.2070\\\ 0.7416\\\ 0.9170\\\ 1.5956\\\ 1.7080\\\ 1.2746\\\ \end{array}$ | $\begin{array}[]{c}2.7534\\\ 2.2994\\\ 1.8643\\\ 2.6989\\\ 2.6636\\\ 1.9426\\\ 2.2266\\\ 1.1280\\\ 1.5068\\\ 1.8677\\\ \end{array}$ | $\begin{array}[]{c}7.468441\\\ 6.860586\\\ 63.452235\\\ 7.00335\\\ 7.997457504\\\ 0.50391\\\ 0.96367\\\ 0.8801221\\\ 10.682771\\\ 0.9801045\\\ \end{array}$ | $\begin{array}[]{c}64877.06229\\\ 87471.42567\\\ 12758.0129\\\ 96288.03528\\\ 41591.26201\\\ 38267.78314\\\ 49382\\\ 20287.13295\\\ 36778.8836\\\ 37255.24414\\\ \end{array}$
Table 2: ${E_{nl}}$ (eV) of DFP for CO molecule. |  | ${N=3}$ | ${N=3}$ | ${\delta=1}$
---|---|---|---|---
${n}$ | ${l}$ | ${\delta=0.2}$ | ${\delta=0.5}$ | ${\delta=0.8}$ | ${\delta=1}$ | ${N=4}$ | ${N=5}$ | ${N=6}$
$\begin{array}[]{c}0\\\ 1\\\ \\\ 2\\\ \\\ \\\ 3\\\ \\\ \\\ \\\ \end{array}$ | $\begin{array}[]{c}0\\\ 0\\\ 1\\\ 0\\\ 1\\\ 2\\\ 0\\\ 1\\\ 2\\\ 3\\\ \end{array}$ | $\begin{array}[]{c}0.03089963\\\ 0.09208239\\\ 0.09263068\\\ 0.15310157\\\ 0.15364899\\\ 0.15474383\\\ 0.21395707\\\ 0.21450363\\\ 0.21559675\\\ 0.21723640\\\ \end{array}$ | $\begin{array}[]{c}0.08051605\\\ 0.24044444\\\ 0.24099063\\\ 0.39912230\\\ 0.39966624\\\ 0.40075410\\\ 0.55655092\\\ 0.55709261\\\ 0.55817599\\\ 0.55980101\\\ \end{array}$ | $\begin{array}[]{c}0.12410601\\\ 0.36996537\\\ 0.37050972\\\ 0.61277480\\\ 0.61331569\\\ 0.61439745\\\ 0.85254182\\\ 0.85307925\\\ 0.85415410\\\ 0.85576633\\\ \end{array}$ | $\begin{array}[]{c}0.14236930\\\ 0.42399536\\\ 0.42453895\\\ 0.70157568\\\ 0.70211529\\\ 0.70319450\\\ 0.97512297\\\ 0.97565862\\\ 0.97672990\\\ 0.97833679\\\ \end{array}$ | $\begin{array}[]{c}0.14257464\\\ 0.42419921\\\ 0.42501458\\\ 0.70177804\\\ 0.70258745\\\ 0.70393644\\\ 0.97532384\\\ 0.97612730\\\ 0.97746640\\\ 0.97934108\\\ \end{array}$ | $\begin{array}[]{c}0.14291688\\\ 0.42453895\\\ 0.42562610\\\ 0.70211529\\\ 0.70319450\\\ 0.70481328\\\ 0.97565862\\\ 0.97672990\\\ 0.97833679\\\ 0.98047926\\\ \end{array}$ | $\begin{array}[]{c}0.14339599\\\ 0.42501458\\\ 0.42637351\\\ 0.70258745\\\ 0.70393644\\\ 0.70582499\\\ 0.97612730\\\ 0.97746640\\\ 0.97934108\\\ 0.98175131\\\ \end{array}$
Table 3: ${E_{nl}}$ (eV) of DFP for I2 molecule. |  | ${N=3}$ | ${N=3}$ | ${\delta=1}$
---|---|---|---|---
${n}$ | ${l}$ | ${\delta=0.2}$ | ${\delta=0.5}$ | ${\delta=0.8}$ | ${\delta=1}$ | ${N=4}$ | ${N=5}$ | ${N=6}$
$\begin{array}[]{c}0\\\ 1\\\ \\\ 2\\\ \\\ \\\ 3\\\ \\\ \\\ \\\ \end{array}$ | $\begin{array}[]{c}0\\\ 0\\\ 1\\\ 0\\\ 1\\\ 2\\\ 0\\\ 1\\\ 2\\\ 3\\\ \end{array}$ | $\begin{array}[]{c}0.00290217\\\ 0.00866328\\\ 0.00868398\\\ 0.01441374\\\ 0.01443444\\\ 0.01447584\\\ 0.02015357\\\ 0.02017427\\\ 0.02021566\\\ 0.02027774\\\ \end{array}$ | $\begin{array}[]{c}0.00757848\\\ 0.02266887\\\ 0.02268957\\\ 0.03768510\\\ 0.03770578\\\ 0.03774715\\\ 0.05262715\\\ 0.05264783\\\ 0.05268918\\\ 0.05275121\\\ \end{array}$ | $\begin{array}[]{c}0.01168805\\\ 0.03492991\\\ 0.03495060\\\ 0.05799444\\\ 0.05801512\\\ 0.05805647\\\ 0.08088166\\\ 0.08090232\\\ 0.08094365\\\ 0.08100563\\\ \end{array}$ | $\begin{array}[]{c}0.01341022\\\ 0.04005490\\\ 0.04007558\\\ 0.06646580\\\ 0.06648647\\\ .0.06652781\\\ 0.09264296\\\ 0.09266361\\\ 0.09270493\\\ 0.09276689\\\ \end{array}$ | $\begin{array}[]{c}0.01341798\\\ 0.04006266\\\ 0.04009368\\\ 0.06647355\\\ 0.06650456\\\ 0.06655623\\\ 0.09265070\\\ 0.09268169\\\ 0.09273333\\\ 0.09280562\\\ \end{array}$ | $\begin{array}[]{c}0.01343092\\\ 0.04007558\\\ 0.04011695\\\ 0.06648647\\\ 0.06652781\\\ 0.06658982\\\ 0.09266361\\\ 0.09270493\\\ 0.09276689\\\ 0.09284952\\\ \end{array}$ | $\begin{array}[]{c}0.01344903\\\ 0.04009368\\\ 0.04014539\\\ 0.06650456\\\ 0.06655623\\\ 0.06662857\\\ 0.09268169\\\ 0.09273333\\\ 0.09280562\\\ 0.09289857\\\ \end{array}$
Table 4: ${E_{nl}}$ (eV) of DFP for NO molecule. |  | ${N=3}$ | ${N=3}$ | ${\delta=1}$
---|---|---|---|---
${n}$ | ${l}$ | ${\delta=0.2}$ | ${\delta=0.5}$ | ${\delta=0.8}$ | ${\delta=1}$ | ${N=4}$ | ${N=5}$ | ${N=6}$
$\begin{array}[]{c}0\\\ 1\\\ \\\ 2\\\ \\\ \\\ 3\\\ \\\ \\\ \\\ \end{array}$ | $\begin{array}[]{c}0\\\ 0\\\ 1\\\ 0\\\ 1\\\ 2\\\ 0\\\ 1\\\ 2\\\ 3\\\ \end{array}$ | $\begin{array}[]{c}0.02958454\\\ 0.08794744\\\ 0.08849444\\\ 0.14610481\\\ 0.14665108\\\ 0.14774361\\\ 0.20405658\\\ 0.20460211\\\ 0.20569316\\\ 0.20732971\\\ \end{array}$ | $\begin{array}[]{c}0.07685720\\\ 0.22923250\\\ 0.22977770\\\ 0.38010256\\\ 0.38064583\\\ 0.38173235\\\ 0.52946844\\\ 0.53000977\\\ 0.53109243\\\ 0.53271639\\\ \end{array}$ | $\begin{array}[]{c}0.11839178\\\ 0.35239340\\\ 0.35293702\\\ 0.58275773\\\ 0.58329836\\\ 0.58437963\\\ 0.80949102\\\ 0.81002867\\\ 0.81110397\\\ 0.81271689\\\ \end{array}$ | $\begin{array}[]{c}0.13579494\\\ 0.40372140\\\ 0.40426436\\\ 0.66683706\\\ 0.66737660\\\ 0.66845566\\\ 0.92515259\\\ 0.92568870\\\ 0.92676092\\\ 0.92836923\\\ \end{array}$ | $\begin{array}[]{c}0.13599984\\\ 0.40392501\\\ 0.40473945\\\ 0.66703939\\\ 0.66784869\\\ 0.66919750\\\ 0.92535363\\\ 0.92615780\\\ 0.92749807\\\ 0.92937441\\\ \end{array}$ | $\begin{array}[]{c}0.13634133\\\ 0.40426436\\\ 0.40535028\\\ 0.66737660\\\ 0.66845566\\\ 0.67007422\\\ 0.92568870\\\ 0.92676092\\\ 0.92836923\\\ 0.93051361\\\ \end{array}$ | $\begin{array}[]{c}0.13681943\\\ 0.40473945\\\ 0.40609683\\\ 0.66784869\\\ 0.66919750\\\ 0.67108582\\\ 0.92615780\\\ 0.92749807\\\ 0.92937441\\\ 0.93178682\\\ \end{array}$
Table 5: ${E_{nl}}$ (eV) of DFP for N2 molecule. |  | ${N=3}$ | ${N=3}$ | ${\delta=1}$
---|---|---|---|---
${n}$ | ${l}$ | ${\delta=0.2}$ | ${\delta=0.5}$ | ${\delta=0.8}$ | ${\delta=1}$ | ${N=4}$ | ${N=5}$ | ${N=6}$
$\begin{array}[]{c}0\\\ 1\\\ \\\ 2\\\ \\\ \\\ 3\\\ \\\ \\\ \\\ \end{array}$ | $\begin{array}[]{c}0\\\ 0\\\ 1\\\ 0\\\ 1\\\ 2\\\ 0\\\ 1\\\ 2\\\ 3\\\ \end{array}$ | $\begin{array}[]{c}0.03680223\\\ 0.10957917\\\ 0.11019288\\\ 0.18214266\\\ 0.18275555\\\ 0.18398134\\\ 0.25449260\\\ 0.25510469\\\ 0.25632884\\\ 0.25816506\\\ \end{array}$ | $\begin{array}[]{c}0.09579693\\\ 0.28598430\\\ 0.28659602\\\ 0.47458659\\\ 0.47519618\\\ 0.47641535\\\ 0.66160500\\\ 0.66221245\\\ 0.66342735\\\ 0.66524968\\\ \end{array}$ | $\begin{array}[]{c}0.14763020\\\ 0.43994427\\\ 0.44055425\\\ 0.72841609\\\ 0.72902278\\\ 0.73023614\\\ 1.01305263\\\ 1.01365603\\\ 1.01486282\\\ 1.01667299\\\ \end{array}$ | $\begin{array}[]{c}0.16934841\\\ 0.50415932\\\ 0.50476857\\\ 0.83388317\\\ 0.83448865\\\ 0.83569959\\\ 1.15853183\\\ 1.15913354\\\ 1.16033694\\\ 1.16214202\\\ \end{array}$ | $\begin{array}[]{c}0.16957830\\\ 0.50438779\\\ 0.50530166\\\ 0.83411023\\\ 0.83501843\\\ 0.83653210\\\ 1.15875747\\\ 1.15966002\\\ 1.16116427\\\ 1.16327018\\\ \end{array}$ | $\begin{array}[]{c}0.16996144\\\ 0.50476857\\\ 0.50598707\\\ 0.83448865\\\ 0.83569959\\\ 0.83751597\\\ 1.15913354\\\ 1.16033694\\\ 1.16214202\\\ 1.16454875\\\ \end{array}$ | $\begin{array}[]{c}0.17049785\\\ 0.50530166\\\ 0.50682477\\\ 0.83501843\\\ 0.83653210\\\ 0.83865120\\\ 1.15966002\\\ 1.16116427\\\ 1.16327018\\\ 1.16597773\\\ \end{array}$
Table 6: ${-E_{nl}}$ (eV) of SDFP for CO molecule. |  | ${N=3}$ | ${N=3}$ | ${\delta=1}$
---|---|---|---|---
${n}$ | ${l}$ | ${\delta=0.2}$ | ${\delta=0.5}$ | ${\delta=0.8}$ | ${\delta=1}$ | ${N=4}$ | ${N=5}$ | ${N=6}$
$\begin{array}[]{c}0\\\ 1\\\ \\\ 2\\\ \\\ \\\ 3\\\ \\\ \\\ \\\ \end{array}$ | $\begin{array}[]{c}0\\\ 0\\\ 1\\\ 0\\\ 1\\\ 2\\\ 0\\\ 1\\\ 2\\\ 3\\\ \end{array}$ | $\begin{array}[]{c}10.8141740\\\ 10.7529912\\\ 10.7524429\\\ 10.6919720\\\ 10.6914246\\\ 10.6903298\\\ 10.6311165\\\ 10.6305700\\\ 10.6294768\\\ 10.6278372\\\ \end{array}$ | $\begin{array}[]{c}10.7645575\\\ 10.6046292\\\ 10.6040830\\\ 10.4459513\\\ 10.4454074\\\ 10.4443195\\\ 10.2885227\\\ 10.2879810\\\ 10.2868976\\\ 10.2852726\\\ \end{array}$ | $\begin{array}[]{c}10.7209676\\\ 10.4751082\\\ 10.4745639\\\ 10.2322988\\\ 10.2317579\\\ 10.2306761\\\ 9.99253181\\\ 9.99199438\\\ 9.99091954\\\ 9.98930730\\\ \end{array}$ | $\begin{array}[]{c}10.7027043\\\ 10.4210782\\\ 10.4205347\\\ 10.1434979\\\ 10.1429583\\\ 10.1418791\\\ 9.86995067\\\ 9.86941502\\\ 9.86834374\\\ 9.86673685\\\ \end{array}$ | $\begin{array}[]{c}10.7024990\\\ 10.4208744\\\ 10.4200591\\\ 10.1432956\\\ 10.1424862\\\ 10.1411372\\\ 9.86974980\\\ 9.86894634\\\ 9.86760724\\\ 9.86573256\\\ \end{array}$ | $\begin{array}[]{c}10.7021568\\\ 10.4205347\\\ 10.4194475\\\ 10.1429583\\\ 10.1418791\\\ 10.1402604\\\ 9.86941502\\\ 9.86834374\\\ 9.86673685\\\ 9.86459438\\\ \end{array}$ | $\begin{array}[]{c}10.7016776\\\ 10.4200591\\\ 10.4187001\\\ 10.1424862\\\ 10.1411372\\\ 10.1392486\\\ 9.86894634\\\ 9.86760724\\\ 9.86573256\\\ 9.86332233\\\ \end{array}$
Table 7: ${-E_{nl}}$ (eV) of SDFP for I2 molecule. |  | ${N=3}$ | ${N=3}$ | ${\delta=1}$
---|---|---|---|---
${n}$ | ${l}$ | ${\delta=0.2}$ | ${\delta=0.5}$ | ${\delta=0.8}$ | ${\delta=1}$ | ${N=4}$ | ${N=5}$ | ${N=6}$
$\begin{array}[]{c}0\\\ 1\\\ \\\ 2\\\ \\\ \\\ 3\\\ \\\ \\\ \\\ \end{array}$ | $\begin{array}[]{c}0\\\ 0\\\ 1\\\ 0\\\ 1\\\ 2\\\ 0\\\ 1\\\ 2\\\ 3\\\ \end{array}$ | $\begin{array}[]{c}1.55269782\\\ 1.54693672\\\ 1.54691602\\\ 1.54118625\\\ 1.54116556\\\ 1.54112416\\\ 1.53544642\\\ 1.53542573\\\ 1.53538434\\\ 1.53532225\\\ \end{array}$ | $\begin{array}[]{c}1.54802152\\\ 1.53293112\\\ 1.53291043\\\ 1.51791490\\\ 1.51789421\\\ 1.51785284\\\ 1.50297285\\\ 1.50295217\\\ 1.50291081\\\ 1.50284878\\\ \end{array}$ | $\begin{array}[]{c}1.54391195\\\ 1.52067008\\\ 1.52064939\\\ 1.49760555\\\ 1.49758488\\\ 1.49754353\\\ 1.47471833\\\ 1.47469767\\\ 1.47465635\\\ 1.47459436\\\ \end{array}$ | $\begin{array}[]{c}1.54218978\\\ 1.51554509\\\ 1.51552441\\\ 1.48913420\\\ 1.48911353\\\ 1.48907219\\\ 1.46295704\\\ 1.46293638\\\ 1.46289507\\\ 1.46283310\\\ \end{array}$ | $\begin{array}[]{c}1.54218201\\\ 1.51553734\\\ 1.51550631\\\ 1.48912645\\\ 1.48909544\\\ 1.48904377\\\ 1.46294929\\\ 1.46291831\\\ 1.46286667\\\ 1.46279437\\\ \end{array}$ | $\begin{array}[]{c}1.54216908\\\ 1.51552441\\\ 1.51548304\\\ 1.48911353\\\ 1.48907219\\\ 1.48901018\\\ 1.46293638\\\ 1.46289507\\\ 1.46283310\\\ 1.46275048\\\ \end{array}$ | $\begin{array}[]{c}1.54215097\\\ 1.51550631\\\ 1.51545460\\\ 1.48909544\\\ 1.48904377\\\ 1.48897142\\\ 1.46291831\\\ 1.46286667\\\ 1.46279437\\\ 1.46270142\\\ \end{array}$
Table 8: ${-E_{nl}}$ (eV) of SDFP for NO molecule. |  | ${N=3}$ | ${N=3}$ | ${\delta=1}$
---|---|---|---|---
${n}$ | ${l}$ | ${\delta=0.2}$ | ${\delta=0.5}$ | ${\delta=0.8}$ | ${\delta=1}$ | ${N=4}$ | ${N=5}$ | ${N=6}$
$\begin{array}[]{c}0\\\ 1\\\ \\\ 2\\\ \\\ \\\ 3\\\ \\\ \\\ \\\ \end{array}$ | $\begin{array}[]{c}0\\\ 0\\\ 1\\\ 0\\\ 1\\\ 2\\\ 0\\\ 1\\\ 2\\\ 3\\\ \end{array}$ | $\begin{array}[]{c}8.01414531\\\ 7.95578242\\\ 7.95523541\\\ 7.89762504\\\ 7.89707877\\\ 7.89598625\\\ 7.83967328\\\ 7.83912775\\\ 7.83803670\\\ 7.83640014\\\ \end{array}$ | $\begin{array}[]{c}7.96687265\\\ 7.81449736\\\ 7.81395215\\\ 7.66362729\\\ 7.66308403\\\ 7.66199750\\\ 7.51426141\\\ 7.51372008\\\ 7.51263743\\\ 7.51101347\\\ \end{array}$ | $\begin{array}[]{c}7.92533807\\\ 7.69133645\\\ 7.69079283\\\ 7.46097213\\\ 7.46043149\\\ 7.45935023\\\ 7.23423884\\\ 7.23370118\\\ 7.23262589\\\ 7.23101296\\\ \end{array}$ | $\begin{array}[]{c}7.90793492\\\ 7.64000845\\\ 7.63946549\\\ 7.37689279\\\ 7.37635326\\\ 7.37527420\\\ 7.11857727\\\ 7.11804115\\\ 7.11696893\\\ 7.11536062\\\ \end{array}$ | $\begin{array}[]{c}7.90773002\\\ 7.63980484\\\ 7.63899040\\\ 7.37669047\\\ 7.37588117\\\ 7.37453235\\\ 7.11837622\\\ 7.11757206\\\ 7.11623179\\\ 7.11435544\\\ \end{array}$ | $\begin{array}[]{c}7.90738852\\\ 7.63946549\\\ 7.63837958\\\ 7.37635326\\\ 7.37527420\\\ 7.37365563\\\ 7.11804115\\\ 7.11696893\\\ 7.11536062\\\ 7.11321624\\\ \end{array}$ | $\begin{array}[]{c}7.90691043\\\ 7.63899040\\\ 7.63763302\\\ 7.37588117\\\ 7.37453235\\\ 7.37264404\\\ 7.11757206\\\ 7.11623179\\\ 7.11435544\\\ 7.11194304\\\ \end{array}$
Table 9: ${-E_{nl}}$ (eV) of SDFP for N2 molecule. |  | ${N=3}$ | ${N=3}$ | ${\delta=1}$
---|---|---|---|---
${n}$ | ${l}$ | ${\delta=0.2}$ | ${\delta=0.5}$ | ${\delta=0.8}$ | ${\delta=1}$ | ${N=4}$ | ${N=5}$ | ${N=6}$
$\begin{array}[]{c}0\\\ 1\\\ \\\ 2\\\ \\\ \\\ 3\\\ \\\ \\\ \\\ \end{array}$ | $\begin{array}[]{c}0\\\ 0\\\ 1\\\ 0\\\ 1\\\ 2\\\ 0\\\ 1\\\ 2\\\ 3\\\ \end{array}$ | $\begin{array}[]{c}11.9013916\\\ 11.8286147\\\ 11.8280009\\\ 11.7560512\\\ 11.7554383\\\ 11.7542125\\\ 11.6837012\\\ 11.6830891\\\ 11.6818650\\\ 11.6800288\\\ \end{array}$ | $\begin{array}[]{c}11.8423969\\\ 11.6522095\\\ 11.6515978\\\ 11.4636072\\\ 11.4629976\\\ 11.4617785\\\ 11.2765888\\\ 11.2759814\\\ 11.2747665\\\ 11.2729441\\\ \end{array}$ | $\begin{array}[]{c}11.7905636\\\ 11.4982495\\\ 11.4976396\\\ 11.2097777\\\ 11.2091710\\\ 11.2079577\\\ 10.9251412\\\ 10.9245378\\\ 10.9233310\\\ 10.9215208\\\ \end{array}$ | $\begin{array}[]{c}11.7688454\\\ 11.4340345\\\ 11.4334252\\\ 11.1043106\\\ 11.1037052\\\ 11.1024942\\\ 10.7796620\\\ 10.7790603\\\ 10.7778569\\\ 10.7760518\\\ \end{array}$ | $\begin{array}[]{c}11.7686155\\\ 11.4338060\\\ 11.4328922\\\ 11.1040836\\\ 11.1031754\\\ 11.1016617\\\ 10.7794364\\\ 10.7785338\\\ 10.7770296\\\ 10.7749236\\\ \end{array}$ | $\begin{array}[]{c}11.7682324\\\ 11.4334252\\\ 11.4322067\\\ 11.1037052\\\ 11.1024942\\\ 11.1006778\\\ 10.7790603\\\ 10.7778569\\\ 10.7760518\\\ 10.7736451\\\ \end{array}$ | $\begin{array}[]{c}11.7676960\\\ 11.4328922\\\ 11.4313690\\\ 11.1031754\\\ 11.1016617\\\ 11.0995426\\\ 10.7785338\\\ 10.7770296\\\ 10.7749236\\\ 10.7722161\\\ \end{array}$

(a)

(b)

Figure 2: Energy levels of I2 molecule with ${n}$ at numerous values of
${\delta}$ for (a) DFP and (b) SDFP.

(a)

(b)

Figure 3: Energy levels of I2 molecule with ${l}$ at numerous values of
${\delta}$ for (a) DFP and (b) SDFP..

(a)

(b)

Figure 4: Energy levels of I2 molecule with ${\alpha}$ at numerous values of
${\delta}$ for (a) DFP and (b) SDFP.

(a)

(b)

Figure 5: Energy levels of I2 molecule with ${r_{e}}$ at numerous values of
${\delta}$ for (a) DFP and (b) SDFP.

(a)

(b)

Figure 6: Energy levels of I2 molecule with ${\mu}$ at numerous values of ${\delta}$ for (a) DFP and (b) SDFP. Table 10: ${E_{nl}}$ (eV) of DFP for LiH and ScH molecules.  |  | LiH | ScH
---|---|---|---
${n}$ | ${l}$ | Present | [44] | [50] | Present | [44] | [50]
$\begin{array}[]{c}0\\\ 1\\\ \\\ 2\\\ \\\ \\\ 3\\\ \\\ \\\ \\\ 4\\\ \\\ \\\ \\\ \\\ 5\\\ \\\ \\\ \\\ \\\ \\\ \end{array}$ | $\begin{array}[]{c}0\\\ 0\\\ 1\\\ 0\\\ 1\\\ 2\\\ 0\\\ 1\\\ 2\\\ 3\\\ 0\\\ 1\\\ 2\\\ 3\\\ 4\\\ 0\\\ 1\\\ 2\\\ 3\\\ 4\\\ 5\\\ \end{array}$ | $\begin{array}[]{c}0.103334307\\\ 0.302004929\\\ 0.303837626\\\ 0.490684151\\\ 0.492449049\\\ 0.495977286\\\ 0.669598629\\\ 0.671297257\\\ 0.674692995\\\ 0.679782810\\\ 0.838967494\\\ 0.840601332\\\ 0.843867529\\\ 0.848763128\\\ 0.855283702\\\ 0.999002656\\\ 1.000573134\\\ 1.003712649\\\ 1.008418320\\\ 1.014685833\\\ 1.022509444\\\ \end{array}$ | $\begin{array}[]{c}0.103334650\\\ 0.302005955\\\ 0.303838653\\\ 0.490685861\\\ 0.492450759\\\ 0.495978997\\\ 0.669601019\\\ 0.671299648\\\ 0.674695388\\\ 0.679785205\\\ 0.838970564\\\ 0.840604402\\\ 0.843870601\\\ 0.848766203\\\ 0.855286782\\\ 0.999006401\\\ 1.000576880\\\ 1.003716397\\\ 1.008422072\\\ 1.014689589\\\ 1.022513206\\\ \end{array}$ | $\begin{array}[]{c}0.103397595\\\ 0.302068901\\\ 0.303901531\\\ 0.490748806\\\ 0.492513638\\\ 0.496041760\\\ 0.669663964\\\ 0.671362527\\\ 0.674758151\\\ 0.679847770\\\ 0.839033509\\\ 0.840667281\\\ 0.843933363\\\ 0.848828768\\\ 0.855349091\\\ 0.999069346\\\ 1.000639759\\\ 1.003779159\\\ 1.008484637\\\ 1.014751899\\\ 1.022575209\\\ \end{array}$ | $\begin{array}[]{c}0.104850693\\\ 0.306246537\\\ 0.307704128\\\ 0.496950687\\\ 0.498367396\\\ 0.501200315\\\ 0.677093198\\\ 0.678469522\\\ 0.681221677\\\ 0.685348676\\\ 0.846801672\\\ 0.848138097\\\ 0.850810460\\\ 0.854817790\\\ 0.860158632\\\ 1.006201313\\\ 1.007498315\\\ 1.010091841\\\ 1.013980933\\\ 1.019164157\\\ 1.025639600\\\ \end{array}$ | $\begin{array}[]{c}0.104850694\\\ 0.306246538\\\ 0.307704129\\\ 0.496950687\\\ 0.498367397\\\ 0.501200316\\\ 0.677093198\\\ 0.678469522\\\ 0.681221677\\\ 0.685348677\\\ 0.846801673\\\ 0.848138097\\\ 0.850810460\\\ 0.854817791\\\ 0.860158632\\\ 1.006201313\\\ 1.007498315\\\ 1.010091841\\\ 1.013980933\\\ 1.019164157\\\ 1.025639693\\\ \end{array}$ | $\begin{array}[]{c}0.104938646\\\ 0.306334489\\\ 0.307791993\\\ 0.497038637\\\ 0.498455260\\\ 0.501288002\\\ 0.677181146\\\ 0.678557384\\\ 0.681309362\\\ 0.685436100\\\ 0.846889620\\\ 0.848225957\\\ 0.850898144\\\ 0.854905213\\\ 0.860245696\\\ 1.006289259\\\ 1.007586174\\\ 1.010179523\\\ 1.014068354\\\ 1.019251220\\\ 1.025726220\\\ \end{array}$
Table 11:  ${E_{nl}}$ (eV) of DFP for HCl and CO molecules.  |  | HCl | CO
---|---|---|---
${n}$ | ${l}$ | Present | [44] | [40] | Present | [50]
$\begin{array}[]{c}0\\\ 1\\\ \\\ 2\\\ \\\ \\\ 3\\\ \\\ \\\ \\\ 4\\\ \\\ \\\ \\\ \\\ 5\\\ \\\ \\\ \\\ \\\ \\\ \end{array}$ | $\begin{array}[]{c}0\\\ 0\\\ 1\\\ 0\\\ 1\\\ 2\\\ 0\\\ 1\\\ 2\\\ 3\\\ 0\\\ 1\\\ 2\\\ 3\\\ 4\\\ 0\\\ 1\\\ 2\\\ 3\\\ 4\\\ 5\\\ \end{array}$ | $\begin{array}[]{c}0.201983505\\\ 0.590745819\\\ 0.593535604\\\ 0.960007699\\\ 0.962718245\\\ 0.968138299\\\ 1.310023186\\\ 1.312655523\\\ 1.317919173\\\ 1.325812091\\\ 1.641041232\\\ 1.643596366\\\ 1.648705629\\\ 1.656367005\\\ 1.666577476\\\ 1.953305816\\\ 1.955784737\\\ 1.960741588\\\ 1.968174386\\\ 1.978080159\\\ 1.990454948\\\ \end{array}$ | $\begin{array}[]{c}0.201984174\\\ 0.590747827\\\ 0.593537612\\\ 0.960011044\\\ 0.962721591\\\ 0.968141645\\\ 1.310027865\\\ 1.312660203\\\ 1.317923855\\\ 1.325816775\\\ 1.641047243\\\ 1.643602379\\\ 1.648711644\\\ 1.656373023\\\ 1.666583499\\\ 1.953313156\\\ 1.955792078\\\ 1.960748932\\\ 1.968181734\\\ 1.978087513\\\ 1.990462308\\\ \end{array}$ | $\begin{array}[]{c}0.202139134\\\ 0.590902787\\\ 0.593692401\\\ 0.960166004\\\ 0.962876380\\\ 0.968296136\\\ 1.310182826\\\ 1.312814994\\\ 1.318078347\\\ 1.325970803\\\ 1.641202203\\\ 1.643757170\\\ 1.648866136\\\ 1.656527052\\\ 1.666736911\\\ 1.953468116\\\ 1.955946871\\\ 1.960903424\\\ 1.968335762\\\ 1.978240925\\\ 1.990614950\\\ \end{array}$ | $\begin{array}[]{c}0.144850105\\\ 0.431437877\\\ 0.431981483\\\ 0.713979856\\\ 0.714519556\\\ 0.715598945\\\ 0.992488526\\\ 0.993024333\\\ 0.994095934\\\ 0.995703306\\\ 1.266976312\\\ 1.267508237\\\ 1.268572075\\\ 1.270167801\\\ 1.272295381\\\ 1.537455578\\\ 1.537983633\\\ 1.539039731\\\ 1.540623848\\\ 1.542735948\\\ 1.545375984\\\ \end{array}$ | $\begin{array}[]{c}0.144969072\\\ 0.430178995\\\ 0.430719631\\\ 0.711910800\\\ 0.712447561\\\ 0.713521071\\\ 0.989632352\\\ 0.990165249\\\ 0.991231033\\\ 0.992829678\\\ 1.263355977\\\ 1.263885023\\\ 1.264943104\\\ 1.266530195\\\ 1.268646261\\\ 1.533093944\\\ 1.533619151\\\ 1.534669552\\\ 1.536245125\\\ 1.538345834\\\ 1.540971631\\\ \end{array}$
Table 12: ${E_{nl}}$ (eV) of DFP for HF, O2 and H2 molecules. |  | HF |  O2 |  H2
---|---|---|---|---
${n}$ | ${l}$ | Present | [40] | Present | [40] | Present | [40]
$\begin{array}[]{c}0\\\ 1\\\ \\\ 2\\\ \\\ \\\ 3\\\ \\\ \\\ \\\ 4\\\ \\\ \\\ \\\ \\\ 5\\\ \\\ \\\ \\\ \\\ \\\ \end{array}$ | $\begin{array}[]{c}0\\\ 0\\\ 1\\\ 0\\\ 1\\\ 2\\\ 0\\\ 1\\\ 2\\\ 3\\\ 0\\\ 1\\\ 2\\\ 3\\\ 4\\\ 0\\\ 1\\\ 2\\\ 3\\\ 4\\\ 5\\\ \end{array}$ | $\begin{array}[]{c}0.291247434\\\ 0.848733535\\\ 0.853910381\\\ 1.374650333\\\ 1.379631880\\\ 1.389590976\\\ 1.869636743\\\ 1.874426923\\\ 1.884003381\\\ 1.898358314\\\ 2.334312319\\\ 2.338914946\\\ 2.348116388\\\ 2.361909031\\\ 2.380281458\\\ 2.769277951\\\ 2.773696723\\\ 2.782530546\\\ 2.795771983\\\ 2.813409888\\\ 2.835429422\\\ \end{array}$ | $\begin{array}[]{c}0.296644754\\\ 0.848791378\\\ 0.853968904\\\ 1.374741414\\\ 1.379723603\\\ 1.389683982\\\ 1.869756985\\\ 1.874547769\\\ 1.884125435\\\ 1.898482178\\\ 2.334457762\\\ 2.339060957\\\ 2.348263535\\\ 2.362057878\\\ 2.380432567\\\ 2.769444752\\\ 2.773864056\\\ 2.782698944\\\ 2.795941975\\\ 2.813582001\\\ 2.835604179\\\ \end{array}$ | $\begin{array}[]{c}0.349980220\\\ 0.996777053\\\ 1.010323238\\\ 1.580248366\\\ 1.592700793\\\ 1.617539648\\\ 2.104086156\\\ 2.115507769\\\ 2.138289195\\\ 2.172307398\\\ 2.571680443\\\ 2.582129083\\\ 2.602968445\\\ 2.634083222\\\ 2.675301759\\\ 2.986148433\\\ 2.995677323\\\ 3.014680784\\\ 3.043050669\\\ 3.080625975\\\ 3.127194276\\\ \end{array}$ | $\begin{array}[]{c}0.365141571\\\ 0.998213655\\\ 1.011736491\\\ 1.582601950\\\ 1.595035333\\\ 1.619836697\\\ 2.107302268\\\ 2.118708590\\\ 2.141459973\\\ 2.175434452\\\ 2.575691305\\\ 2.586127919\\\ 2.606943719\\\ 2.638024369\\\ 2.679199648\\\ 2.990875171\\\ 3.000394872\\\ 3.019380399\\\ 3.047724487\\\ 3.085267435\\\ 3.131798515\\\ \end{array}$ | $\begin{array}[]{c}0.101919228\\\ 0.302577426\\\ 0.303043144\\\ 0.499057714\\\ 0.499520138\\\ 0.500444979\\\ 0.691370318\\\ 0.691829458\\\ 0.692747730\\\ 0.694125117\\\ 0.879525427\\\ 0.879981291\\\ 0.880893009\\\ 0.882260567\\\ 0.884083941\\\ 1.063533190\\\ 1.063985785\\\ 1.064890966\\\ 1.066248719\\\ 1.068059018\\\ 1.070321833\\\ \end{array}$ | $\begin{array}[]{c}0.102043850\\\ 0.3015750557\\\ 0.3020376575\\\ 0.4974222630\\\ 0.4978816054\\\ 0.4988002821\\\ 0.6891297288\\\ 0.6895858196\\\ 0.6904979933\\\ 0.6918662339\\\ 0.8767075389\\\ 0.8771603861\\\ 0.8780660724\\\ 0.8794245822\\\ 0.8812358915\\\ 1.0601657396\\\ 1.0606153510\\\ 1.0615145658\\\ 1.0628633684\\\ 1.0646617349\\\ 1.0669096339\\\ \end{array}$
Table 13: ${-E_{nl}}$ (eV) of SDFP for CO and HCl molecules.  |  | CO | HCl
---|---|---|---
${n}$ | ${l}$ | Present | [46] | [49] | Present | [46] | [49]
$\begin{array}[]{c}0\\\ \\\ \\\ 5\\\ \\\ \\\ 7\\\ \\\ \\\ \end{array}$ | $\begin{array}[]{c}0\\\ 5\\\ 10\\\ 0\\\ 5\\\ 10\\\ 0\\\ 5\\\ 10\\\ \end{array}$ | $\begin{array}[]{c}11.08074990\\\ 11.07253746\\\ 11.05064208\\\ 9.688144422\\\ 9.680224016\\\ 9.659107308\\\ 9.159162287\\\ 9.151357442\\\ 9.130548864\\\ \end{array}$ | $\begin{array}[]{c}11.08075178\\\ 11.07253985\\\ 11.05064581\\\ 9.688146187\\\ 9.680226284\\\ 9.659110919\\\ 9.159164003\\\ 9.151359661\\\ 9.130552425\\\ \end{array}$ | $\begin{array}[]{c}11.08074989\\\ 11.07360695\\\ 11.05456479\\\ 9.688144422\\\ 9.681355403\\\ 9.663257003\\\ 9.159162288\\\ 9.152513419\\\ 9.134788734\\\ \end{array}$ | $\begin{array}[]{c}4.417047400\\\ 4.374033861\\\ 4.259723835\\\ 2.665725089\\\ 2.628575957\\\ 2.529874313\\\ 2.096504515\\\ 2.061597436\\\ 1.968863331\\\ \end{array}$ | $\begin{array}[]{c}4.417077001\\\ 4.374065784\\\ 4.259761948\\\ 2.665748019\\\ 2.628601192\\\ 2.529905688\\\ 2.096524802\\\ 2.061620020\\\ 1.968892038\\\ \end{array}$ | $\begin{array}[]{c}4.417047400\\\ 4.378422991\\\ 4.275898727\\\ 2.665725089\\\ 2.633922445\\\ 2.549565348\\\ 2.096504516\\\ 2.067315382\\\ 1.989918464\\\ \end{array}$
Table 14: ${-E_{nl}}$ (eV) of SDFP for LiH and ScH molecules.  |  | LiH | ScH
---|---|---|---
${n}$ | ${l}$ | Present | [46] | [49] | Present | [46] | [48]
$\begin{array}[]{c}0\\\ \\\ \\\ 5\\\ \\\ \\\ 7\\\ \\\ \\\ \end{array}$ | $\begin{array}[]{c}0\\\ 5\\\ 10\\\ 0\\\ 5\\\ 10\\\ 0\\\ 5\\\ 10\\\ \end{array}$ | $\begin{array}[]{c}2.411932904\\\ 2.383459166\\\ 2.308127882\\\ 1.516264555\\\ 1.492757768\\\ 1.430598166\\\ 1.223382139\\\ 1.201712023\\\ 1.144423818\\\ \end{array}$ | $\begin{array}[]{c}2.411949045\\\ 2.383476249\\\ 2.308147473\\\ 1.516277294\\\ 1.492771433\\\ 1.430614300\\\ 1.223393538\\\ 1.201724343\\\ 1.144438594\\\ \end{array}$ | $\begin{array}[]{c}2.411932904\\\ 2.384580437\\\ 2.312283846\\\ 1.516264555\\\ 1.494210023\\\ 1.435969432\\\ 1.223382139\\\ 1.203290365\\\ 1.150257886\\\ \end{array}$ | $\begin{array}[]{c}2.145149306\\\ 2.122682696\\\ 2.062960535\\\ 1.243798687\\\ 1.224360400\\\ 1.172700118\\\ 0.955436772\\\ 0.937159725\\\ 0.888590944\\\ \end{array}$ | $\begin{array}[]{c}2.1451493060\\\ 2.1226840150\\\ 2.0629653710\\\ 1.2437986870\\\ 1.2243617190\\\ 1.1727049540\\\ 0.9554367725\\\ 0.9371610441\\\ 0.8885957808\\\ \end{array}$ | $\begin{array}[]{c}2.145149306\\\ 2.122682696\\\ \-----\\\ \-----\\\ \-----\\\ \-----\\\ \-----\\\ \-----\\\ \-----\\\ \end{array}$
Table 15: ${-E_{nl}}$ (eV) of SDFP for H2 and I2 molecules. |  |  H2 |  I2
---|---|---|---
${n}$ | ${l}$ | Present | [46] | [49] | Present | [46] | [48]
$\begin{array}[]{c}0\\\ \\\ \\\ 5\\\ \\\ \\\ 7\\\ \\\ \\\ \end{array}$ | $\begin{array}[]{c}0\\\ 5\\\ 10\\\ 0\\\ 5\\\ 10\\\ 0\\\ 5\\\ 10\\\ \end{array}$ | $\begin{array}[]{c}4.394619779\\\ 4.176613157\\\ 3.621820491\\\ 1.758451567\\\ 1.617405724\\\ 1.260433707\\\ 1.077636993\\\ 0.961809891\\\ 0.669826131\\\ \end{array}$ | $\begin{array}[]{c}4.394619779\\\ 4.176618048\\\ 3.621838424\\\ 1.758451567\\\ 1.617410615\\\ 1.260451640\\\ 1.077636993\\\ 0.961814782\\\ 0.669844065\\\ \end{array}$ | $\begin{array}[]{c}4.394619776\\\ 4.180734836\\\ 3.637874575\\\ 1.758451566\\\ 1.624065481\\\ 1.285780199\\\ 1.077636991\\\ 0.969361194\\\ 0.698426418\\\ \end{array}$ | $\begin{array}[]{c}1.542189775\\\ 1.541879304\\\ 1.541051381\\\ 1.411303767\\\ 1.410994355\\\ 1.410169257\\\ 1.360584943\\\ 1.360275954\\\ 1.359451985\\\ \end{array}$ | $\begin{array}[]{c}1.542189775\\\ 1.541879340\\\ 1.541051513\\\ 1.411303767\\\ 1.410994391\\\ 1.410169388\\\ 1.360584943\\\ 1.360275990\\\ 1.359452116\\\ \end{array}$ | $\begin{array}[]{c}1.542189775\\\ 1.541879304\\\ \-----\\\ \-----\\\ \-----\\\ \-----\\\ \-----\\\ \-----\\\ \-----\\\ \end{array}$

##  4 Results and discussion

In this part, we employ the previously obtained solutions to analyze the
energy spectra of numerous DMs for both the DFP and the SDFP under the
influence of the fractional parameter (${\delta}$) in ${N}$-dimensional space.
In our computations, we utilize the spectroscopic values reported in Table (1)
for selected DMs: HCl, CO, O2 and NO besides the following replacements [71]:
${\hbar c=1973.29}$ eV ${\overset{o}{A}}$, ${1cm^{-1}=1.239841875\times
10^{-4}}$ eV, and ${1amu=931.494028}$ MeV/ ${c^{2}}$. In Fig. (1), the
variation of the DFP and its shifted is indicated at ${r=r_{e}}$ for some DMs.
In Tables (2\- 5), the numerical results of the energy spectra for various
DMs: CO, I2, NO and N2 with the DFP are displayed at diverse values of the
fractional parameter (${\delta}$) and the dimensional number (${N}$). It is
found that increasing the fractional parameter (${\delta}$) causes a
significant increase in the energy eigenvalues. Concurrently, as the
dimensional number (${N}$) rises, so do the energy eigenvalues. Furthermore,
the impact of fractional parameter (${\delta}$) and the dimensional number
(${N}$) on the energy eigenvalues for the SDFP is explored in Tables (6\- 9).
We also observe here that the energy eigenvalues move to higher values as
${\delta}$ and ${N}$ rise. The influence of the fractional parameter on the
behavior of the energy spectra for both the DFP and SDFP with varied potential
parameters is depicted in Figs. (2\- 6) for a clearer appreciation. For
various values of the fractional parameter
${\delta=0.1,\delta=0.3,\delta=0.5,\delta=0.7,\delta=0.9}$ and ${\delta=1}$,
the variations of the energy spectra with the vibrational (${n}$) and
rotational (${l}$) quantum numbers are shown in Fig. (2) and Fig. (3)
respectively. Fig. (2) reveals that the energy spectra gradually grow as the
vibrational quantum number (${n}$) increases. In line with this inspection,
Fig. 3 illustrates a progressive rise also in the energy spectra as the
rotational quantum number (${l}$) rises. In Fig. (4), we display the behavior
of the energy spectra with the screening parameter (${\alpha}$) for different
values of the fractional parameter. As seen in Fig. (4), the energy spectra
elevate in a consistent manner as the screening parameter is increased. In
Fig. (5), the energy spectra are plotted with the equilibrium bond length
(${r_{e}}$) at ${\delta=0.1,\delta=0.3,\delta=0.5,\delta=0.7,\delta=0.9}$ and
${\delta=1}$. As shown in Fig. (5), the energy spectra decrease by increasing
the equilibrium bond length. The behavior of the energy spectra with the
reduced mass (${\mu}$) is depicted in Fig. (6). Obviously, increasing the
reduced mass leads to a drop in the energy spectra.

Due to our investigation being innovative, we are unable to compare the
obtained results for different values of the fractional parameter. As a
consequence, we recover the classical solutions of the DFP and its shifted by
setting ${\delta=1}$ and comparing our results to those found in the
literature. In Tables (10\- 12), we report the numerical results for the ro-
vibrational energy spectra of the DFP for several DMs: LiH, ScH, HCl, CO, O2
and H2 in comparison with the found in Refs. [40], [44], [50]. Furthermore,
the ro-vibrational energy spectra of the SDFP for numerous DMs: LiH, ScH, HCl,
CO, H2 and I2 are displayed in Tables (13\- 15) compared to the findings in
Refs. [46], [48], [49]. As can be shown in Tables (10\- 15), the ro-
vibrational energy spectra of all selected DMs rise as the vibrational and
rotational quantum numbers increase. Importantly, one can see that our
estimates are perfectly consistent with prior works that used other
techniques. To our knowledge, such an investigation has never been done
previously. We hope that the current findings will be helpful in future
studies.

##  5 Conclusion

In this paper, we have employed the generalized fractional derivative (GFD) to
derive the bound state solutions of the ${N}$-dimensional SE with the DFP
within the framework of the generalized fractional NU method. The formulas of
energy eigenvalues and corresponding eigenfunctions for the DFP as a function
of the fractional parameter have been produced for arbitrary values of the
vibrational and rotational quantum numbers in the ${N}$-dimensional space. As
an application, we have evaluated the ro-vibrational energy levels of varied
diatomic molecules (DMs) for both the DFP and its shifted potential at
different values of the fractional parameter. The effect of the fractional
parameter on the energy levels has also been graphically illustrated. We noted
that decreasing fractional parameter (${\delta}$) lowers the energy
eigenvalues significantly. As a consequence, we conclude that the diatomic
molecule’s energy is more bounded at lower fractional parameter values than in
the classical case (${\delta=1}$). Furthermore, we estimated the energy levels
of numerous DMs in three-dimensional space as well as higher dimensions. It’s
notable that as the number of dimensions grows, the energy spectrum grows as
well. The variation of the energy levels of the DFP and its shifted potential
for the I2 DM has been graphically depicted with the reduced mass, screening
parameter, equilibrium bond length, rotational and vibrational quantum
numbers. To prove the validity of our findings, we calculated the energy
levels of numerous DMs at ${\delta=1}$ and compared the results to previous
studies. As can be observed, our findings are in perfect accord with those of
others. We infer that the fractional parameter has a significant impact on the
ro-vibrational energy levels of DMs. Consequently, exploring the solutions of
the ${N}$-dimensional SE within the framework of the GFD, and studying the
impact of the fractional number on various features of diverse systems in
molecular and atomic physics should be investigated. Finally, we indicate that
this work would be extended in the future to derive the solutions of the
${N}$-dimensional SE within the scope of the GFD for further molecular
potentials such as the Kratzer potential, Morse potential, Pöschl-Teller
potential, Hulthén potential, and so on.

## References

  * [1] R. Hilfer, Applications of fractional calculus in physics, World Scientific, Singapore, 2000.
  * [2] M. Dalir, M. Bashour, Appl. Math. Sci. 4 (21) (2010) 1021-1032.
  * [3] Y. A. Rossikhin, M. V. Shitikova, App. Mech. Rev. 50 (1) (1997) 15-67.
  * [4] G. Jumarie, Comput. Math. Appl. 51 (2006) 1367-1376.
  * [5] K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, New York, 1993.
  * [6] M. Caputo, Geophysical. J. Int. 13 (5) (1967) 529-539.
  * [7] T. Das, U. Ghosh, S. Sarkar, S. Das, J. Math. Phys. 59 (2) (2018) 022111.
  * [8] T. Das, U. Ghosh, S. Sarkar, S. Das, Pramana J. Phys. 93 (2019) 76.
  * [9] T. Das, U. Ghosh, S. Sarkar, S. Das, Pramana J. Phys. 94 (2020) 33.
  * [10] M. Al-Raeei, M. S. El-Daher, Pramana J. Phys. 94 (2020) 157.
  * [11] M. Al-Raeei, M. S. El-Daher, Heliyon 6 (7) (2020) e04495.
  * [12] M. Al-Raeei, M. S. El-Daher, Phys. Lett. A 383 (26) (2019) 125831.
  * [13] M. Al-Raeei, M. S. El-Daher, AIP Adv. 10 (3) (2020) 035305.
  * [14] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, J. Comput. Appl. Math. 264 (2014) 65-70.
  * [15] T. Abdeljawad, J. Comput. Appl. Math. 279 (2015) 57-66.
  * [16] H. Karayer, D. Demirhan, F. Büyükkılıç, Commun. Theor. Phys. 66 (2016) 12-18.
  * [17] W. S. Chung, S. Zare, H. Hassanabadi, Commun. Theor. Phys. 67 (2017) 250-254.
  * [18] F. S. Mozaffari, H. Hassanabadi, H. Sobhani, J. Kor. Phys. Soc. 72 (9) (2017) 980-986.
  * [19] H. H. Karayer, A. D. Demirhan, F. Büyükkılıç, Turk. J. Phys. 41 (2017) 551-559.
  * [20] N. Jamshir, B. Lari, H. Hassanabadi, Physica A: Stat. Mech. Appl. 565 (2021) 125616.
  * [21] M. Abu-Shady, Int. J. Mod. Phys. A 34 (31) (2019) 1950201.
  * [22] M. Abu-Shady· Sh. Y. Ezz-Alarab, Few-Body Syst. 62 (2) (2021) 1-8.
  * [23] M. Abu-Shady, A. I. Ahmadov, H. M. Fath-Allah, V. H. Badalov, (2021) arXiv:2105.01607
  * [24] A. Al-Jamel, Int. J. Mod. Phys. A 34 (10) (2019) 1950054.
  * [25] M. M. Hammad, A. S. Yaqut, M. A. Abdel-Khalek, S. B. Doma, Nuc. Phys. A 1015 (2021) 122307.
  * [26] M. Abu-Shady, Fur. App. Math. 1 (2) (2021) 34-42.
  * [27] U. S. Okorie, A. N. Ikot, P. O. Amadi, G. J. Rampho, (2020) arXiv:2103.14064.
  * [28] U. S. Okorie et al., Mod. Phys. Lett. A 36 (7) (2021) 2150041.
  * [29] O. Bayrak, I. Boztosun, H. Ciftci, Int. J. Quant. Chem. 107 (3) (2007) 540-544.
  * [30] D. A. Morales, Chem. Phys. Lett. 394 (1-3) (2004) 68-75.
  * [31] S. H. Dong, R. Lemus, Int. J. Quant. Chem. 86 (3) (2002) 265-272.
  * [32] A. K. Roy, Mod. Phys. Lett. (29) 9 (2014) 1450042.
  * [33] X. Y. Gu, J. Q. Sun, J. Math. Phys. 51 (2) (2010) 022106.
  * [34] D. Schiöberg, Mol. Phys. 59 (5) (1986) 1123-1137.
  * [35] A. K. Roy, J. Math. Chem. 52 (5) (2014) 1405-1413.
  * [36] Z. H. Deng, Y. P. Fan, Shandong Univ. J. 7 (1957) 162.
  * [37] S. H. Dong, X. Y. Gu, J. Phys. Conf. Ser. 96 (1) (2008) 012109.
  * [38] S. H. Dong, Commun. Theor. Phys. 55 (2011) 969–971.
  * [39] L. H. Zhang, X. P. Li, C. S. Jia, Int. J. Quant. Chem. 111 (9) (2011) 1870–1878.
  * [40] B. J. Falaye, S. M. Ikhdair, M. Hamzavi, Z. Naturforsch. 70 (2) (2015) 85–90.
  * [41] S. M. Ikhdair, J. Math. Phys. 52 (5) (2011) 052303.
  * [42] O. J. Oluwadare, K. J. Oyewumi, C. O. Akoshile, O. A. Babalola, Phys. Scr. 86 (3) (2012) 035002.
  * [43] O. Mustafa, J. Phys. B At. Mol. Opt. Phys. 48 (6) (2015) 065101.
  * [44] K. J. Oyewumi et al., J. Math. Chem. 51 (3) (2013) 976–991.
  * [45] M. Hamzavi, S. M. Ikhdair, K. E. Thylwe, J. Math. Chem. 51 (1) (2013) 227–238.
  * [46] K. J. Oyewumi et al., Mol. Phys. 112 (1) (2014) 127-141.
  * [47] A. K. Roy, Int. J. Quant. Chem. 114 (6) (2014) 383–391.
  * [48] O. J. Oluwadare, K. J. Oyewumi, Eur. Phys. J. Plus 133 (10) (2018) 422.
  * [49] H. Boukabcha, M. Hachamab, A. Diaf, App. Math. Comp. 321 (2018) 121–129.
  * [50] E. Omugbe, Asian J. Phys. Chem. Sci. 8 (1) (2020) 26-36.
  * [51] D. Nath, A. K. Roy, Int. J. Quant. Chem. 121 (10) (2021) e26616.
  * [52] C. P. Onyenegecha et al., (2021) https://doi.org/10.21203/rs.3.rs-763444/v1
  * [53] E. Omugbe, O. E. Osafile, E. P. Inyang, A. Jahanshir, Phys. Scr. 96 (12) (2021) 125408.
  * [54] A. N. Ikot et al., Few-Body Syst. 62 (2021) 101.
  * [55] E. M. Khokha, M. Abu-Shady, T. A. Abdel-Karim, Int. J. Theor. Appl. Math. 2 (2) (2016) 86-92.
  * [56] M. Abu-Shady, T. A. Abdel-Karim, E. M. Khokha, Adv. High Energy Phys. 2018 (2018) 7356843.
  * [57] M. Abu-Shady, T. A. Abdel-Karim, E. M. Khokha, SciFed J. Quant. Phys. 2 (2018) 1000017.
  * [58] M. Abu-Shady, E. M. Khokha, Adv. High Energy Phys. 2018 (2018) 7032041.
  * [59] M. Abu-Shady, T. A. Abdel-Karim, Sh. Y. Ezz-Alarab, J. Egyp. Math. Soc. 27 (2019) 14.
  * [60] M. Abu-Shady, Sh. Y. Ezz-Alarab, Few-Body Syst. 60 (4) (2019) 1-11.
  * [61] M. Abu-Shady, H. M. Fath-Allah, Int. J. Mod. Phys. A 35 (21) (2020) 2050110.
  * [62] M. Abu-Shady, H. M. Fath-Allah, J. Found. Appl. Phys. 6 (2019) 163.
  * [63] M. Abu-Shady, A. N. Ikot, Eur. Phys. J. Plus, 134 (7) (2019) 321.
  * [64] M. Abu-Shady, J. Egyp. Math. Soc. 25 (1) (2019) 86-89.
  * [65] M. Abu-Shady, Adv. Math. Phys. 2016 (2016) 4935940.
  * [66] M. Abu-Shady, Int. J. Appl. Math. Theor. Phys. 2 (2) (2016) 16.
  * [67] M. Abu-Shady, M. K. A. Kaabar, Adv. High Energy Phys. 2021 (2021) 9444803.
  * [68] I. Podlubny, Fractional Differential Equations, Academic Press, USA, 1999.
  * [69] A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations, in: Math. Studies., North-Holland, New York, 2006.
  * [70] A. F. Nikivorof, V. B. Uvarov, Special Functions of Mathematical. Physics–A Unified Introduction with Applications, Basel, Birkhäuser, 1988.
  * [71] U. S. Okorie et al., Int. J. Thermophys. 41 (7) (2020) 1-15.


# Discovery of the Superconductivity in 3d Element Ni doped IrTe2

Qiang Jing1,2<EMAIL_ADDRESS>Ping Li4 Yunze Long5 Jingfeng Wang2 Fei
Jiao7 Xiaoxiong Wang5 Xiaomin Cui2,6 Wenxiang Jiang1 Guohua Wang1 Yunlong Li1
Gan Liu8 Cao Wang2 Bo Liu2 Dong Qian1,3<EMAIL_ADDRESS>1Key Laboratory of
Artificial Structures and Quantum Control (Ministry of Education), Shenyang
National Laboratory for Materials Science, School of Physics and Astronomy,
Shanghai Jiao Tong University, Shanghai 200240, China  2Laboratory of
Functional Molecules and Materials, School of Physics and Optoelectronic
Engineering, Shandong University of Technology, 266 Xincun Xi Road, Zibo,
255000, China  3Tsung-Dao Lee Institute, Shanghai Jiao Tong University,
Shanghai 200240  4Anhui Jianzhu University, School of Mathematics and Physics,
Hefei 230601, China  5College of Physics, Qingdao University, No.308 Ningxia
Road, Qingdao 266071, China  6Laboratory of Solid State Microstructures and
Innovation Center of Advanced Microstructures, Nanjing University, Nanjing
210093, China).  7School of Materials Science and Engineering, Shandong
University of Technology, 266 Xincun Xi Road, Zibo, 255000, China  8National
Laboratory of Solid State Microstructures, Department of Physics and
Collaborative Innovation Center of Advanced Microstructures, Nanjing
University, 22 Hankou Road, Nanjing 210093, People’s Republic of China

###### Abstract

IrTe2 with large spin-orbital coupling (SOC) shows a CDW-like first order
structural phase transition from high-temperature trigonal phase to low-
temperature monoclinic phase at 270 K, accompanying with a large jump in
transport and magnetic measurement as well as in heat capacity. Here, the 3d
element Ni has been doped into IrTe2 by growing Ir1-xNixTe2 single crystals.
Both XRD and XPS results reveal that the Ni atoms have substituted for Ir,
which is consistent with the calculation result. Like the CDW behaviour, the
structural phase transition shows competition and coexistence with the
superconductivity. The monoclinic phase transition has been suppressed
gradually with the increase of the doping amount of Ni, at last giving rise to
the stabilization of the trigonal phase with superconductivity. Within
0.1$\leq$x$\leq$0.2, Ir1-xNixTe2 shows the superconductive behaviour with Tc
around 2.6K. The superconductivity shows anisotropy with dimensionless
anisotropy parameter $\gamma$=$\xi_{//}$$/$$\xi_{\perp}$$\sim$ 2\. Even Ni
element shows ferromagnetic behaviour, Ir1-xNixTe2 only shows weak
paramagnetism, no ferromagnetic order is observed in it, which is coincident
with the calculation result that their up and down spin density of states
compensate each other well. In addition, for other 3d elements Fe, Co and Mn
doped IrTe2, only Ir1-xMnxTe2 owns magnetism with magnetic moment of 3.0$\mu$B
to the supercell, theoretically.

Keywords: IrTe2; structural phase transition; superconductivity; magnetic
elements doping

††preprint: APS/123-QED

##  I Introduction

One of major challenges in condensed matter physics is to comprehend the
interaction between superconductivity and other modulated electronic ground
states, e.g., magnetism, charge/orbital ordering, structure distortion,
spin/charge-density wave (SDW/CDW)[1]. Charge density wave (CDW) is a kind of
periodic electronic charge modulation, which is accompanied by a distortion of
the underlying lattice with the same periodicity [2, 3]. It is a collective
state that often appears in low-dimensional electronic systems. In many cases
it is also related with superconductivity, another collective phenomenon[4].
The driving force of a CDW could be a Fermi surface nesting instability, the
formation of local bound states[5, 6] or the indirect Jahn-Teller effect[7].
For the interplay between CDW and superconductivity, there exists competition
and coexistence of CDW and superconductivity in quasi-two-dimensional
transition-metal dichalcogenides [8, 9, 10]. When the CDW phase is suppressed,
the superconductivity emerges. It is believed that this phenomenon is related
with high-Tc superconductivity [11, 12, 13, 14, 15]. For the interplay between
magnetism and superconductivity, it was believed that elements with a large
magnetic moment were harmful to the emergence of superconductivity because the
magnetism originating from the static ordering of magnetic moments would
compete with superconductivity, which is realized through the formation of
Cooper pairs. However, the pairing process is a kind of dynamic process
realized by two conducting electrons with opposite spin[16]. Spin-orbital
coupling (SOC) is proportional to Z4, where Z is the atomic number. Large SOC
can result in unique quantum states such as Jeff=1/2 Mott insulators[17],
topological insulator[18, 19]. Furthermore, nonconventional superconductivity
pairing can be present in the so-called topological superconductors with large
SOC[20, 21, 22, 23]. For IrTe2, both Ir and Te have large atomic number, which
makes it own large SOC. In another aspect, IrTe2 undergoes a CDW-like first
order structural phase transition at about Ts= 270 K to a phase with a charge-
ordered state characterized by a wave vector qn = [1/(3n+2), 0, 1/(3n+2)] with
respect to the room-temperature unit-cell vectors[24, 25]. The phase
transition is accompanied by a change of its symmetry from trigonal to
monoclinic [25]. A large jump appears in transport and magnetic measurement
[25]as well as in heat capacity [26]. Later, it was suggested that the
structural phase transition was driven by the energy gain due to Ir and Te
dimerization, with Ir orbitals playing a key role [27, 28, 29]. For the
superconductive property of IrTe2, IrTe2 thin flake shows superconductive
behaviour with Tc=3.2K. It was suggested that the CDW-like normal-to-striped
charge order transition was suppressed in the thin flake, resulting in the
metastable superconductivity[30]. Through element doping (Pd, Pt, Rh and Cu)
[31, 32, 33, 34, 35], IrTe2 can become a superconductor too. Then, it offers
an appropriate platform to investigate structure/charge/orbital fluctuations
and superconductivity under strong SOC. Here, we try to add another physical
quantity i.e. ferromagnetism to IrTe2 system to investigate the interaction
between structure/charge/orbital fluctuations, superconductivity, strong SOC
and ferromagnetism. J.-Q. Yan et al. has investigated the doping effect of
several 3d elements by synthesizing M0.5IrTe2 (M = Mn, Fe, Co, and Ni)
polycrystals. No superconductivity was observed down to 1.80 K in any of the
doped compositions. Furthermore, no phase transition was observed and the
trigonal structure was stable down to 1.8 K[36]. Here, experimentally, the
doping effect of nickel on the properties of IrTe2 is investigated.
Differently from MxTiSe2 (M = Cr, Mn, Fe, Ni), whose CDW phase transition
disappears at x$\sim$0.1, whereas when x$\geq$0.25, the CDW state reappears
[37], the suppression on the phase transition of IrTe2 strengthens
proportionally to the doping amount and no reappearance of the phase
transition is observed within x $\leq$ 0.2. The superconductivity is observed,
while no ferromagnetism is observed even Ni is a kind of ferromagnetic
element. Theoretically, the doping effect of Mn, Fe, Co, and Ni on the
magnetic property of IrTe2 is investigated too.

##  II Experimental and calculational details

Figure 1: (a)The crystal structure of IrTe2 at the room temperature (RT). The
crystal structure of IrTe2 below the 1/5 dimerization: (b) the triangular Ir
and Te layers and (c) the layers stacking along c axis. (a, b, and c) and (a∗,
b∗, and c∗) are the crystallographic axes at high and low
temperature,respectively. The pink box represents the unit cell of the 1/5
phase, and the numeric labels denote the inequivalent Ir sites. The
Ir(3)-Ir(3) dimers and their stripes are marked with green bonds and yellow
shading, respectively. Figure 2: (a)The powder XRD pattern of Ir1-xNixTe2 at
room temperature. The inserted part is the crystal morphology of
Ir0.8Ni0.2Te2. The background scale is in millimeters for the single crystal.
(b) The normalized core-level photoemission spectrum of Ni in Ir0.8Ni0.2Te2.
Figure 3: (a)The temperature dependence of normalized
resistivity$\rho$/$\rho$300K for Ir1-xNixTe2 (x=0, 0.06, 0.08,0.1, 0.15 and
0.2). Here, the curve with “De” was obtained during the decreasing temperature
process, while the curve with “In” was obtained during the increasing
temperature process. (b) shows the superconducting transition part of
Ir1-xNixTe2. The double lines correspond to the lines gotten during the
increasing and decreasing temperature process, respectively. (c) The
temperature dependent DC magnetic susceptibility of Ir1-xNixTe2 with x=0,
0.03, 0.05, 0.06, 0.08, 0.1, 0.15 and 0.2, respectively, from 2 K to 300 K
under a field of 2 T, parallel to ab-plane. (d) Low temperature DC magnetic
susceptibility 4$\pi$$\chi$ versus temperature for Ir1-xNixTe2 with x=0.1,0.15
and 0.2, respectively, obtained in a magnetic field of 30 Oe, parallel to ab-
plane. Figure 4: Determination of the upper critical magnetic fields for
Ir0.8Ni0.2Te2. (a) and (b) Resistance as a function of magnetic field applied
out of and in ab plane from 1.9 to 2.8 K. (c)Extracted upper critical magnetic
fields Hc2 for H//ab and H$\perp$ab as a function of temperature. The solid
line is the fitted line according to WHH theory. Figure 5: M-H loop of
Ir1-xNixTe2 at 1.8K under applied magnetic fields up to 6T. Figure 6: Local
atomic geometries of (a) substitutional MIr, (b) unrelaxed Mi,t, (c) relaxed
Mi,t, and (d) Mi,o. Figure 7: Calculated density of states (DOSs). Black lines
denote total DOS, red ones denote the projected DOSs of the substitutional
magnetic atoms.  Figure 8: Electronic phase diagram of Ir1-xNixTe2 as a
function of x. Red symbols correspond to the first order structural phase
transition temperatures collected during heating process, and blue symbols
represent the superconducting transition temperatures. The superconductivity
and structural phase transition coexist at x=0.1

Single crystals of Ir1-xNixTe2 have been successfully grown via self-flux
method. Ir, Ni and Te powder was mixed in an atomic ratio of (1-x): x: 5, then
was placed in an Al2O3 crucible and sealed in an evacuated quartz tube. The
mixture was initially heated up to 950∘C and kept for several hours, then to
1160∘C for one day, and finally cooled down slowly to 900∘ C at a rate of
2∘C/h. At last, the remanent Te was separated from single crystals by
centrifuging. The DC transport data was collected by the PPMS of Quantum
Design company with a four-probe method. The magnetic susceptibility was
performed in a quantum design superconducting quantum interference device
vibrating sample magnetometer system (SQUID-VSM). XPS was performed on Axis
Ultra of Kratos Analytical with AlK$\alpha$ (h$\nu$=1486.6eV) as the
monochromatic X-ray radiation source. The power of the X-ray was about 150 W.
All peaks had been fitted using a Shirley background and Voigt (mixed
Lorentzian-Gaussian) line shapes. First-principles calculations were performed
with the PBE pseudopotentials [38] as implemented in the VASP codes [39]. The
van de Waals (vdW) correction was involved using the optB88-vdW scheme[40],
and spin polarization was taken into account. More details have been listed in
the supplementary materials.

##  III Result and Discussion

As shown by Fig. 1(a) that, at room temperature IrTe2 has a layered structure
with the space group of P$\overline{3}$m1 belonging to trigonal system. Ir
ions are coordinated octahedrally with six Te ions. Differently from typical
layered transitional metal dichalcogenides, IrTe2 layers bond with each other
through Te-Te bonding rather than weak van der Waals force[41, 36]. Below the
phase transition temperature, IrTe2 would enter into monoclinic system [37].
As illustrated in Fig. 1(b), the 1/5 phase is stable at lower temperature [42,
43, 44, 45]. In this phase, Ir-Ir dimerizes between Ir(3)-Ir(3) sites along
the a axis together with Te-Te dimerization (not shown), accompanied by charge
transfer from Ir to Te atoms [43, 46, 45]. The Ir ions with a relatively
larger valence (Ir4+ like) form dimers running along the b axis (a∗ axis),
while those with smaller valence (Ir3+like) do not participate in
dimerization. As shown by Fig. 1 (c) that, the dimerization leads to stripes
of Ir atoms with different charges, and the sequence along the a axis is
“33344,” where “3”(“4”) denotes the Ir3+like (Ir4+like) site [45]. In the
neighboring layer, the location of these stripes shifts by a unit cell,
resulting in a staircaselike arrangement with a modulation vector q1=(1/5, 0,
-1/5)[45].

Table 1: The lattice parameters of Ir1-xNixTe2 | a(Å) | c(Å)
---|---|---
IrTe2 | 3.93055 | 5.38946
Ir0.9Ni0.1Te2 | 3.92853 | 5.38401
Ir0.85Ni0.15Te2 | 3.9269 | 5.38399
Ir0.8Ni0.2Te2 | 3.92232 | 5.36866
NiTe2 | 3.8550 | 5.2659 [47]

Fig.2 (a) shows the powder X-ray diffraction patterns of Ir1-xNixTe2, with
nominal composition, at room temperature. The Ir1-xNixTe2 powder were gotten
by smashing the single crystals. The main observed reflections could be
indexed according to the P$\overline{3}$m1 space group of trigonal system. The
other impurity peaks mainly come from flux agent tellurium. Table I lists the
refined lattice parameters of Ir1-xTexNi2. It shows that both a axis and c
axis shrink slightly with the increase of the doping amount of Ni, which
coincides with the fact that both the lattice parameters of a axis and c axis
of NiTe2 are smaller than those of IrTe2. Fig. 2(b) shows two spin-orbit
doublets and two shakeup satellites (labelled as “S1”&“S2”) of Ni element in
Ir0.8Ni0.2Te2. Peak 1 and Peak 3 centered at 853.3eV and 870.3 eV,
respectively, coinciding with the binding energy of Ni2+ in NiTe2[48, 49],
which means that the Ni atoms have substituted for Ir atoms successfully. Peak
2 and Peak 4 centered at 853.3eV and 870.3 eV, respectively, can be designated
to Ni2+ arising from inevitable surface oxidation of Ir0.8Ni0.2Te2[48, 50]. S1
and S2 locating at the position of 861.00eV and 878.80eV are their
corresponding shakeup satellites, respectively.

The structural transition temperature Ts could be determined from both
transport and magnetic measurements. Fig. 3 (a) shows the normalized dc
resistivity $\rho$/$\rho$300K of Ir1-xNixTe2 over the temperature range of 2
K-300 K. For pure IrTe2, steep jumps at 275 K (reaching maximum at 262 K) on
cooling and 286 K (maximum at 278 K) on heating are observed, respectively.
The significant hysteresis could be ascribed to a first order phase
transition. The resistivity $\rho$(T) keeps decreasing with lowering
temperature after the phase transition. However, similar with the transport
behaviour of Ir1-xPtxTe2, an upward curvature is observed below 50 K, which
may be due to electron-phonon interactions [10]. The doping of Ni could
suppress the structural phase transition. It could be observed from Fig.3 (a)
and (c) that, the transition temperature Ts decreases with the increase of the
doping amount. The structural phase transition is almost suppressed at x=0.2
according to the transport result. With the suppression of the structural
phase transition, the superconducting phase emerges for samples with x$\geq$
0.10. The superconductivity and structural phase transition coexist at x=0.1.
Even no superconducting transition is observed for pure IrTe2 in the transport
measurement, when a magnetic field parallel to ab-plane is applied, a weak
diamagnetic signal is observed (see Fig.S1(a) in the Supplemental Materials).
It could be ascribed to the filament superconductivity, possibly originated
from Ir vacancies or excess Te in the samples [10, 51]. The diamagnetic signal
could also be observed in Ir0.94Ni0.06Te2 and Ir0.92Ni0.08Te2 samples (see
Fig.S1(b) and (c)), but they are not the bulk superconductivity according to
the transport measurement and superconductivity volume calculation result.
Bulk superconductivity could be observed in Ir1-xNixTe2 samples with x=0.1,
0.15 and 0.2, as shown by Fig.3(b) and (d). Their onset superconducting
transition temperatures are 2.55K, 2.6K and 2.7K, respectively. As shown by
Fig.3(d) that, the onset diamagnetic transition temperatures of three samples
coincide well with the transport result. The upper limit of the
superconducting volume fraction, measured in the ZFC process, reaches 19$\%$,
35$\%$ and 52$\%$, respectively, which means that they are bulk
superconductivity. Simultaneously, the minor distinction between FC and ZFC
processes implies weak superconducting vortex pinning, which indicates the
good quality of our single crystals. Fig.3 (c) shows the magnetic
susceptibilities of Ir1-xNixTe2 single crystals in an applied magnetic field
H=1T parallel to ab–plane. For samples with Ni-doping amount of 0 $\leq$ x
$\leq$ 0.1, the diamagnetism which is in independent of the temperature is
observed above their structural phase transitions, and then drops suddenly at
the transition temperatures, resulting in much more diamagnetism. For
Ir0.85Ni0.15Te2 and Ir0.8Ni0.2Te2 samples, the sudden drop of the magnetic
susceptibility marking the phase transition has disappeared, which coincides
with the transport result, as the resistivity jump becomes such even too.
However, compared with the magnetic result, the resistivity jump marking the
phase transition vanishes hysteretically. It could be due to the fact that the
transport measurement is more sensitive to the structural phase transition or
the field applied to the samples is not large enough. The upturn of the
magnetic susceptibility at low temperature is likely from paramagnetic
impurities.

Fig.4(a) and (b) show the resistance of Ir0.8Ni0.2Te2 as a function of
magnetic field applied out of (H$\perp$ab) and in ab plane (H//ab) from 1.9 to
2.8 K, respectively. It could be observed from the curves that the
superconductivity of Ir0.8Ni0.2Te2 shows the anisotropic behaviour. The
superconducting state is much more robust against in-plane magnetic field
(H//ab) than the out-of-plane magnetic field (H$\perp$ab). In Fig.4(a) and
(b), the upper critical field Hc2 is defined as the superconducting onset
temperature. Fig. 4(c) summarizes the upper critical field Hc2 as a function
of temperature. The linear temperature dependence close to Tc is obtained for
Hc2, suggesting the dominance of only one type of bulk carrier. The solid line
is the fitted line based on Werthamer-Helfand-Hohenberg (WHH) theory. We
obtained the upper critical field at zero temperature Hc2,⟂(0)=0.14 T and
Hc2,//(0) = 0.24 T.Based on Hc2,⟂, the coherence length
$\xi$//=$\sqrt{{\Phi}_{0}/{2{\pi}H_{{c2},\perp}}}$= 48 nm is obtained, while
$\xi$⟂=28 nm is gotten according to
$\xi$//$\xi$⟂=${\Phi}_{0}/{2{\pi}H_{{c2,//}}}$.Then the dimensionless
anisotropy parameter $\gamma$=$\xi$///$\xi$⟂$\sim$ 2,which is similar with the
reported value of Ir0.95Pd0.05Te2[41].

In order to confirm its magnetic property, the M-H loop of Ir1-xNixTe2 is
measured at 1.8K. Neither ferromagnetic hysteresis loop nor obvious remanence
is observed from Fig.5, which means that Ir1-xNixTe2 does not show
ferromagnetic order at 1.8K. The diamagnetic behaviour of Ir1-xNixTe2
(x$\neq$0) under low field could be ascribed to the superconductive
diamagnetism. All of them demonstrate paramagnetic behaviour and show
saturated at high field, with the saturation magnetization of 0.010, 0.016,
0.022 and 0.056 $\mu$B/f.u. for x=0, 0.1, 0.15 and 0.2, respectively. As shown
by the inset of Fig.5(a) that below x$\leq$ 0.15, the saturation magnetization
versus x shows almost linear behaviour, while when x=0.2, it deviates largely.
This could be interpreted as following: the paramagnetism may be from the tiny
amount of Ni atoms that have not taken up the position of Ir as substitution
doping (the substitution doping of Ni brings no magnetism which is elaborated
in the calculation part ). Furthermore, according to the inductively coupled
plasma (ICP) result, the maximal doping ratio of Ni in IrTe2 single crystals
is around 20% i.e. x=0.2. With x=0.2, the Ni atoms that could not occupy the
substitution position increase dramatically which makes its magnetization
deviate the linear behaviour largely.

To investigate the most possible doping position of M (M=Mn, Fe, Co and Ni),
the substitutional impurities on the Ir site, interstitials on the tetragonal
and octagonal sites (see Fig. 6) are considered. The calculated formation
energies in Ir-rich and Te-rich conditions are listed in Table S1 and Table S2
of the supplementary materials, respectively, and the magnetic moments of the
doped supercells are listed in Table S3. According to calculations, the
tetragonal interstitials are always not stable. Impurity atoms initially
placed on the tetragonal site spontaneously relax downward to the Te triangle
center, as shown by Fig.6 (b). On the contrary, they could stay on the
octagonal site after relaxation. All interstitials have high formation energy,
ranging from 0.15 to 1.31 eV, thus they are very unlikely to occur in
considerable equilibrium concentrations. Co, Ni and Mn impurities
energetically prefer to substitute on the Ir site from Te-rich to Ir-rich
growth conditions. Fe impurities prefer the Ir site in Te-rich conditions, but
they are energetically more favorable on the octagonal interstitial site in
Ir-rich conditions. The substitutional FeIr and CoIr always have high
formation energy, hence high equilibrium concentrations cannot be expected.
The substitutional NiIr and MnIr have very low formation energy. This is true
especially in Te-rich conditions. Both impurities have negative formation
energy, meaning that they can be easily formed in equilibrium conditions. From
Te-rich to Ir-rich conditions, the formation energy increases. But even in
extremely Ir-rich conditions, the formation energy of NiIr is just slightly
above zero (0.02 eV). Obviously, Te-rich conditions are energetically more
favorable for the substitutional doping of Ni and Mn. To study the electronic
structure of Ir1-xNixTe2 (M = Mn, Fe, Co, and Ni), the density of states is
plotted in Fig. 7. Here, the Fermi level is taken as reference. For Fe, Co and
Ni doped IrTe2, their up and down spin density of states compensate each other
well, which makes them show no magnetism. However, for Mn doped IrTe2,the up
and down spin density of states do not compensate each other which introduces
a magnetic moment of 3.0$\mu$B to the supercell.

Our results are summarized in the electronic phase diagram presented in Fig.
8. The structural phase transition temperature of IrTe2 is strongly suppressed
with Ir substitution. As soon as the phase transition is significantly
suppressed, the superconductivity appears. The superconducting state appears
at x=0.1, and superconducting Tc reaches the maximum of 2.7 K for x=0.2. The
structural phase transition and superconductivity coexist at x=0.1.

##  IV Conclusions

In conclusion, in this study, Ir1-xNixTe2 single crystals have been
synthesized. The Ni atoms have substituted for Ir in Ir1-xNixTe2. Like the CDW
behaviour, the structural phase transition shows competition and coexistence
with the superconductivity. The monoclinic phase transition has been
suppressed gradually with the increase of Ni-doping amount, giving rise to the
stabilization of the trigonal phase with superconductivity. Within
0.1$\leq$x$\leq$0.2, Ir1-xNixTe2 shows the superconductive behaviour with Tc
around 2.6K. The superconductivity shows anisotropy. Ir1-xNixTe2 only shows
weak paramagnetism and no ferromagnetic order is observed in it, which could
be ascribed to the well compensation of their up and down spin density of
states. In addition, according to the calculation result, Co, Ni and Mn
impurities energetically prefer to substitute on the Ir site from Te-rich to
Ir-rich growth conditions. Fe impurities prefer the Ir site in Te-rich
conditions, but they are energetically more favorable on the octagonal
interstitial site in Ir-rich conditions. For Fe, Co and Ni doped IrTe2, they
do not show magnetism. However, for Mn doped IrTe2, it has a magnetic moment
of 3.0$\mu$B to the supercell.

###### Acknowledgements.

This work was supported by the Ministry of Science and Technology of China
(Grant Nos. 2016YFA0301003 and 2016YFA0300501) and the National Natural
Science Foundation of China (NNSFC) (Grant Nos. U1632272, 11574201,
11804194,U1632102, 11674224, 11974243, 11774223, U1732154, and 11521404). D.Q.
acknowledges support from the Changjiang Scholars Program. D.Q., J.M., and
W.Z. acknowledge additional support from a Shanghai talent program.

## References

  * [1] J.X. Dai, K. Haule, J. J. Yang, and W. Wu et al.  Phys. Rev. B., 90:235121, 2014.
  * [2] G. Grüner.  Rev. Mod. Phys., 60:1129, 1988.
  * [3] P. Hsu, T. Mauerer, M. Bode, and W. Wu et al.  Phys. Rev. Lett., 111:266401, 2013.
  * [4] Z. Zhu, Y. Cheng, and U. Schwingenschlog et al.  Phys. Rev. B., 85:245133, 2012.
  * [5] D. Jérome, T. M. Rice, and W. Kohn.  Phys. Rev., 158:462, 1967.
  * [6] P. B. Littlewood and J. van Wezel.  Phys., 3(87), 2010.
  * [7] H. A. Jahn and E. Teller.  Proc. R. Soc. A, 161:220, 1937.
  * [8] J. J. Yang, Y. J. Choi, Y. S. Oh, and A. Hogan et al.  Phys. Rev. Lett., 108:116402, 2012.
  * [9] S. Pyon, K. Kudo, and J. M. Nohara.  Phys. Soc. Jpn., 81:053701, 2012.
  * [10] A. F. Fang, G. Xu, P. Zheng, and T. Dong et al.  Sci. Rep., 3:1153, 2020.
  * [11] R. L. Withers and J. A. Wilson.  J. Phys. Condens. Matter, 19:4809, 1986.
  * [12] J. A. Wilson and A. D. Yoffe.  Adv. Phys, 28:193, 1969.
  * [13] J. A. Wilson, F. J. DiSalvo, and S. Mahajan.  Adv. Phys., 24:117, 1975.
  * [14] K. Rossnagel.  J. Phys.: Condens. Matter, 23:213001, 2011.
  * [15] E. Morosan, H. W. Zandbergen, B. S. Dennis, and J. W. G. Bos et al.  Nat. Phys., 2:544, 2006.
  * [16] H. Hosono, A. Yamamoto, H. Hiramatsu, and Y. Ma et al.  Mater. Toda, 21:278.
  * [17] B. J. Kim, H. Jin, S. J. Moon, V. Durairaj, G. Cao, , and E. Rotenberg et al.  Phys. Rev. Lett., 101:076402, 2008.
  * [18] Y. Xia, D. Qian, D. Hsieh, and M. Z. Hasan et al.  Nat. Mater, 5:398, 2009.
  * [19] H.J. Zhang, C. X. Liu, and S. C. Zhang et al.  Nat. Phys., 5:438, 2009.
  * [20] A. P. Schnyder, S. Ryu, and Andreas W. W. Ludwig et al.  Phys. Rev. B, 78:195125, 2008.
  * [21] X. L. Qi, T. L. Hughes, and S. C. Zhang et al.  Phys. Rev. Lett., 102:187001, 2019.
  * [22] J. Linder, Y. Tanaka, and A. Sudb et al.  Phys. Rev. Lett., 104:067001, 2010.
  * [23] J. J. Yang, Y. J. Choi, Y. S. Oh, and A. Hogan et al.  Phys. Rev. Lett., 108:116402, 2012.
  * [24] C. Monne, A. Schuler, T. Jaouen, and M. Hengsberger et al.  Phys. Rev. B, 97:075110, 2018.
  * [25] N. Matsumotoa, K. Taniguchi, H. Takano, and S.Nagata et al.  J. Low Temp. Phys., 117:1129, 1999.
  * [26] A. F. Fang, G. Xu, T. Dong, and N. L. Wang et al.  Sci. Rep., 3:1153, 2013.
  * [27] G. L. Pascut, K. Haule, M. J. Gutmann, and S. A. Barnett et al.  Phys. Rev. Lett., 112:086402, 2014.
  * [28] G. L. Pascut, T. Birol, M. J. Gutmann, and V. Kiryukhin et al.  Phys. Rev. B, 90:195122, 2014.
  * [29] M. Jin Eom, K. Kim, Y. J. Jo, and J. S. Kim et al.  Phys. Rev. Lett., 113:266406, 2014.
  * [30] M. Yoshida, K. Kudo, M. Nohara, and Y. Iwasa et al.  Nano Lett., 18:3113–3117, 2018.
  * [31] J. J. Yang, Y. J. Choi, Y. S. Oh, and A. Hogan et al.  Phys. Rev. Lett., 108:116402, 2012.
  * [32] S. Pyon, K. Kudoand, and M. Nohara.  J. Phys. Soc. Jpn., 81:053701, 2012.
  * [33] D. Ootsuki, Y. Wakisaka, K. Kudo, and M. Nohara et al.  Phys. Rev. B., 44(4):1530, 1991.
  * [34] K. Kudo and M. Kobayashi et al.  J. Phys. Soc. Jpn., 82:085001, 2013.
  * [35] H. Li, S.S. Liu, C. Liu, and J.S. Zhang et al.  Chem. Phys., 22(2):556–563, 2020.
  * [36] J.-Q. Yan, A. S. Sefat, B. Saparov, and D. G. Mandrus et al.  Phys. Rev. B., 88:134502, 2013.
  * [37] N. V. Baranov, V. I. Maksimov, J. Mesot, and N. V. Selezneva et al.  J. Phys. Condens. Matter, 19:016005, 2007.
  * [38] J. P. Perdew, K. Burke, and M. Ernzerhof et al.  Phys. Rev. Lett., 77:3865, 1996.
  * [39] G. Kresse and Furthmuller et al.  Phys. Rev. B, 54:11169, 1996.
  * [40] J. Klimes, D. R. Bowler, and A. Michaelides.  J. Phys.: Cond. Matt., 22:022201, 2010.
  * [41] D. J. Yu, F. Yang, L. Miao, and J. Jia et al.  Phys. Rev. B., 89:100501, 2014.
  * [42] H. Cao, B. C. Chakoumakos, X. Chen, and D. Mandrus et al.  Phys. Rev. B., 88:115122, 2013.
  * [43] G. L. Pascut et al.  Phys. Rev. Lett., 112:086402, 2014.
  * [44] T. Toriyama, M. Kobori, T. Konishi, and M. Nohara et al.  J. Phys. Soc. Jpn., 83:033701, 2014.
  * [45] M. Jin Eom, K. Kim, Y. J. Jo, and J.H. Park et al.  Phys. Rev. Lett., 113:266406, 2014.
  * [46] D. Ootsuki, Y. Wakisaka, S. Pyon, and T. Mizokawa et al.  Phys. Rev. B, 86:014519, 2012.
  * [47] C. Xu, Bin Li, R. Sankar, and X. Xu et al.  Chem. Mater., 30:4823, 2018.
  * [48] Z. Wang and L. Zhang.  AIP Adv., 5(5):1153–1158, 2018.
  * [49] X. Chia, Z. Sofer, J. Luxa, and M. Pumera.  Chem. Eur. J., 23:11719–11726, 2017.
  * [50] C. Tang, R. Zhang, A. M. Asiri, and X. Sun et al.  Angew. Chem. Int. Ed., 56:842–846, 2017.
  * [51] X. Zhang, J.Wang, Y. Liu, and J.Wang et al.  J Phys Chem Solids, 128:245–250.