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Analytical calculations of thermodynamic functions of lithium dimer using modified Tietz and Badawi-Bessis-Bessis potentials
Accepted Manuscript Analytical calculations of thermodynamic functions of lithium dimer using modified Tietz and Badawi-Bessis-Bessis potentials R. Khordad, A. Ghanbari PII: DOI: Reference:
S2210-271X(19)30081-7 https://doi.org/10.1016/j.comptc.2019.03.019 COMPTC 12454
To appear in:
Computational & Theoretical Chemistry
Received Date: Revised Date: Accepted Date:
19 September 2017 16 March 2019 16 March 2019
Please cite this article as: R. Khordad, A. Ghanbari, Analytical calculations of thermodynamic functions of lithium dimer using modified Tietz and Badawi-Bessis-Bessis potentials, Computational & Theoretical Chemistry (2019), doi: https://doi.org/10.1016/j.comptc.2019.03.019
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Analytical calculations of thermodynamic functions of lithium dimer using modified Tietz and Badawi-Bessis-Bessis potentials R. Khordad* and A. Ghanbari Department of Physics, College of Sciences, Yasouj University, Yasouj, Iran
Abstract We have analytically solved the Klein-Gordon (KG) equation for two different potentials. We have used the modified Tietz (MT) and the Badawi-Bessis-Bessis (BBB) diatomic molecular potentials and derived the analytical expressions for the energy spectra. Then, we have analytically obtained the thermodynamic functions for a3
state of the 7Li2 molecule using
the two potentials. For this goal, we have calculated the partition function, mean energy, specific heat, and entropy of the molecule as a function of temperature.
Keywords: Lithium molecule, Thermodynamic functions, Entropy
*Corresponding author, E-mail: [email protected]
1
1. Introduction It is fully known that the determination of exact solutions of the non-relativistic or relativistic wave equations plays a key role in quantum systems because the solutions contain important information regarding the system. One of the most famous wave equations is the KleinGordon (KG) equation. The equation is a relativistic wave equation correctly describing the spin-zero particles such as spinless pion and a composite particle. The KG equation has received considerable attention in the literature [1-5]. We can consider the KG equation as a relativistic version of the Schrödinger equation. It is noteworthy that the calculation of analytical solutions is possible only in a few simple cases such as the hydrogen atom and the harmonic oscillator [6-10]. Due to the complicated form of several potentials, the calculation of analytical solutions appears to be a nontrivial task which requires considerable theoretical effort. Examples of the potentials are Tietz-Wei, Deng-Fan, Hulthen, Morse, Manning-Rosen, and Tietz [11-17]. Hitherto, the KG equation has been solved for several potentials. For example, Sun et al [18] have calculated the bound state solutions of the relativistic Klein-Gordon equation with the Tietz-Wei (TW) diatomic molecular potential. Hassanabadi et al [19] have obtained the spectra and the eigenfunctions of the relativistic Klein-Gordon equation for Deng-Fan potential. Hamzavi and Amirfakhrian [20] have solved the KG equation for Deng-Fan potential in arbitrary N-dimension. They have employed an improved approximation scheme to the centrifugal term. Rezaei Akbarieh and Motavali [21] have reported the exact solutions of the one-dimensional KG equation for the Rosen-Morse type potential with equal scalar and vector potentials. Arda and Sever [22] have solved the radial part of the KG equation for the generalized Woods-Saxon potential using the Nikiforov-Uvarov method. Ikhdair [23] has obtained the approximate bound-state rotational-vibrational energy levels. Jia et al [24] have used the Manning-Rosen to investigate the vibrational levels and the interatomic interaction potential for the a3
state of 7Li2 in terms of the Rydberg-Klein-Rees (RKR) method. To
obtain more information, the reader can refer to [25-30]. Recently, the lithium dimer (7Li2 or 6Li2 ) has attracted considerable attention because it is the second smallest stable homonuclear molecule next to H2. So far, many researchers have employed different procedures to study the vibrational levels of the a3
state of the 7Li2.
Examples of the methods are Rydberg-Klein-Rees method and ab initio approach [31-35]. Recently, Jia et al [36] have reported temperature-dependent thermodynamic functions of the 7
Li2 molecular gas using the Manning-Rosen (MR) potential energy. They have solved the 2
Schrödinger equation with the MR potential to obtain the energy levels. In this work, we have investigated the thermodynamic properties of the a3
state of the 7Li2 molecular gas
using two different potentials. Since the MT and BBB potentials have the same behaviors in comparing with Manning-Rosen (MR) potential (see Fig. 1), we have MT and BBB potentials to determine thermodynamic functions of 7Li2 molecular. It is fully known that modeling diatomic potential energy through analytical functions is of fundamental importance. Examples of the potential functions are Rosen-Morse, Tietz, improved Tietz, Wei, and Badawi-Bessis-Bessis potential [29, 37-42].
2. Theory and model 2.1. Modified Tietz potential In 1963, Tietz proposed a potential energy function for diatomic molecules. The Tietz potential is one of the very best analytical potential models for the vibrational energy of diatomic molecules [29, 37]:
where
is the dissociation energy. The parameters , , , and
can be determined so as to
fit calculated values of spectroscopic parameters to observed ones. Another diatomic potential is Tietz-Wei (TW) potential. The TW potential is much more realistic than the Morse potential in describing molecular dynamics at moderate and high rotational and vibrational quantum numbers. Kunc and Vazquez obtained the analytical expressions for the rotational-vibrational energy levels of diatomic TW potential using the Hamilton-Jacobi theory and the Bohr-Sommerfeld quantization rule [38]. In 2012, Jia et al [13] proposed a more convenient form of the original Tietz potential function. The modified Tietz (MT) potential is given by [43]
where
is the internuclear separation,
energy, and
is the equilibrium bond length,
is the dissociation
is an adjustable parameter which governs the range of the interaction. It is well
known that the original expression of the Tietz potential function Eq. (1) is defined in terms of five parameters whereas the modified Tietz potential only has four parameters. Recently, 3
Tang et al [44] have obtained the rotation-vibrational energy spectra and the un-normalized wave functions of the modified Tietz potential with solving the Schrödinger equation. Also, Zhang et al [45] have solved the Schrödinger equation with the modified Tietz potential model in D spatial dimensions. They have used the supersymmetric shape invariance approach and obtained the D-dimensional rotation-vibrational energy spectra. In these works [44, 45], the authors have studied Na2 system by solving the Schrödinger equation. It is to be noted that Liu et al [43] have solved the Klein-Gordon equation with the modified Tietz (MT) potential in terms of the supersymmetric shape invariance approach. They have calculated the relativistic vibrational transition frequencies for the Na 2 molecule. Also, Liu et al [46] have solved the Klein-Gordon equation with the improved Tietz potential in higher spatial dimensions. They have calculated relativistic vibrational energies for the ground electronic state of the CO molecule. In this work, we attempt to solve the Klein-Gordon equation with the modified Tietz (MT) potential in terms of the function analysis method. The Klein–Gordon equation for a quantum system with the rest mass and a vector potential
in a scalar potential
is given by
where we have set
. Using the separation of variable method for the wave functions , the Klein-Gordon equation of the s-wave for equal scalar and
vector potentials is given by
where
and
represent the rest mass and energy of the spinless particle, respectively. Also,
we have applied
. For a diatomic system, the rest mass
reduced mass of the diatomic molecule. The scalar potential
can be regarded as the and vector potential
are chosen as equal to modified Tietz potentials as defined in Eq. (2). The potential
is
called vector potential because it treated as the fourth component of a Lorentz-vector. The KG equation has square terms causing a great complexity for finding the exact solutions especially when unequal scalar and vector potentials are studied. Under the choice of equal scalar and vector potentials, Eq. (4) reduces to the Schrödinger equation for the potential in the nonrelativistic limit. Here, we have chosen
4
. In this
study, we use the modified Tietz potential for calculating the vibrational energy levels and ignore the rotational terms. To solve Eq. (4) for the modified Tietz potential, we consider the following separation of variable
where
. Therefore, we have
Inserting above relations into Eq. (4), we can obtain
where
In order to obtain the solution to Eq. (6), we take the wave functions of the form
where
With substituting of Eq. (8) into Eq. (6), we can obtain the following differential equation
5
whose solutions are nothing but the hypergeometric functions Here
and
Now, we should apply the properties of the hypergeometric functions. It is to be noted that the series of
given in Eq. (11) approaches infinity unless
is a negative integer like
.
To obtain the properties of the hypergeometric functions, the reader can refer to [47]. Therefore
where
We can obtain the energy levels as
2.2. Badawi-Bessis-Bessis potential In 1972, Badawi et al [41] introduced a general potential energy expression for several empirical potential functions as
6
where
, and
are five adjustable parameters. It is worth mentioning that Du et al
[40] have shown that the BBB potential is equivalent to the Tietz potential for a diatomic molecule. Applying separation of variable method for the wave functions, we can solve Eq. (4) with considering the scalar potential
and vector potential V(r) as equal to BBB potential [Eq.
(17)]. For this purpose, we use the following change of variable
Therefore, we have
Now, Eq. (4) can be written as
where
We consider the following wave function
where
With inserting Eq. (21) into Eq. (19), the following differential equation is achieved
where the solutions of the equation are the hypergeometric functions 7
Here
and
With respect to properties of the hypergeometric functions, it should be noted the series of given in Eq. (24) approaches infinity unless
where
is a negative integer like –
[47].
We can obtain the energy levels as
To obtain the partition function of a given system, one can need the energy levels. Now, one can determine the partition function with using the energy levels Eqs. (16) and (29).
3. Thermodynamic properties A starting point to derive thermodynamic properties system is the partition function. The quantity is obtained by direct summation over all possible states of the system,
where
,
is the Boltzmann's constant and
is the temperature. To obtain the
partition function for two potentials, we insert the energy levels from Eqs. (16) and (29) into Eq. (30). To calculate the summation in the partition function, we employ the Poisson summation formula [48] 8
The above relation under the lowest-order approximation can be written as
In our numerical calculations, we have chosen the number of states
.
After obtaining the partition function, one can calculate the thermodynamic function of the system using the following relations, (i)
Mean energy
(ii)
Specific heat
(iii)
Free energy
(iv)
Entropy
, , ,
3.1. Thermodynamic functions of modified Tietz potential The partition function of the modified Tietz potential is given by
=
)
+ (
+
+
+
where used parameters in above equation are in the following form
9
+
The mean energy of modified Tietz potential is given by
= [-
+
+ (
-
where
+
-
)
+ (
+
+
+
-
and
=[
-
+
)
+
+
].
The specific heat of modified Tietz potential is expressed as
=[
-
+ (-
-
+
) +
=[
-
+ (
)
+
+
+
+
+
].
The entropy of modified Tietz potential is given by
=[
+
],
)
where
+ (
where
+
+
+
10
+ ],
= [-
+
+ (
-
+
-
)
-
+ +
-
Recently, Jia et al [49] have obtained the non-relativistic vibrational partition function of the improved Tietz potential. Also, some authors [50, 51] have successfully predicted the molar entropy values for the gases CO, HCl, HF, DF, and gaseous BBr, and also successfully predicted the molar enthalpy values for the gases CO, HCl, and BF. In recent years, some authors [52, 53] have calculated enthalpy and Gibbs free energy of gaseous phosphorus dimer. In this work, our purpose was only to obtain entropy, specific heat and mean energy of 7Li2 molecule using two different potentials, MT and BBB models.
3.2. Thermodynamic functions of BBB potential The partition function of BBB potential is obtained as
=
-
+ [
+
+
+
The applied parameters in above equation are in the following form
The mean energy of BBB potential is given by
11
where
+
=
( +
+
-
)
-
+
-
-
+ (
+
+
+
+
]
] =[
)
The specific heat of BBB potential is expressed as
=[
-
where
+ (
+
+
)
+ =[
-
= [-
+
+
+
+ (
=[
)
-
+ + (
)
-
]
+
+
+
]
-
)
-
]
+ (
+
+
+
+
]
The entropy of BBB potential is given by
where
12
=[
)
= [-
+
+ (
+
+
+ + (
-
+ ]
-
)
+
]
4. Results and discussion In this paper, we have determined the numerical results for a3 7
state (excited state) of the
Li2 molecule. The experimental data of molecular constants of the target is
=65.130 cm-1
[35, 48]. Other parameters for the two potentials have been presented in tables 1 and 2. Using the parameters, thermodynamic properties of the 7Li2 molecule as functions of temperature are calculated. For this goal, the partition functions for the two potentials model are obtained using Eqs. (33) and (34). Recently, Barklem and Collet [54] have presented partition functions and dissociation equilibrium constants for 291 diatomic molecules for temperatures in the range from near absolute zero to 10000 K. They have performed their calculations based on molecular spectroscopic data from the book of Huber and Herzberg [55] with significant improvements from the literature. At first step, to check our results, we have compared the partition function calculated in this work with available data of Ref. [55] at several temperatures. Comparison has been presented in table 3. It is seen from the table that the partition functions obtained by the two potential models are in fairly agreement with Ref. [55] at low temperatures. But, at high temperatures, the obtained results in this work have discrepancy with Ref. [55]. Since the obtained results from Ref. [55] are exact, we can say that our models are not good candidates at high temperatures. In table 4, we have presented the vibrational energy values of Li2 molecule for
to
using the modified Tietz (MT) and Badawi-Bessis-Bessis (BBB) potentials. Our results have been compared with the available data of Ref. [36] for the Manning-Rosen (MR) potential. 13
Fig. 1 presents the potential curves for MT, BBB and MR potentials as a function of distance r. In this figure, the qualitative behaviors of the potentials have been compared. In Fig. 2, the variation of partition function with temperature has been presented for the two potentials, modified Tietz (MT) and BBB models. From the figure, it is seen that the partition function (for both potential models) is monotonically increased when the temperature is enhanced. From
to
K, the difference between the partition function values of
the two potentials is small. But, this difference grows with increasing the temperature. It is clear that the partition function values of BBB model are higher than MT model when the temperature enhances. The mean energy is plotted as a function of temperature for the two potential models in Fig. 3. It is observed from the figure that the mean energy, for the two models, is increased as the temperature is enhanced. The difference between the mean energy values of the two potentials at low (0< <100 K) and high temperature ( =700 K) is small. The mean energy values of the BBB model are lower than MT model. Fig. 4 displays the variation of the specific heat of the 7Li2 molecule as a function of temperature for the MT, BBB and MR potential models. It is seen from the figure that the specific heat shows a peak structure. The peaks for three potentials occur at different temperatures. Although the specific heat passes through a maximum, it is a smooth function of temperature as
. It is observed from the figure that the peak value of the specific heat
for BBB model is higher than MT and MR potential models. The entropy variation of the 7Li2 molecule with temperature is presented in Fig. 5 for the MT, BBB and MR potential models. As seen from the figure that entropy is monotonically enhanced with increasing temperature. When the temperature increases, the entropy for three models approaches to a constant value. The entropy value for the MT potential is higher than BBB and MR potentials. As
, the entropy values of three potentials are approximately
the same.
5. Conclusions In the present work, we have considered two potential models like modified Tietz and Badawi-Bessis-Bessis potentials. We have solved the Klein-Gordon equation using the two potentials and obtained the analytical expressions for the energy spectra. We have employed the Poisson summation formula to calculate the partition function analytically. Using the
14
result, we have analytically obtained thermodynamic functions of a3
state of the 7Li2
molecule. The temperature-dependent of thermodynamic function of the molecule are presented. It is to be noted that the used potentials in this work are analytical potential models to study the thermodynamic properties of diatomic molecules. Actually, the MT potential for state of the Na2 molecule is realistic [44]. Using the potentials, one can obtain some information about the transport properties of diatomic molecules. It is to be noted that such potentials can be used in some physical system. For example, we have recently employed the Tietz potential to study optical properties of spherical quantum dots and also singlet-triplet transition of a two-electron quantum dot [56, 57].
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18
[53] X. L. Peng, R. Jiang, C. S. Jia, L. H. Zhang, Y. L. Zhao, Gibbs free energy of gaseous phosphorus dimer. Chem. Engin. Sci. 190 (2018) 122-125. [54] P. S. Barklem, R. Collet, Partition functions and equilibrium constants for diatomic molecules and atoms of astrophysical interest. Astron. Astrophys. 588 (2016) A96-A99. [55] K. P. Huber, G. Herzberg, Constants of Diatomic Molecules, Van Nostrand Reinhold (1979). [56] R. Khordad, B. Mirhosseini, Application of Tietz potential to study optical properties of spherical quantum dots. Pramana J. Phys. 85 (2015) 723-737. [57] R. Khordad, B. Mirhosseini, Application of Tietz potential to study singlet-triplet transition of a two-electron quantum dot. Commun. Theor. Phys. 62 (2014) 77-80.
Table 1. The used parameters for Badawi-Bessis-Bessis (BBB) potential. (eV)
(eV)
(eV)
19
(cm-1)
Table 2. The used parameters for the modified Tietz (MT) potential. (A0)
(eV)
(A0)-1
Table 3. The partition function of Li2 molecule at several temperatures. The results for modified Tietz (MT) and Badawi-Bessis-Bessis (BBB) potentials have been compared with the available data of Ref. [54]. MT potential
BBB potential
Table 4. Vibrational energy values (cm-1) of Li2 molecule for
Ref. [54]
to
. The
results have been obtained for the modified Tietz (MT) and Badawi-Bessis-Bessis (BBB) potentials. Our results have been compared with the available data of Ref. [36] for the Manning-Rosen (MR) potential. MR
BBB
MT
1
169.9138
103.3232
135.8942
2
167.14334
108.3662
136.4196
3
166.0646
113.1848
136.8480
4
164.6776
119.6052
137.0380
5
162.9824
124.8458
137.8424
6
160.9791
129.8752
139.0170
7
158.6675
134.7169
140.1817
8
156.0477
140.1544
143.8454
9
153.1197
146.1210
144.8816
10
149.8835
149.7295
146.7992
20
14
BBB potential Tietz potential MR potential
12
8 6 4 2 0 0
2
4
6
8
10
12
14
r (nm)
Fig. 1
10
MT BBB 8
6
Q
22
V(r) * 10 (J)
10
4
2
0 0
20
40
60
T(K)
21
80
100
Fig. 2
6.00E-021
MT BBB
U(J)
4.00E-021
2.00E-021
0.00E+000 0
100
200
300
400
T(K) Fig. 3
22
500
600
700
1.60E-022
MT potential BBB potential MR potential
1.40E-022
C (J*K^-1)
1.20E-022 1.00E-022 8.00E-023 6.00E-023 4.00E-023 2.00E-023 0.00E+000
0
50
100
150
200
T(K) Fig. 4
Fig. 5 23
250
300
350
400
Caption of Figures:
Fig. 1: The variation of three potentials (MT, BBB and MR) as a function of . The unit of potential and distance are eV and nm, respectively. Fig. 2: The variations of the partition function of the 7Li2 dimer as a function of temperature. Our results have been compared with the Manning-Rosen potential [36]. Fig. 3: The variations of the mean energy of the 7Li2 molecules as a function of temperature. Fig. 4: The variations of the specific heat of the 7 Li2 molecule as a function of temperature. Our results have been compared with the Manning-Rosen potential [36]. Fig. 5: The variations of the entropy of the 7Li2 molecule as a function of temperature. Our results have been compared with the Manning-Rosen potential [36].
24
Highlights The Klein-Gordon equation with two different potentials has been analytically solved.
We have used the modified Tietz and the Badawi-Bessis-Bessis diatomic molecular potentials.
We have employed the Poisson summation formula to calculate the partition function analytically. We have analytically obtained thermodynamic functions of a3 molecule.
25
state of the 7Li2
We have employed two different potentials to study thermodynamic functions of a3
state of the 7Li2 molecule.
Modified Tietz and Badawi-Bessis-Bessis potentials are used
The partition function is analytically obtained. The partition function is used to calculate
Mean energy
Entropy
Specific heat
6.00E-021
MT BBB
U(J)
4.00E-021
2.00E-021
0.00E+000 100
200
300
400
500
600
700
T(K)
1.60E-022
MT BBB
1.40E-022 1.20E-022
C (J*K^-1)
0
1.00E-022 8.00E-023 6.00E-023 4.00E-023 2.00E-023 0.00E+000
0
50
100
150
200
T(K)
250
300
350
400
S2210-271X(19)30081-7 https://doi.org/10.1016/j.comptc.2019.03.019 COMPTC 12454
To appear in:
Computational & Theoretical Chemistry
Received Date: Revised Date: Accepted Date:
19 September 2017 16 March 2019 16 March 2019
Please cite this article as: R. Khordad, A. Ghanbari, Analytical calculations of thermodynamic functions of lithium dimer using modified Tietz and Badawi-Bessis-Bessis potentials, Computational & Theoretical Chemistry (2019), doi: https://doi.org/10.1016/j.comptc.2019.03.019
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Analytical calculations of thermodynamic functions of lithium dimer using modified Tietz and Badawi-Bessis-Bessis potentials R. Khordad* and A. Ghanbari Department of Physics, College of Sciences, Yasouj University, Yasouj, Iran
Abstract We have analytically solved the Klein-Gordon (KG) equation for two different potentials. We have used the modified Tietz (MT) and the Badawi-Bessis-Bessis (BBB) diatomic molecular potentials and derived the analytical expressions for the energy spectra. Then, we have analytically obtained the thermodynamic functions for a3
state of the 7Li2 molecule using
the two potentials. For this goal, we have calculated the partition function, mean energy, specific heat, and entropy of the molecule as a function of temperature.
Keywords: Lithium molecule, Thermodynamic functions, Entropy
*Corresponding author, E-mail: [email protected]
1
1. Introduction It is fully known that the determination of exact solutions of the non-relativistic or relativistic wave equations plays a key role in quantum systems because the solutions contain important information regarding the system. One of the most famous wave equations is the KleinGordon (KG) equation. The equation is a relativistic wave equation correctly describing the spin-zero particles such as spinless pion and a composite particle. The KG equation has received considerable attention in the literature [1-5]. We can consider the KG equation as a relativistic version of the Schrödinger equation. It is noteworthy that the calculation of analytical solutions is possible only in a few simple cases such as the hydrogen atom and the harmonic oscillator [6-10]. Due to the complicated form of several potentials, the calculation of analytical solutions appears to be a nontrivial task which requires considerable theoretical effort. Examples of the potentials are Tietz-Wei, Deng-Fan, Hulthen, Morse, Manning-Rosen, and Tietz [11-17]. Hitherto, the KG equation has been solved for several potentials. For example, Sun et al [18] have calculated the bound state solutions of the relativistic Klein-Gordon equation with the Tietz-Wei (TW) diatomic molecular potential. Hassanabadi et al [19] have obtained the spectra and the eigenfunctions of the relativistic Klein-Gordon equation for Deng-Fan potential. Hamzavi and Amirfakhrian [20] have solved the KG equation for Deng-Fan potential in arbitrary N-dimension. They have employed an improved approximation scheme to the centrifugal term. Rezaei Akbarieh and Motavali [21] have reported the exact solutions of the one-dimensional KG equation for the Rosen-Morse type potential with equal scalar and vector potentials. Arda and Sever [22] have solved the radial part of the KG equation for the generalized Woods-Saxon potential using the Nikiforov-Uvarov method. Ikhdair [23] has obtained the approximate bound-state rotational-vibrational energy levels. Jia et al [24] have used the Manning-Rosen to investigate the vibrational levels and the interatomic interaction potential for the a3
state of 7Li2 in terms of the Rydberg-Klein-Rees (RKR) method. To
obtain more information, the reader can refer to [25-30]. Recently, the lithium dimer (7Li2 or 6Li2 ) has attracted considerable attention because it is the second smallest stable homonuclear molecule next to H2. So far, many researchers have employed different procedures to study the vibrational levels of the a3
state of the 7Li2.
Examples of the methods are Rydberg-Klein-Rees method and ab initio approach [31-35]. Recently, Jia et al [36] have reported temperature-dependent thermodynamic functions of the 7
Li2 molecular gas using the Manning-Rosen (MR) potential energy. They have solved the 2
Schrödinger equation with the MR potential to obtain the energy levels. In this work, we have investigated the thermodynamic properties of the a3
state of the 7Li2 molecular gas
using two different potentials. Since the MT and BBB potentials have the same behaviors in comparing with Manning-Rosen (MR) potential (see Fig. 1), we have MT and BBB potentials to determine thermodynamic functions of 7Li2 molecular. It is fully known that modeling diatomic potential energy through analytical functions is of fundamental importance. Examples of the potential functions are Rosen-Morse, Tietz, improved Tietz, Wei, and Badawi-Bessis-Bessis potential [29, 37-42].
2. Theory and model 2.1. Modified Tietz potential In 1963, Tietz proposed a potential energy function for diatomic molecules. The Tietz potential is one of the very best analytical potential models for the vibrational energy of diatomic molecules [29, 37]:
where
is the dissociation energy. The parameters , , , and
can be determined so as to
fit calculated values of spectroscopic parameters to observed ones. Another diatomic potential is Tietz-Wei (TW) potential. The TW potential is much more realistic than the Morse potential in describing molecular dynamics at moderate and high rotational and vibrational quantum numbers. Kunc and Vazquez obtained the analytical expressions for the rotational-vibrational energy levels of diatomic TW potential using the Hamilton-Jacobi theory and the Bohr-Sommerfeld quantization rule [38]. In 2012, Jia et al [13] proposed a more convenient form of the original Tietz potential function. The modified Tietz (MT) potential is given by [43]
where
is the internuclear separation,
energy, and
is the equilibrium bond length,
is the dissociation
is an adjustable parameter which governs the range of the interaction. It is well
known that the original expression of the Tietz potential function Eq. (1) is defined in terms of five parameters whereas the modified Tietz potential only has four parameters. Recently, 3
Tang et al [44] have obtained the rotation-vibrational energy spectra and the un-normalized wave functions of the modified Tietz potential with solving the Schrödinger equation. Also, Zhang et al [45] have solved the Schrödinger equation with the modified Tietz potential model in D spatial dimensions. They have used the supersymmetric shape invariance approach and obtained the D-dimensional rotation-vibrational energy spectra. In these works [44, 45], the authors have studied Na2 system by solving the Schrödinger equation. It is to be noted that Liu et al [43] have solved the Klein-Gordon equation with the modified Tietz (MT) potential in terms of the supersymmetric shape invariance approach. They have calculated the relativistic vibrational transition frequencies for the Na 2 molecule. Also, Liu et al [46] have solved the Klein-Gordon equation with the improved Tietz potential in higher spatial dimensions. They have calculated relativistic vibrational energies for the ground electronic state of the CO molecule. In this work, we attempt to solve the Klein-Gordon equation with the modified Tietz (MT) potential in terms of the function analysis method. The Klein–Gordon equation for a quantum system with the rest mass and a vector potential
in a scalar potential
is given by
where we have set
. Using the separation of variable method for the wave functions , the Klein-Gordon equation of the s-wave for equal scalar and
vector potentials is given by
where
and
represent the rest mass and energy of the spinless particle, respectively. Also,
we have applied
. For a diatomic system, the rest mass
reduced mass of the diatomic molecule. The scalar potential
can be regarded as the and vector potential
are chosen as equal to modified Tietz potentials as defined in Eq. (2). The potential
is
called vector potential because it treated as the fourth component of a Lorentz-vector. The KG equation has square terms causing a great complexity for finding the exact solutions especially when unequal scalar and vector potentials are studied. Under the choice of equal scalar and vector potentials, Eq. (4) reduces to the Schrödinger equation for the potential in the nonrelativistic limit. Here, we have chosen
4
. In this
study, we use the modified Tietz potential for calculating the vibrational energy levels and ignore the rotational terms. To solve Eq. (4) for the modified Tietz potential, we consider the following separation of variable
where
. Therefore, we have
Inserting above relations into Eq. (4), we can obtain
where
In order to obtain the solution to Eq. (6), we take the wave functions of the form
where
With substituting of Eq. (8) into Eq. (6), we can obtain the following differential equation
5
whose solutions are nothing but the hypergeometric functions Here
and
Now, we should apply the properties of the hypergeometric functions. It is to be noted that the series of
given in Eq. (11) approaches infinity unless
is a negative integer like
.
To obtain the properties of the hypergeometric functions, the reader can refer to [47]. Therefore
where
We can obtain the energy levels as
2.2. Badawi-Bessis-Bessis potential In 1972, Badawi et al [41] introduced a general potential energy expression for several empirical potential functions as
6
where
, and
are five adjustable parameters. It is worth mentioning that Du et al
[40] have shown that the BBB potential is equivalent to the Tietz potential for a diatomic molecule. Applying separation of variable method for the wave functions, we can solve Eq. (4) with considering the scalar potential
and vector potential V(r) as equal to BBB potential [Eq.
(17)]. For this purpose, we use the following change of variable
Therefore, we have
Now, Eq. (4) can be written as
where
We consider the following wave function
where
With inserting Eq. (21) into Eq. (19), the following differential equation is achieved
where the solutions of the equation are the hypergeometric functions 7
Here
and
With respect to properties of the hypergeometric functions, it should be noted the series of given in Eq. (24) approaches infinity unless
where
is a negative integer like –
[47].
We can obtain the energy levels as
To obtain the partition function of a given system, one can need the energy levels. Now, one can determine the partition function with using the energy levels Eqs. (16) and (29).
3. Thermodynamic properties A starting point to derive thermodynamic properties system is the partition function. The quantity is obtained by direct summation over all possible states of the system,
where
,
is the Boltzmann's constant and
is the temperature. To obtain the
partition function for two potentials, we insert the energy levels from Eqs. (16) and (29) into Eq. (30). To calculate the summation in the partition function, we employ the Poisson summation formula [48] 8
The above relation under the lowest-order approximation can be written as
In our numerical calculations, we have chosen the number of states
.
After obtaining the partition function, one can calculate the thermodynamic function of the system using the following relations, (i)
Mean energy
(ii)
Specific heat
(iii)
Free energy
(iv)
Entropy
, , ,
3.1. Thermodynamic functions of modified Tietz potential The partition function of the modified Tietz potential is given by
=
)
+ (
+
+
+
where used parameters in above equation are in the following form
9
+
The mean energy of modified Tietz potential is given by
= [-
+
+ (
-
where
+
-
)
+ (
+
+
+
-
and
=[
-
+
)
+
+
].
The specific heat of modified Tietz potential is expressed as
=[
-
+ (-
-
+
) +
=[
-
+ (
)
+
+
+
+
+
].
The entropy of modified Tietz potential is given by
=[
+
],
)
where
+ (
where
+
+
+
10
+ ],
= [-
+
+ (
-
+
-
)
-
+ +
-
Recently, Jia et al [49] have obtained the non-relativistic vibrational partition function of the improved Tietz potential. Also, some authors [50, 51] have successfully predicted the molar entropy values for the gases CO, HCl, HF, DF, and gaseous BBr, and also successfully predicted the molar enthalpy values for the gases CO, HCl, and BF. In recent years, some authors [52, 53] have calculated enthalpy and Gibbs free energy of gaseous phosphorus dimer. In this work, our purpose was only to obtain entropy, specific heat and mean energy of 7Li2 molecule using two different potentials, MT and BBB models.
3.2. Thermodynamic functions of BBB potential The partition function of BBB potential is obtained as
=
-
+ [
+
+
+
The applied parameters in above equation are in the following form
The mean energy of BBB potential is given by
11
where
+
=
( +
+
-
)
-
+
-
-
+ (
+
+
+
+
]
] =[
)
The specific heat of BBB potential is expressed as
=[
-
where
+ (
+
+
)
+ =[
-
= [-
+
+
+
+ (
=[
)
-
+ + (
)
-
]
+
+
+
]
-
)
-
]
+ (
+
+
+
+
]
The entropy of BBB potential is given by
where
12
=[
)
= [-
+
+ (
+
+
+ + (
-
+ ]
-
)
+
]
4. Results and discussion In this paper, we have determined the numerical results for a3 7
state (excited state) of the
Li2 molecule. The experimental data of molecular constants of the target is
=65.130 cm-1
[35, 48]. Other parameters for the two potentials have been presented in tables 1 and 2. Using the parameters, thermodynamic properties of the 7Li2 molecule as functions of temperature are calculated. For this goal, the partition functions for the two potentials model are obtained using Eqs. (33) and (34). Recently, Barklem and Collet [54] have presented partition functions and dissociation equilibrium constants for 291 diatomic molecules for temperatures in the range from near absolute zero to 10000 K. They have performed their calculations based on molecular spectroscopic data from the book of Huber and Herzberg [55] with significant improvements from the literature. At first step, to check our results, we have compared the partition function calculated in this work with available data of Ref. [55] at several temperatures. Comparison has been presented in table 3. It is seen from the table that the partition functions obtained by the two potential models are in fairly agreement with Ref. [55] at low temperatures. But, at high temperatures, the obtained results in this work have discrepancy with Ref. [55]. Since the obtained results from Ref. [55] are exact, we can say that our models are not good candidates at high temperatures. In table 4, we have presented the vibrational energy values of Li2 molecule for
to
using the modified Tietz (MT) and Badawi-Bessis-Bessis (BBB) potentials. Our results have been compared with the available data of Ref. [36] for the Manning-Rosen (MR) potential. 13
Fig. 1 presents the potential curves for MT, BBB and MR potentials as a function of distance r. In this figure, the qualitative behaviors of the potentials have been compared. In Fig. 2, the variation of partition function with temperature has been presented for the two potentials, modified Tietz (MT) and BBB models. From the figure, it is seen that the partition function (for both potential models) is monotonically increased when the temperature is enhanced. From
to
K, the difference between the partition function values of
the two potentials is small. But, this difference grows with increasing the temperature. It is clear that the partition function values of BBB model are higher than MT model when the temperature enhances. The mean energy is plotted as a function of temperature for the two potential models in Fig. 3. It is observed from the figure that the mean energy, for the two models, is increased as the temperature is enhanced. The difference between the mean energy values of the two potentials at low (0< <100 K) and high temperature ( =700 K) is small. The mean energy values of the BBB model are lower than MT model. Fig. 4 displays the variation of the specific heat of the 7Li2 molecule as a function of temperature for the MT, BBB and MR potential models. It is seen from the figure that the specific heat shows a peak structure. The peaks for three potentials occur at different temperatures. Although the specific heat passes through a maximum, it is a smooth function of temperature as
. It is observed from the figure that the peak value of the specific heat
for BBB model is higher than MT and MR potential models. The entropy variation of the 7Li2 molecule with temperature is presented in Fig. 5 for the MT, BBB and MR potential models. As seen from the figure that entropy is monotonically enhanced with increasing temperature. When the temperature increases, the entropy for three models approaches to a constant value. The entropy value for the MT potential is higher than BBB and MR potentials. As
, the entropy values of three potentials are approximately
the same.
5. Conclusions In the present work, we have considered two potential models like modified Tietz and Badawi-Bessis-Bessis potentials. We have solved the Klein-Gordon equation using the two potentials and obtained the analytical expressions for the energy spectra. We have employed the Poisson summation formula to calculate the partition function analytically. Using the
14
result, we have analytically obtained thermodynamic functions of a3
state of the 7Li2
molecule. The temperature-dependent of thermodynamic function of the molecule are presented. It is to be noted that the used potentials in this work are analytical potential models to study the thermodynamic properties of diatomic molecules. Actually, the MT potential for state of the Na2 molecule is realistic [44]. Using the potentials, one can obtain some information about the transport properties of diatomic molecules. It is to be noted that such potentials can be used in some physical system. For example, we have recently employed the Tietz potential to study optical properties of spherical quantum dots and also singlet-triplet transition of a two-electron quantum dot [56, 57].
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18
[53] X. L. Peng, R. Jiang, C. S. Jia, L. H. Zhang, Y. L. Zhao, Gibbs free energy of gaseous phosphorus dimer. Chem. Engin. Sci. 190 (2018) 122-125. [54] P. S. Barklem, R. Collet, Partition functions and equilibrium constants for diatomic molecules and atoms of astrophysical interest. Astron. Astrophys. 588 (2016) A96-A99. [55] K. P. Huber, G. Herzberg, Constants of Diatomic Molecules, Van Nostrand Reinhold (1979). [56] R. Khordad, B. Mirhosseini, Application of Tietz potential to study optical properties of spherical quantum dots. Pramana J. Phys. 85 (2015) 723-737. [57] R. Khordad, B. Mirhosseini, Application of Tietz potential to study singlet-triplet transition of a two-electron quantum dot. Commun. Theor. Phys. 62 (2014) 77-80.
Table 1. The used parameters for Badawi-Bessis-Bessis (BBB) potential. (eV)
(eV)
(eV)
19
(cm-1)
Table 2. The used parameters for the modified Tietz (MT) potential. (A0)
(eV)
(A0)-1
Table 3. The partition function of Li2 molecule at several temperatures. The results for modified Tietz (MT) and Badawi-Bessis-Bessis (BBB) potentials have been compared with the available data of Ref. [54]. MT potential
BBB potential
Table 4. Vibrational energy values (cm-1) of Li2 molecule for
Ref. [54]
to
. The
results have been obtained for the modified Tietz (MT) and Badawi-Bessis-Bessis (BBB) potentials. Our results have been compared with the available data of Ref. [36] for the Manning-Rosen (MR) potential. MR
BBB
MT
1
169.9138
103.3232
135.8942
2
167.14334
108.3662
136.4196
3
166.0646
113.1848
136.8480
4
164.6776
119.6052
137.0380
5
162.9824
124.8458
137.8424
6
160.9791
129.8752
139.0170
7
158.6675
134.7169
140.1817
8
156.0477
140.1544
143.8454
9
153.1197
146.1210
144.8816
10
149.8835
149.7295
146.7992
20
14
BBB potential Tietz potential MR potential
12
8 6 4 2 0 0
2
4
6
8
10
12
14
r (nm)
Fig. 1
10
MT BBB 8
6
Q
22
V(r) * 10 (J)
10
4
2
0 0
20
40
60
T(K)
21
80
100
Fig. 2
6.00E-021
MT BBB
U(J)
4.00E-021
2.00E-021
0.00E+000 0
100
200
300
400
T(K) Fig. 3
22
500
600
700
1.60E-022
MT potential BBB potential MR potential
1.40E-022
C (J*K^-1)
1.20E-022 1.00E-022 8.00E-023 6.00E-023 4.00E-023 2.00E-023 0.00E+000
0
50
100
150
200
T(K) Fig. 4
Fig. 5 23
250
300
350
400
Caption of Figures:
Fig. 1: The variation of three potentials (MT, BBB and MR) as a function of . The unit of potential and distance are eV and nm, respectively. Fig. 2: The variations of the partition function of the 7Li2 dimer as a function of temperature. Our results have been compared with the Manning-Rosen potential [36]. Fig. 3: The variations of the mean energy of the 7Li2 molecules as a function of temperature. Fig. 4: The variations of the specific heat of the 7 Li2 molecule as a function of temperature. Our results have been compared with the Manning-Rosen potential [36]. Fig. 5: The variations of the entropy of the 7Li2 molecule as a function of temperature. Our results have been compared with the Manning-Rosen potential [36].
24
Highlights The Klein-Gordon equation with two different potentials has been analytically solved.
We have used the modified Tietz and the Badawi-Bessis-Bessis diatomic molecular potentials.
We have employed the Poisson summation formula to calculate the partition function analytically. We have analytically obtained thermodynamic functions of a3 molecule.
25
state of the 7Li2
We have employed two different potentials to study thermodynamic functions of a3
state of the 7Li2 molecule.
Modified Tietz and Badawi-Bessis-Bessis potentials are used
The partition function is analytically obtained. The partition function is used to calculate
Mean energy
Entropy
Specific heat
6.00E-021
MT BBB
U(J)
4.00E-021
2.00E-021
0.00E+000 100
200
300
400
500
600
700
T(K)
1.60E-022
MT BBB
1.40E-022 1.20E-022
C (J*K^-1)
0
1.00E-022 8.00E-023 6.00E-023 4.00E-023 2.00E-023 0.00E+000
0
50
100
150
200
T(K)
250
300
350
400