Molecular Spinless Energies of the Modified Rosen-Morse Potential Energy Model

Bulletin of the Korean Chemical Society.
2014.
Sep,
35(9):
2699-2703

- Received : May 01, 2014
- Accepted : May 04, 2014
- Published : September 20, 2014

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We solve the Klein-Gordon equation with the modified Rosen-Morse potential energy model. The bound state energy equation has been obtained by using the supersymmetric shape invariance approach. The relativistic vibrational transition frequencies for the 6
^{1}
Π
_{u}
state of the
^{7}
Li
_{2}
molecule have been computed by using the modified Rosen-Morse potential model. The calculated relativistic vibrational transition frequencies are in good agreement with the experimental RKR values.
^{1,2}
Rosen-Morse potential,
^{3-5}
Manning-Rosen potential,
^{6}
Pöschl-Teller potential,
^{7 - 9}
and Deng-Fan potential.
^{10,11}
As far as we known, the authors in these works did not quantitatively investigated relativistic rotation-vibrational energies for actual diatomic molecules. In recent years, one of present authors and co-workers
^{12-15}
studied solutions of the Klein-Gordon equation with the improved Manning- Rosen potential, improved Rosen-Morse potential, improved Tietz potential, and Morse potential, and calculated relativistic vibrational transition frequencies for the a
^{3}
Σ
_{u}
^{+}
state of the
^{7}
Li
_{2}
molecule, the 3
^{3}
Σ
_{g}
^{+}
state of the Cs
_{2}
molecule, the C
^{1}
П
_{u}
state of the Na
_{2}
molecule, and the X
^{1}
Σ
^{+}
state of the ScI molecule.
In 2012, Zhang
et al
.
^{16}
introduced the effect of inner-shell radii of two atoms for diatomic molecules into the original Rosen-Morse potential,
^{17}
and proposed a modified Rosen- Morse potential energy model,
where
,
D_{e}
denotes the dissociation energy,
r_{e}
denotes the equilibrium bond length, and
K_{e}
presents the equilibrium harmonic vibrational force constant. The parameter
K
is a dimensionless constant,
K
= 4.00.
^{18}
The ability of a potential energy model to reproduce the potential energy curve is judged by agreement with the known potential energy curve determined by the Rydberg-Klein-Rees (RKR) approach.
^{19-21}
The modified Rosen-Morse potential is superior to the Morse potential and original Rosen-Morse potential in reproducing potential energy curves for six molecule states examined in Ref..
^{16}
Recently, with the use of the standard function analysis method, Tang
et al
.
^{22}
solved the Schrödinger equation with the modified Rosen-Morse potential model to obtain rotation-vibrational energy spectra for a diatomic molecule, and calculated the vibrational energy levels calculated for the 6
^{1}
Π
_{u}
state of the
^{7}
Li
_{2}
molecule. In terms of the RKR method and the multireference configuration interaction approach, many authors have carried out on investigation on the vibrational levels and the interaction potential curves for the lithium dimer.
^{23-25}
In this work, we employ the basic concept of the supersymmetric shape invariance approach to study the bound state solutions of the Klein-Gordon equation with the modified Rosen-Morse potential energy model. We also attempt to calculate the relativistic vibrational transition frequencies of the 6
^{1}
Π
_{u}
state of the
^{7}
Li
_{2}
molecule and compare the present calculated values with the RKR data.
S
(
r
) and a vector potential
V
(
r
) for the nuclear motion of a diatomic molecule with reduced mass
μ
is given by
^{12}
where ▽
^{2}
is the Laplace operator,
E
denotes the relativistic energy of the quantum system,
c
is the speed of light, and
=
h/2π, h
is the Planck constant. We express the wave function as Ψ(
r,θ,φ
) (
u_{vJ}(r)/r
)
Y_{Jm}
= (
θ,φ
), where
Y_{Jm}
(
θ,φ
) is the spherical harmonic function. Substituting this expression into Eq. (2), we obtain the radial part of the Klein-Gordon equation,
where
v
and
J
are the vibrational and rotational quantum numbers, respectively. Taking the equal scalar and vector potentials,
S
(
r
) =
V
(
r
), Eq. (3) turns to the form
Under the nonrelativistic limit, Eq. (4) becomes a Schrödinger equation with the interaction potential 2
V
(
r
). In order to make the interaction potential as
V
(
r
), not 2
V
(
r
) in nonrelativistic limit, we take the scheme proposed by Alhaidari
et al
.
^{26}
to rescale the scalar potential
S
(
r
) and vector potential
V
(
r
), and write Eq. (4) in the form of
Considering the scalar and vector potentials as the modified Rosen-Morse potential,
S
(
r
) =
V
(
r
) =
U
_{MRM}
(
r
), we obtain the following second-order Schrödinger-like equation,
where
λ
=
. In the case of
J
= 0, we can exactly solve the above equation. When
J
≠ 0 , one can only approximately solve it. We apply the Pekeris approximation scheme to deal with the centrifugal term.
^{27}
The Pekeris approximation approach has been widely used to investigate the analytical solutions of the Klein-Gordon equation with various molecular potential models.
^{11-13}
We replace the centrifugal potential energy term by the following form
^{22}
where
r
=(
J
(
J
+1)
c
^{2}
)/
, and the coefficients
c
_{0}
,
c
_{1}
, and
c
_{2}
are given by
22
Substituting approximation expression (7) into Eq. (6), we rewrite it in the following form
where the parameters
A
,
B
,
ε_{vJ}
are defined as
With the use of the supersymmetric shape invariance approach,
^{28-30}
we solve Eq. (11). The ground-state wave function
u
_{0,J}
(
r
) is written as
where
W
(
r
) is called a superpotential in supersymmetric quantum mechanics.
^{28}
Substituting expression (15) into Eq. (11) yields the following relation satisfied by the superpotential
W
(
r
),
where
ε
_{0, J}
presents the ground-state energy. We take the superpotential
W
(
r
) in the form
where
C
_{1}
and
C
_{2}
are two constants. Substituting the above expression into expression (15), we obtain the following expression for the ground-state wave function
u
_{0, J}
(
r
)
For the bound state solutions, the wave function
u
_{vJ}
(
r
) should satisfy the boundary conditions:
u
_{vJ}
(∞) = 0 and
u_{vJ}
(0) is limitary. These regularity conditions leads us to have
C
_{1}
< 0 and
C
_{2}
< 0.
In terms of expression (17), we can construct a pair of supersymmetric partner potentials
U
_{−}
(
r
) and
U
_{+}
(
r
),
Making a comparison of Eq. (19) with Eq. (16), we have the following three relationships
Solving Eqs. (22) and (23), we obtain
Substituting expression
into expressions (19) and (20) and using Eq. (22), we can rewrite the two supersymmetric partner potentials
U
_{−}
(
r
) and
U
_{+}
(
r
) as follows
With the use of expressions (26) and (27), we can yield the following relationship
where
a
_{0}
=
C
_{2}
,
a
_{1}
is a function of
a
_{0}
,
i.e
.,
a
_{1}
=
h
(
a
_{0}
) =
a
_{0}
+
αλ
, and the reminder
R
(
a
_{1}
) is independent of
. Eq. (28) implies that the supersymmetric partner potentials
U
_{−}
(
r
) and
U
_{+}
(
r
) possess the shape invariance. Using the shape invariance approach,
^{29 }
one can exactly determine their energy spectra. The energy spectra of the potential
U
_{−}
(
r
) are given by
where the quantum number
v
= 0, 1, 2, ···. From Eqs. (11), (16) and (26), we obtain the following relationship for
ε_{vJ}
,
Substituting expressions (30) and (21) into Eqs. (31) and considering
C
_{1}
=
A
/2
C
_{2}
−
C
_{2}
/2
λ
, we obtain the following expression for
ε_{vJ}
,
Substituting expression (25) into expression (32), we have
Substituting expressions (12) and (13) into expression (33) and using
, we obtain the relativistic rotation-vibrational energy eigenvalue equation for the diatomic molecule in the presence of equal scalar and vector modified Rosen-Morse potential energy models,
where
v
= 0, 1, 2, 3,...,
λ
=
, and we have made a replacement for
When
J
= 0, we obtain the relativistic vibrational energy eigenvalue equation for the diatomic molecule with equal scalar and vector modified Rosen-Morse potentials,
With the use of the superpotential given in expression (17) and the ground-state wave function given in expression (18), one can determine the excited state wave functions by employing the explicit recursion operator approach.
^{31,32}
k_{e}
is defined as the second derivative of the potential energy function
U
(
r
) for a diatomic molecule, namely
Using this definition and the relation
, we deduce the following expression for the potential parameter
α
appearing in the modified Rosen- Morse potential,
where
ω_{e}
denotes the equilibrium harmonic vibrational reciprocal wavelength, and
W
is the Lambert
W
function, which satisfies
We consider the 6
^{1}
Π
_{u}
state of the
^{7}
Li
_{2}
molecule. The experimental values of
D_{e}
,
r_{e}
, and
ω_{e}
for the
^{7}
Li
_{2}
6
^{1}
Π
_{u}
state are taken from the literature
^{23}
: cm
^{−1}
, Å, and cm
^{−1}
. Taking these experimental data as inputs and applying expression (36), we can determine the value of the parameter
α
in the modified Rosen-Morse potential. The experimental RKR data points for the 6
^{1}
Π
_{u}
state of
^{7}
Li
_{2}
are shown in
Figure 1
, which also contains the curves reproduced by employing the four three-parameter potential models: modified Rosen- Morse potential, Rosen-Morse potential,
^{17,33}
Morse potential,
^{34}
and Frost-Musulin potential.
^{33,35}
The Rosen-Morse, Morse, and Frost-Musulin potential functions are given by, respectively
(Color online) RKR data points and four empirical potential energy models for the 6^{1}Π_{u} state of the ^{7}Li_{2} molecule.
An outlook for the range covered by the experimental RKR data points tells us that the modified Rosen-Morse potential is superior to the Rosen-Morse, Morse, and Frost-Musulin potential functions for the 6
^{1}
Π
_{u}
state of the
^{7}
Li
_{2}
molecule.
An available potential function should well model the experimental RKR potential curve, and satisfy the Lippincott criterion, i.e., an average absolute deviation of less than 1% of the dissociation energy
D_{e}
.
^{36}
The average absolute deviation is defined as
where is the number of experimental data points,
U
_{exp}
(
r
) and
U
_{calc}
(
r
) are the experimentally determined potential and the empirical potential, respectively. The average absolute deviations of the modified Rosen-Morse potential, Rosen- Morse potential, Morse potential, and Frost-Musulin potential for the 6
^{1}
Π
_{u}
state of the
^{7}
Li
_{2}
molecule from the RKR potential reported by Grochola
et al
.
^{23}
are 0.648% of
D_{e}
, 1.053% of
D_{e}
, 1.901% of
D_{e}
, and 2.64% of
D_{e}
, respectively. These deviation values show that the modified Rosen-Morse potential is best for the examined four potential models in reproducing the potential energy curve for the 6
^{1}
Π
_{u}
state of the
^{7}
Li
_{2}
molecule.
By employing energy eigenvalue Eq. (35), we calculate relativistic vibrational transition frequencies for the 6
^{1}
Π
_{u}
state of the
^{7}
Li
_{2}
molecule. The present calculated values are given in
Table 1
, in which we also present the RKR values taken from the literature.
^{23}
From
Table 1
, we observe that the relativistic vibrational transition frequencies obtained by using the modified Rosen-Morse potential model are in good agreement with the RKR data.
A comparison of the calculated relativistic vibrational transition frequencies and experimental RKR values for the 6^{1}Π_{u} state of the ^{7}Li_{2} molecule (in units of cm^{−1})
^{1}
Π
_{u}
state of the
^{7}
Li
_{2}
molecule. The relativistic vibrational transition frequencies predicted with the modified Rosen-Morse potential model are in good agreement with the experimental RKR data.

Klein-Gordon equation
;
Modified Rosen-Morse potential model
;
Vibrational transition
;
Lithium dimer

Introduction

There has been the continuous interest in investigating solutions of the Klein-Gordon equation with some diatomic molecule potential energy models, such as the Morse potential,
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Bound State Solutions

The Klein-Gordon equation with a scalar potential
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Discussion

The force constant
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A comparison of the calculated relativistic vibrational transition frequencies and experimental RKR values for the 61Πustate of the7Li2molecule (in units of cm−1)

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Conclusions

In this work, we have investigated the bound state solutions of the Klein-Gordon equation with the Morse potential energy model. The energy eigenvalue equation has been obtained using the supersymmetric shape invariance method. We calculate the relativistic vibrational transition frequencies for the 6
Acknowledgements

Publication cost of this paper was supported by the Korean Chemical Society.

Sun H.
2011
Bull. Korean Chem. Soc.
32
4233 -
** DOI : 10.5012/bkcs.2011.32.12.4233**

Bayrak O.
,
Soylu A.
,
Boztosun I.
2010
J. Math. Phys.
51
112301 -
** DOI : 10.1063/1.3503413**

Yi L. Z.
,
Diao Y. F.
,
Liu J. Y.
,
Jia C. S.
2004
Phys. Lett. A
333
212 -
** DOI : 10.1016/j.physleta.2004.10.054**

Ibrahim T. T.
,
Oyewumi K. J.
,
Wyngaardt S. M.
2012
Eur. Phys. J. Plus
127
100 -
** DOI : 10.1140/epjp/i2012-12100-5**

Qiang W. C.
,
Sun G. H.
,
Dong S. H.
2012
Ann. Phys. (Berlin)
524
360 -
** DOI : 10.1002/andp.201200030**

Wei G. F.
,
Zhen Z. Z.
,
Dong S. H.
2009
Eur. J. Phys.
7
175 -

Qiang W. C.
,
Dong S. H.
2008
Phys. Lett. A
372
4789 -
** DOI : 10.1016/j.physleta.2008.05.020**

Koçak G.
,
Taşkin F.
2010
Ann. Phys. (Berlin)
522
802 -
** DOI : 10.1002/andp.201000031**

Xu Y.
,
He S.
,
Jia C. S.
2010
Phys. Scr.
81
045001 -
** DOI : 10.1088/0031-8949/81/04/045001**

Dong S. H.
2011
Commun. Theor. Phys.
55
969 -
** DOI : 10.1088/0253-6102/55/6/05**

Oluwadare O. J.
,
Oyewumi K. J.
,
Akoshile C. O.
,
Babalola O. A.
2012
Phys. Scr.
86
035002 -
** DOI : 10.1088/0031-8949/86/03/035002**

Jia C. S.
,
Cao S. Y.
2013
Bull. Korean Chem. Soc.
34
3425 -
** DOI : 10.5012/bkcs.2013.34.11.3425**

Liu J. Y.
,
Du J. F.
,
Jia C. S.
2013
Eur. Phys. J. Plus
128
139 -
** DOI : 10.1140/epjp/i2013-13139-4**

Chen T.
,
Lin S. R.
,
Jia C. S.
2013
Eur. Phys. J. Plus
128
69 -
** DOI : 10.1140/epjp/i2013-13069-1**

Jia C. S.
,
Chen T.
,
He S.
2013
Phys. Lett. A
377
682 -
** DOI : 10.1016/j.physleta.2013.01.016**

Zhang G. D.
,
Liu J. Y.
,
Zhang L. H.
,
Zhou W.
,
Jia C. S.
2012
Phys. Rev. A
86
062510 -
** DOI : 10.1103/PhysRevA.86.062510**

Rosen N.
,
Morse P. M.
1932
Phys. Rev.
42
210 -
** DOI : 10.1103/PhysRev.42.210**

Frost A. A.
,
Musulin B.
1954
J. Am. Chem. Soc.
76
2045 -
** DOI : 10.1021/ja01637a005**

Rydberg R.
1933
Z. Phys.
80
514 -
** DOI : 10.1007/BF02057312**

Klein O.
1932
Z. Phys.
76
226 -
** DOI : 10.1007/BF01341814**

Rees A. L. G.
1947
Proc. Phys. Soc.
59
998 -
** DOI : 10.1088/0959-5309/59/6/310**

Tang H. M.
,
Liang G. C.
,
Zhang L. H.
,
Zhao F.
,
Jia C. S.
2014
Can. J. Chem.
92
341 -
** DOI : 10.1139/cjc-2013-0563**

Grochola A.
,
Jastrzebski W.
,
Kowalczyk P.
2008
Mol. Phys.
106
1375 -
** DOI : 10.1080/00268970802275595**

Jędrzejewski-Szmek Z.
,
Grochola A.
,
Jastrzebski W.
,
Kowalczyk P.
2007
Chem. Phys. Lett.
444
229 -
** DOI : 10.1016/j.cplett.2007.07.042**

Li D.
,
Xie F.
,
Li L.
,
Lazoudis A.
,
Lyyra A. M.
2007
J. Mol. Spectrosc.
246
180 -
** DOI : 10.1016/j.jms.2007.09.008**

Alhaidari A. D.
,
Bahlouli H.
,
Al-Hasan A.
2006
Phys. Lett. A
349
87 -
** DOI : 10.1016/j.physleta.2005.09.008**

Pekeris C. L.
1934
Phys. Rev.
45
98 -
** DOI : 10.1103/PhysRev.45.98**

Witten E.
1981
Nucl. Phys. B
185
513 -

Gendenshtein L. E.
1983
Sov. Phys.-JETP Lett.
38
356 -

Cooper F.
,
Khare A.
,
Sukhatme U.
1995
Phys. Rep.
251
267 -
** DOI : 10.1016/0370-1573(94)00080-M**

Dabrowska J. W.
,
Khare A.
,
Sukhatme U. P.
1988
J. Phys. A: Math. Gen.
21
L195 -
** DOI : 10.1088/0305-4470/21/4/002**

Jia C. S.
,
Wang X. G.
,
Yao X. K.
,
Chen P. C.
,
Xiao W.
1998
J. Phys. A: Math. Gen.
31
4763 -
** DOI : 10.1088/0305-4470/31/20/013**

Morse M.
1929
Phys. Rev.
34
57 -
** DOI : 10.1103/PhysRev.34.57**

Frost A. A.
,
Musulin B.
1954
J. Chem. Phys.
22
1017 -
** DOI : 10.1063/1.1740254**

Jia C. S.
,
Diao Y. F.
,
Liu X. J.
,
Wang P. Q.
,
Liu J. Y.
,
Zhang G. D.
2012
J. Chem. Phys.
137
014101 -
** DOI : 10.1063/1.4731340**

Steele D.
,
Lippincott E. R.
,
Vanderslice J. T.
1962
Rev. Mod. Phys.
34
239 -
** DOI : 10.1103/RevModPhys.34.239**

Citing 'Molecular Spinless Energies of the Modified Rosen-Morse Potential Energy Model
'

@article{ JCGMCS_2014_v35n9_2699}
,title={Molecular Spinless Energies of the Modified Rosen-Morse Potential Energy Model}
,volume={9}
, url={http://dx.doi.org/10.5012/bkcs.2014.35.9.2699}, DOI={10.5012/bkcs.2014.35.9.2699}
, number= {9}
, journal={Bulletin of the Korean Chemical Society}
, publisher={Korean Chemical Society}
, author={Jia, Chun-Sheng
and
Peng, Xiao-Long
and
He, Su}
, year={2014}
, month={Sep}