J. Nig. Soc. Phys. Sci. 3 (2021) 282–286

Journal of the
Nigerian Society

of Physical
Sciences

Eigensolution to Morse potential for Scandium and Nitrogen
monoiodides

C. A. Onatea,b,∗, G. O. Egharevbaa, D. T. Bankolea

aDepartment of Physical Sciences, Landmark University, Omu-Aran, Nigeria.
bLandmark University SDG 4 (Quality Education)

Abstract

The solutions for Morse potential energy function under the influence of Schrödinger equation are examined using supersymmetric approach. The
energy equation obtained was used to generate eigenvalues forX1

∑+state of scandium monoiodide (ScI) andX3 ∑−state of nitrogen monoiodide
(NI) respectively by imputing their respective spectroscopic parameters. The calculated results for the two molecules aligned excellently with the
predicted/observed values.

DOI:10.46481/jnsps.2021.407

Keywords: Eigensolutions, Wave equation, Bound state, Molecules

Article History :
Received: 13 September 2021
Received in revised form: 27 September 2027
Accepted for publication: 28 October 2021
Published: 29 November 2021

c©2021 Journal of the Nigerian Society of Physical Sciences. All rights reserved.
Communicated by: W. A. Yahya

1. Introduction

The study of the interactions for atomic molecules can be
carried out using some physical potential models. Over the
years, the energy eigenvalues of different molecules were ob-
tained for different potential terms and reported. Among the
reported works are, the energy eigenvalues of four molecules
for Kratzer potential by Bayrak et al. [1]. In ref. [2], Ikhdair
obtained energy eigenvalues of eight molecules for Manning-
Rosen potential. Ikhdair and Sever [3], obtained energy eigen-
values of six molecules for Kratzer-type potential, Falaye et
al. [4], under the study of Tietz-Wei diatomic molecular po-
tential function, obtained energy eigenvalues of ten molecules.
In ref. [5], the energy eigenvalues of some molecules were
obtained for pseudoharmonic potential. Onate et al. [6], ob-
tained energy eigenvalues of six molecules for hyperbolic-sinus

∗Corresponding author tel. no:
Email address: oaclems14@physicist.net (C. A. Onate )

potential model. Recently, Farout et al. [7], obtained exact
momentum states of some molecules for improved deformed
exponential-type potential. Despite all the reports given above,
it has not been easy to deduce a potential energy for various
molecules, thus, a challenging issue subsist in the study. How-
ever, some potential energy function that can explain various
diatomic molecules in good agreement with experimental data
have been proposed based on the experimental constants such
as the dissociation energy and equilibrium bond length, e.g.
Deng-Fan, Improved Rosen-Morse, Morse, Tietz-Hua oscilla-
tor, improved generalized Poschl-Teller oscillator [8-13]. The
energies of these potential functions were calculated for cesium
molecule, sodium dimer, nitrogen dimer, hydrogen molecule
and potassium. The result obtained for each molecule aligned
with the experimental data. However, it is noted that only few
potential functions were examined and their results compared
with the experimental results. Motivated by the interest in the
Morse potential function as one of the different potential en-

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Onate et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 282–286 283

Figure 1: The Morse potential function for scandium monoiodide and nitrogen
monoiodide

ergy functions suggested to obtain information about diatomic
molecular structures, this work aims to determine the energy
eigenvalues of scadium monoiodide and nitrogen monoiodide
for a Morse potential energy function. The Morse potential
model is a molecular potential model that is used to describe the
interaction between two atoms in the atomic domain, as well as
interaction closed to the surface. Several works have been re-
ported on the Morse potential with different methods. However,
most of the reports given are on the Morse potential of the form
[17]

V (r) = −2Dee
−α(r−re ) + De

−2α(r−re ), (1)

where Deis the dissociation energy, reis the equilibrium bond
length, ris the internuclear separation and αis a screening pa-
rameter. According to Akanbi et al. [18], the Morse potential
given above has a minimum value at r = re and it is zero at
r = ∞. The authors further emphasized that the barrier is as-
sumed to be outside the influence of the Morse oscillator. In
refs. [19], another form of Morse potential called Shifted Morse
was studied. The form of Morse potential is given as

V (r) = (` + 1)2 − (2` + 3)e−x + e−2x. (2)

In the present study, the interacting Morse potential

V (r) = De(1 − 2e
−α(r−re ) + e−2α(r−re )) (3)

will be considered. The form of Morse potential model was
studied by Desai et al. [20] in one of their papers. However, the
explicit detail analysis for the solutions of the Morse potential
as well as the energy equation were not given. In this study, the
analysis of eigenvalues and eigenfunctions for the Morse po-
tential will be given in detail. Figure 1 below, depicts the shape
of Morse potential function for scandium monoiodide (ScI) and
nitrogen monoiodide (NI).

2. Bound state solution of the Schrödinger equation using
supersymmetric approach

Here, the eigenvalue and eigenfunction for the Morse po-
tential is obtained. The radial Schrödinger equation with an
interacting potential V (r)is given by[

−
~2

2µ
d2

dr2
− En,` + V (r)

]
Rn,`(r) = 0, (4)

where En,`is the non-relativistic energy of the system, ~is a re-
duced Planck’s constant, µ is the reduced mass, Rn,`(r)is the
wave function. The solutions of Eq. (2) can be obtained using
different traditional methodologies. In this work, as said earlier,
the supersymmetric approach will be used for the calculation.
The supersymmetric approach is one of the approximate meth-
ods used to solve wave equations. The method depends on the
proposition of superpotential. To use this method, first we plug
Eq. (3) into Eq. (4) to have

d2Rn(r)
dr2

=
2µ(De − En − 2Dee−α(r−re ) + Dee−2α(r−re ))

~2
Rn(r) = 0. (5)

To solve the equation above using supersymmetric and shape
invariance approach [21-23], the next step is to write a ground
state wave function

R0(r) = ex p
(
−

∫
U(r)dr

)
, (6)

where, U(r)is the superpotential function. Invoking Eq. (6)
onto Eq. (5) gives another relation satisfied by the superpoten-
tial functionU(r) :

U 2(r) −
dU(r)

dr

=
2µ(De − En)

~2
+

2µ(Dee−α(r−re ) − 2De)
~2

e−α(r−re ) (7)

To validate the compatibility of the two sides of Eq. (7) [24],
we express the superpotential function as

U(r) = ρ0 −ρ1e
−αr. (8)

The two terms ρ0 and ρ1 in Eq. (8) are superpotential constants
and their respective values will soon be determined. Plugging
Eq. (8) into Eq. (7) with some mathematical manipulations and
simplifications leads to the following reasonable equations

ρ20 =
2µDe
~2
−

2µEn
~2

, (9)

ρ1 =

√
2µDee2αre

~2
, (10)

ρ0 =

4µDe eαre
~2 −αρ1

2ρ1
. (11)

We consider the bound state solutions that demand the wave
function Rn(r)which satisfy the boundary conditions for Rn(∞) =

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Onate et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 282–286 284

0and Rn(0)is limitary. These regularity conditions suggest that
both ρ0and ρ1are greater than zero. Having determined the two
superpotential constants, the supersymmetric partner potentials
U 2(r) ± dU(r)dr can easily be constructed as follows

V+(r) = U
2(r)+

dU(r)
dr

= ρ20−ρ1(2A−α)e
−αr

+ρ21e
−2αr,(12)

V−(r) = U
2(r)−

dU(r)
dr

= ρ20−ρ1(2A+α)e
−αr

+ρ21e
−2αr.(13)

Eq. (12) and Eq. (13) are related by a simple relation

V+(r, a0) = V−(r, a1) + R(a1), (14)

where a0is an old set of parameters and a1is a new set of param-
eters uniquely determined from the old set of parameters. The
termR(a1)is called a reminder term that do not dependent of the
variable r. In the concept of shape invariance potential, a0 = ρ0
as ρ0 → ρ0 + αn. Using the partner potential V−(r, a1), Eq. (9),
Eq. (10) and Eq. (11), the energy of the Morse potential func-
tion can be obtain following

E(−)n =
∑
k=1

R(ak) = R(a1) + R(a2) + R(a3)

+...... + R(an−1) + R(an) =
(
ρ1 −αa0

2a0

)2
+

(
ρ1 −αa1

2a1

)2
−

(
ρ1 −αa1

2a1

)2
+

(
ρ1 −αa2

2a2

)2
−

(
ρ1 −αa2

2a2

)2
+

(
ρ1 −αa3

2a3

)2
−

(
ρ1 −αa3

2a3

)2
+

(
ρ1 −αa4

2a4

)2
+ ....(

ρ1 −αan−1
2an−1

)2
+

(
ρ1 −αan

2an

)2
=

(
ρ1 −αa0

2a0

)2
+

(
ρ1 −αan

2an

)2
=

(
4µDeeαre −αa0

2a0

)2
+

(
4µDeeαre −αan

2an

)2
(15)

Following the formalism of supersymmetric approach, the com-
plete energy equation is obtain as

En = De −
α2~2

2µ


2µDe eαre
α2~2 −

(
n + 12

) √
2µDe e2αre
α2~2√

2µDe e2αre
α2~2


2

. (16)

2.1. Wave Function

To obtain the wave function, we make a transformation of
the form y = 1eαr and invoke it on Eq. (5) to have

d2Rn(y)
dy2

−
1
y

dRn(y)
dy

+

2µ[En−De +De eαre y(2−eαre y)]
α2~2

y2
Rn(y) = 0.(17)

Following the paper of Tezcan and Sever [25], the radial wave
function for the Morse potential becomes

Rn(y) = Ny

√
2µ(En−De )
α2~2 e

−y

√
2µDe e2αre

α2~2 L
2
√

2µ(En−De )
α2~2

n

(
2
√

2µ(En−De )
α2~2 y

)
.(18)

(a)

(b)

Figure 2: Variation of En (cm−1)against De (cm−1) and αrespectively

3. Discussion

The shape of Morse potential for scandium monoiodide and
nitrogen monoiodide is shown in Figure 1. In Figures 2 (a) and
(b), the variation of energy against the dissociation energy and
screening parameter respectively are shown. The energy and
each of the dissociation energy and screening parameter respec-
tively varies inversely with each other. In each case, the highest
quantum state has the highest energy. The energies at various
quantum state at Deand α = 0respectively are zero, which is
the point of convergence for the energies at different quantum
states. The vibrational energies of the Morse potential for var-
ious values of the screening parameter, quantum number and
dissociation energy are presented in Table 1. The energy of the
system rises with an increase in the quantum number, screen-
ing parameter and dissociation energy respectively. For a unity
value of the dissociation energy, the energy of the system has
turning point as the quantum number and the screening param-

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Onate et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 282–286 285

Table 1: Vibrational energies (incm−1) of Morse potential model with µ = ~ = 1
and re = 0.5Ȧ for three values ofDe(cm1) and various values ofn and α(cm−1).

n α En(De = 1) En(De = 5) En(De = 10)
0

1

2

3

0.05
0.25
0.55
0.75
0.05
0.25
0.55
0.75
0.05
0.25
0.55
0.75
0.05
0.25
0.55
0.75

0.0350428 0.0787444 0.1114909
0.1689642 0.3874722 0.5512045
0.3510962 0.8318137 1.1920249
0.4600176 1.1155416 1.6067385
0.1032535 0.2343583 0.3325977
0.4600176 1.1155416 1.6067385
0.8264137 2.2685666 3.3491997
0.9581778 2.9247499 4.3983404
0.1689642 0.3874722 0.5512045
0.6885710 1.7811110 2.5997725
0.9992312 3.4028193 5.2038744
0.8938379 4.1714581 6.6274424
0.2321749 0.5380861 0.7673113
0.8546244 2.3841805 3.5303065
0.8695486 4.2345720 6.7560492
0.2669981 4.8556664 8.2940444

Table 2: Comparison of the calculated energies (incm−1) forX1
∑+ state of ScI

andX3
∑
− state of NI with the predicted experimental RKR values of the Morse

potential function

n S cI N I
Calculated [27] LTE calculated [27] LTE

0 138.4121 138.3 0.1121 301.6 301.1 0.5000
1 414.7976 413.9 0.4976 898.0135 896.6 1.4135
2 688.6233 687.7 0.9233 1486.4617 1482.3 3.1617
3 961.1402 959.9 1.2402 2062.5594 2058.8 3.7594
4 1232.1101 1230.4 1.7101 2629.9899 2625.9 4.0899
5 1501.6044 1499.3 2.3044 3187.9671 3183.6 4.3671
6 1769.1278 1766.5 2.6278 3736.8788 3731.9 4.9788

eter increases respectively.
The observed data and the calculated energies for X1

∑+state
of ScI and X3

∑
−state of NI obtained using Eq. (15) are given

in Table 2. The relative deviation LTE (calculated result mi-
nus experimental result) of the calculated results from the ex-
perimental results are also given in Table 2. The experimental
values used for this work are taken from ref. [27]. For ScI,
De = 2.858eV, re = 2.6708Å, ωe = 277.18cm−1while that of
NI are De = 1.648eV, re = 1.9653Å, ωe = 604.7cm−1. The
screening parameter is calculated using

α = πcωe

√
2µ
De
. (19)

It is shown that the relative deviation becomes larger as
the vibrational quantum state increases for the two molecules.
However, the relative deviation forX3

∑
−state of NI are higher

compared to the relative deviation forX1
∑+state of ScI. This

simply means that the ScI are more fitted for the calculation
compared to NI. The average absolute percentage deviation for
each of the molecule is calculated using the formula

σav =
100
N

∑
v

∣∣∣∣∣ EV − ERKRERKR
∣∣∣∣∣ . (20)

where ERKR is the observed values, EV is the present results and
is the number of observed data points. The formula seems to be
the revised version of what is given in ref. [18]. The calculated
results in this study are greater than the experimental values,
thus, to avoid negative deviation, we revised the order of the
both the experimental and calculated values. The average abso-
lute percentage deviation for ScI is 0.021% while that of NI is
0.032%. To determine the proximity of the present results to the
predicted values, the percentage error for each of the calculated
result is computed using the formula

errp =

∑
v

LT E

ERKR
× 100. (21)

The percentage error for the results of ScI is 0.14% while that
of the NI is 0.23%.

4. Conclusion

In the present study, we calculated the energies of ScI and
NI for a Morse potential function. The computed results aligned
excellently with the observed results for the two molecules. The
computation of the percentage error shows that the Morse po-
tential is more fitted in the computation for ScI than NI since
the percentage error in ScI is smaller than that of NI.

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