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Equivalence of the three empirical potential energy models for diatomic molecules
Journal of Molecular Spectroscopy 274 (2012) 5–8
Contents lists available at SciVerse ScienceDirect
Journal of Molecular Spectroscopy journal homepage: www.elsevier.com/locate/jms
Equivalence of the three empirical potential energy models for diatomic molecules Ping-Quan Wang, Lie-Hui Zhang, Chun-Sheng Jia ⇑, Jian-Yi Liu State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu 610500, People’s Republic of China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 17 February 2012 In revised form 12 March 2012 Available online 21 March 2012
It is found that the Manning–Rosen potential, Schiöberg potential and Deng–Fan potential are the same solvable empirical potential energy function for diatomic molecules. We calculate the anharmonicity xexe and vibrational rotational coupling parameter ae for 16 molecules by choosing the experimental values of the dissociation energy De, equilibrium bond length re and vibrational frequency xe as inputs. The results show that the Manning–Rosen potential, Deng–Fan potential and Schiöberg potential are not better than the traditional Morse potential. Ó 2012 Published by Elsevier Inc.
Keywords: Empirical diatomic potential function Manning–Rosen potential Deng–Fan potential Schiöberg potential Morse potential
1. Introduction There has been a growing interest in investigating the empirical potential energy function for diatomic molecules in chemistry and molecular physics. The reason is that the potential function provides a quantitative description of the energy–distance relation during the making or breaking of chemical bonds between two atoms [1,2]. Many efforts to construct a universal potential energy function have been made by Morse [3], Rydberg [4], Pöschl and Teller [5], Manning and Rosen [6], Linnett [7], Lippincott [8], Varshni [9], Tietz [10], Deng and Fan [11], Schiöberg [12], Zavitsas [1], and Hajigeorgiou [2], among others. In 1929, Morse [3] proposed the first three-parameter solvable empirical potential energy function for diatomic molecules,
2 U M ðrÞ ¼ De 1 eaðrre Þ ;
ð1Þ
in which De is the dissociation energy, re is the equilibrium bond length, and a denotes the range of the potential. Potential-energy functions with parameters of greater number generally provide a better fit to experimental data than those of fewer parameters, but the latter potential functions have been employed in many computational chemistry softwares, such as CVFF [13], DREIDING [14], and ESFF [15]. In 1933, Manning and Rosen [6] proposed a potential function for diatomic molecules,
U MR ðrÞ ¼
h
"
2 2
8lp2 b
bðb 1Þe2r=b ð1 er=b Þ2
# Aer=b ; 1 er=b
ð2Þ
⇑ Corresponding author. Fax: +86 28 8303 2901. E-mail addresses: [email protected] (P.-Q. Wang), [email protected] (C.-S. Jia). 0022-2852/$ - see front matter Ó 2012 Published by Elsevier Inc. http://dx.doi.org/10.1016/j.jms.2012.03.005
in which b and A are two dimensionless parameters, parameter b is related to the range of the potential and has dimension of length. The Manning–Rosen potential can be used to describe the diatomic molecular vibrations [16–18]. In 1957, Deng and Fan [11] proposed a simple potential model for diatomic molecules,
2 eare 1 U DF ðrÞ ¼ De 1 ar ; e 1
ð3Þ
in which De, re, and a denote the dissociation energy, equilibrium bond length and the range of the potential, respectively. The potential (2) is also called a generalized Morse potential [19]. Deng and Fan [11] suggested that the potential function (3) is better than the Morse potential function (1) in representing diatomic interactions for vibration of diatomic molecules. By employing the Deng–Fan potential, some authors investigated the F–H stretching motion and rotational transitions for HF molecule [20,21], and studied the statistical mechanics of the quasi-one-dimensional system of DNA [22]. In 1986, Schiöberg [12] proposed a new potential function for diatomic molecules,
U S ðrÞ ¼ Dð1 r coth arÞ2 ;
ð4Þ
in which D, r, and a are three adjustable positive parameters, r < 1. Schiöberg [12] suggested that the potential function (4) is more accurate than the Morse potential function in fitting the experimental Rydberg–Klein–Rees (RKR) potential energy curves and the rotational-vibrating levels for some diatomic molecules. The Schiöberg potential has been used to describe diatomic interactions [23–27]. In the present work, we investigate the equivalence of the Manning–Rosen potential, Schiöberg potential and Deng–Fan
6
P.-Q. Wang et al. / Journal of Molecular Spectroscopy 274 (2012) 5–8
potential. We also compare the Morse potential with the Manning– Rosen potential, Deng–Fan potential and Schiöberg potential according to quantitative tests on 16 molecules with diverse values of essential parameters. 2. Equivalence of the three empirical potential functions The empirical potential energy function U(r) should satisfy the following conditions:
dUðrÞ ¼ 0; dr r¼re
ð5Þ
Uð1Þ Uðre Þ ¼ De ;
ð6Þ
2 d UðrÞ 2 dr
ð7Þ
¼ ke ¼ 4p2 lc2 x2e ;
r¼re
A bðb 1Þ ¼ ere =b 1 ; 2 2bðb 1Þ þ1 : r e ¼ b ln A
ð8Þ
ð9Þ
Employing the condition (6) and expression (9), we have the following relationship: 2
32lp
A2 ¼ De : bðb 1Þ
2 b2
ð10Þ
Using Eqs. (8) and (10), we get
A¼
16lp2 b h
2
2
ðere =b 1ÞDe :
ð11Þ
Substituting the expressions (8) and (11) into expression (2) and making some algebraic simplifications, we can rewrite the Manning–Rosen potential in the following form:
2 ere =b 1 U MR ðrÞ ¼ De 1 r=b : e 1
ð12Þ
Here, we have added one term De to the original Manning–Rosen potential energy function given in Eq. (2). This change only yields an energy of zero at the potential minimum, i.e. UMR(re) = 0, and does not affect the physical properties of the potential function. We can rewrite the Schiöberg potential function (4) in the following form:
U S ðrÞ ¼ D 1 r
2r e2ar 1
2
e2are 1 : e2are þ 1
ð14Þ
By using the condition (6), we obtain the following relation
Dð1 rÞ D 1 r 2
2r e2are 1
lim U MR ðrÞ ¼ limU DF ðrÞ ¼ limU S ðrÞ ¼ De a!0
2 De 2are þ1 : e 4
a!0
r r 2 e : r
ð18Þ
The Kratzer–Fues molecular potential has been extensively used for investigating the properties of diatomic molecules [30,31]. 3. Evaluation of spectroscopic parameters ae and xexe On the basis of wave mechanics, Dunham [32] showed that the molecular parameters can be expressed in terms of the derivatives of the potential function. The vibrational rotational coupling parameter ae and anharmonicity parameter xexe are given by
ae ¼
r e f3 ; 1þ xe 3f 2
6B2e
ð19Þ
0 !2 1 Be @ r 2e f4 xe ae A ; xe xe ¼ þ 15 1 þ 8 f2 6B2e
ð20Þ
in which Be ¼ 8p2hclr2 , h is the Planck constant, fi (i = 2, 3, 4) denotes e the derivatives of the potential energy function through
2 d UðrÞ f2 ¼ 2 dr
; r¼r e
3 d UðrÞ f3 ¼ 3 dr
4 d UðrÞ and f 4 ¼ : 4 dr r¼re
r¼r e
ð21Þ
By employing the expressions (19) and (20), we obtain the following expressions for spectroscopic parameters ae and xexe corresponding to the Morse potential and Manning–Rosen potential, respectively,
6B2e
1
a 3 De r e ; 2p2 lc2 x2e
ð22Þ
0 !2 1 Be @ 7a4 De r 2e xe ae ðMÞ A ; xe xe ðMÞ ¼ 2 2 2 þ 15 1 þ 8 2p l c xe 6B2e
ð23Þ
ae ðMÞ ¼
xe
ae ðMRÞ ¼
r e f3 ; 1þ xe 3ke
6B2e
ð24Þ
0 !2 1 Be @ r2e f4 xe ae ðMRÞ A ; xe xe ðMRÞ ¼ þ 15 1 þ 8 ke 6B2e
ð25Þ
where
2 ¼ De :
ð15Þ
Be ¼
Substituting Eq. (14) into Eq. (15), and solving it for D, we obtain
D¼
ð17Þ
Here, we replace 2a by 1/b, this expression becomes exactly the expression (12). By comparing expressions (3) and (17) with expression (12), we can conclude that the Deng–Fan potential and Schiöberg potential are the same empirical diatomic potential function, and they are just the Manning–Rosen potential. When the parameter b in Eq. (2) becomes infinity, and the parameter a in Eqs. (3) and (4) goes to zero, the limits of the Manning–Rosen potential, Deng–Fan potential and Schiöberg potential become the Kratzer–Fues molecular potential [28,29], namely,
ð13Þ
:
Substituting the expression (13) into the condition (5), we have
r¼
2 e2are 1 U S ðrÞ ¼ De 1 2ar : e 1
b!1
in which l is the reduced mass of the diatomic molecule, c is the speed of light, and xe denotes the vibrational frequency. Substituting the Manning–Rosen potential function (2) into the condition (5), we obtain
h
Substituting the expressions (14) and (16) into the potential function (13), we obtain the new expression for the Schiöberg potential function,
ð16Þ
f3 ¼
h ; 8p2 clr2e
ke ¼ 4p2 lc2 x2e ;
12De e3re =b b
3
ðere =b
1Þ
3
þ
6De e2re =b 3
2
b ðere =b 1Þ
ð26Þ
;
ð27Þ
7
P.-Q. Wang et al. / Journal of Molecular Spectroscopy 274 (2012) 5–8 Table 1 Comparison of experimental values of ae with calculated values obtained by using the Morse potential and Manning–Rosen potential. All quantities are in 103 cm1. Molecule
State
12 16
1
+
ae(exp)
ae(M)
Dae (%)
ae(MR)
17.50406 306.807 793.4125 302.203 0.8683
16.42597 310.743 658.5493 251.679 1.4569
6.16 1.28 16.998 16.72 67.79
18.99192 377.316 909.783 348.126 1.6777
Dae (%)
C O H35Cl HF DF Na2
X R X1R+ X1R+ X1R+
7
X1 Rþ g
7.017
7
LiH 39 7 K Li Cs2
X1R+ X1R+
216.391 1.9078 0.022012
193.021 3.15753 0.043499
10.80 65.5 97.6
309.501 3.60793 0.046466
43.03 89.1 111.09
RbCs Na39K AgH MgH NaH 16 O2
X1R+ X1R+ X1R+ X2R+ X1R+
0.03661 0.4485 201.93 177.2981 137.09097 15.910
0.07054 0.7737 212.31 263.6870 135.19571 17.493
92.7 72.5 5.14 48.72 1.38 9.95
0.07583 0.8681 250.44 289.3235 181.25006 18.693
107.1 93.5 24.02 63.18 3.22 17.5
35
X1 Rþ g
1.9027
25.5
X1 Rþ g
Li2
X1 Rþ g
X3 R g
Cl2
10.12
1.5163
43.7
Average
12.54
1.9543
36.4
8.50 22.98 14.667 15.20 93.21 77.7
28.9 50.8
Table 2 Comparison of experimental values of xexe with calculated values obtained by using the Morse potential and Manning–Rosen potential. All quantities are in cm1. Molecule
State
xexe(exp)
xexe(M)
Dxexe
xexe(MR)
Dxexe
12 16
X1 R+ X1 R+ X1 R + X1 R +
13.2880 52.746 89.94511 47.2712 0.7214
12.9814 60.047 86.73872 45.5989 1.0569
0.3066 7.301 3.56 3.54 46.5
14.3466 68.436 106.82988 56.2016 1.1651
1.0586 15.690 18.77 18.89 61.5
C O H35Cl HF DF Na2
X1 Rþ g
7
X1 Rþ g
2.5779
3.5894
39.2
4.1431
60.7
7
LiH 39 7 K Li Cs2
X1 R+ X1 R+
23.1679 1.26617 0.08182
24.3446 1.83123 0.12095
5.08 44.63 47.8
32.6174 2.00857 0.12689
40.79 58.63 55.1
RbCs Na39K AgH MgH NaH 16 O2
X1 R + X1 R+ X1 R+ X2 R + X1 R +
0.10953 0.4827 34.180 29.84682 19.52352 11.981
0.16301 0.7285 40.226 50.16752 21.58835 14.846
48.8 50.9 17.68 68.08 10.58 23.9
0.17177 0.7897 45.023 53.58953 26.12419 15.580
56.8 63.6 31.72 79.55 33.81 30.04
35
X1 Rþ g
Li2
X1 Rþ g
X3 R g
Cl2
2.694271
3.863106
43.38
Average
f4 ¼
28.8
72De e4re =b 4
4
b ðere =b 1Þ
72De e3re =b 4
b ðere =b 1Þ
3
þ
14De e2re =b 4
b ðere =b 1Þ2
:
ð28Þ
According to condition (7), the following relations are satisfied by the potential parameters a and b in the Morse potential and Manning–Rosen potential, respectively,
sffiffiffiffiffiffiffi 2l ; a ¼ pcxe De e2re =b b
2
ðere =b
2
1Þ
¼
ð29Þ 2p2 lc2 x2e : De
ð30Þ
The experimental data of De, re, xe, ae and xexe are taken from the recent literature [2]. Choosing the experimental values of De, re and xe as inputs, we determine the potential parameters a and b from Eqs. (29) and (30). By employing expressions (22)–(25), we calculate the values of vibrational rotational coupling parameter ae and anharmonicity parameter xexe for 16 molecules. The corresponding calculated values and experimental data are listed in Tables 1 and 2, respectively. Average absolute relative deviations from the experimental data for ae are 36.4% and 50.8%, respectively, and are 28.8% and 42.1%, respectively, for xexe. The absolute relative deviation is defined by
Dae ¼
jae ðexpÞ ae ðcalcÞj ; ae ðexpÞ
Dxe xe ¼
jxe xe ðexpÞ xe xe ðcalcÞj ; xe xe ðexpÞ
3.943132
46.35 42.1
ð31Þ
ð32Þ
where ae(exp) and xexe(exp) correspond to the experimental values, and ae(calc) and xexe(calc) correspond to the calculated values. Tables 1 and 2 tell us that the Morse potential is better than the Manning–Rosen potential to model diatomic interactions. In view of the equivalence of Manning–Rosen potential, Deng–Fan potential and Schiöberg potential, we suggest that the three potential functions are not better than the Morse potential function in representing diatomic interactions. By comparing 21 empirical potential functions in terms of fitting experimental RKR data for some diatomic molecule, Royappa et al. [33] also suggested that the Morse potential is better than the Deng–Fan potential. 4. Conclusions In this work, we have shown that the Manning–Rosen potential, Deng–Fan potential and Schiöberg potential are the same empirical potential energy function for diatomic molecules. By choosing the experimental values for the molecular parameters De, re and xe as
8
P.-Q. Wang et al. / Journal of Molecular Spectroscopy 274 (2012) 5–8
inputs, we calculate the spectroscopic parameters ae and xexe. The results show that the Manning–Rosen potential, Deng–Fan potential and Schiöberg potential are not better than the traditional Morse potential in simulating the atomic interaction for diatomic molecules. Acknowledgments This work was supported by the National Science Foundation for Distinguished Scholars of China under Grant No. 51125019, the National Natural Science Foundation of China under Grant No. 51174169, the Science Foundation of Southwest Petroleum University under Grant No. 2010XJZ198 and the Science Foundation of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation of China under Grant No. PLN-ZL001. References [1] [2] [3] [4] [5] [6]
A.A. Zavitsas, J. Am. Chem. Soc. 113 (1991) 4755. P.G. Hajigeorgiou, J. Mol. Spectrosc. 263 (2010) 101. P.M. Morse, Phys. Rev. 34 (1929) 57. R. Rydberg, Z. Phys. 73 (1931) 376. G. Pöschl, E. Teller, Z. Phys. 83 (1933) 143. M.F. Manning, N. Rosen, Phys. Rev. 44 (1933) 953.
[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]
J.W. Linnett, Trans. Faraday Soc. 36 (1940) 1123. E.R. Lippincott, J. Chem. Phys. 21 (1953) 2070. Y.P. Varshni, Rev. Mod. Phys. 29 (1957) 664. T. Tietz, J. Chem. Phys. 38 (1963) 3036. Z.H. Deng, Y.P. Fan, Shandong Univ. J. 7 (1957) 162. D. Schiöberg, Mol. Phys. 59 (1986) 1123. S. Lifson, A.T. Hagler, P. Dauber, J. Am. Chem. Soc. 101 (1979) 5111. S.L. Mayo, B.D. Olafson, W.A. Goddard, J. Phys. Chem. 94 (1990) 8897. S. Barlow, A.A. Rohl, S. Shi, C.M. Freeman, D. O’Hare, J. Am. Chem. Soc. 118 (1996) 7875. A. Diaf, A. Chouchaoui, R.L. Lombard, Ann. Phys. 317 (2005) 354. W.C. Qiang, S.H. Dong, Phys. Lett. A 368 (2007) 13. S.M. Ikhdair, Phys. Scr. 83 (2011) 015010. A.D.S. Mesa, C. Quesne, Y.F. Smirnov, J. Phys. A: Math. Gen. 31 (1998) 321. Z. Rong, H.G. Kjaergaard, M.L. Sage, Mol. Phys. 101 (2003) 2285. L.H. Zhang, X.P. Li, C.S. Jia, Int. J. Quantum. Chem. 111 (2011) 1870. S.L. Nyeo, I.C. Yang, Phys. Rev. A 63 (2001) 046109. J. Lu, Phys. Scr. 72 (2005) 349. C.S. Jia, J.Y. Liu, L. He, L.T. Sun, Phys. Scr. 75 (2007) 388. S.H. Dong, S.G. Miranda, F.M. Enriquez, S.H. Dong, Mod. Phys. Lett. B 22 (2008) 483. S.M. Ikhdair, R. Sever, Ann. Phys. (Berlin) 18 (2009) 189. C. Berkdemir, J. Math. Chem. 46 (2009) 492. A. Kratzer, Z. Phys. 3 (1920) 289. E. Fues, Ann. Physik 80 (1926) 367. C. Berkdemir, A. Berkdemir, J. Han, Chem. Phys. Lett. 417 (2005) 327. P.G. Hajigeorgiou, J. Mol. Spectrosc. 235 (2006) 111. J.L. Dunham, Phys. Rev. 41 (1932) 721. A.T. Royappa, V. Suri, J.R. McDonough, J. Mol. Struct. 787 (2006) 209.
Contents lists available at SciVerse ScienceDirect
Journal of Molecular Spectroscopy journal homepage: www.elsevier.com/locate/jms
Equivalence of the three empirical potential energy models for diatomic molecules Ping-Quan Wang, Lie-Hui Zhang, Chun-Sheng Jia ⇑, Jian-Yi Liu State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu 610500, People’s Republic of China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 17 February 2012 In revised form 12 March 2012 Available online 21 March 2012
It is found that the Manning–Rosen potential, Schiöberg potential and Deng–Fan potential are the same solvable empirical potential energy function for diatomic molecules. We calculate the anharmonicity xexe and vibrational rotational coupling parameter ae for 16 molecules by choosing the experimental values of the dissociation energy De, equilibrium bond length re and vibrational frequency xe as inputs. The results show that the Manning–Rosen potential, Deng–Fan potential and Schiöberg potential are not better than the traditional Morse potential. Ó 2012 Published by Elsevier Inc.
Keywords: Empirical diatomic potential function Manning–Rosen potential Deng–Fan potential Schiöberg potential Morse potential
1. Introduction There has been a growing interest in investigating the empirical potential energy function for diatomic molecules in chemistry and molecular physics. The reason is that the potential function provides a quantitative description of the energy–distance relation during the making or breaking of chemical bonds between two atoms [1,2]. Many efforts to construct a universal potential energy function have been made by Morse [3], Rydberg [4], Pöschl and Teller [5], Manning and Rosen [6], Linnett [7], Lippincott [8], Varshni [9], Tietz [10], Deng and Fan [11], Schiöberg [12], Zavitsas [1], and Hajigeorgiou [2], among others. In 1929, Morse [3] proposed the first three-parameter solvable empirical potential energy function for diatomic molecules,
2 U M ðrÞ ¼ De 1 eaðrre Þ ;
ð1Þ
in which De is the dissociation energy, re is the equilibrium bond length, and a denotes the range of the potential. Potential-energy functions with parameters of greater number generally provide a better fit to experimental data than those of fewer parameters, but the latter potential functions have been employed in many computational chemistry softwares, such as CVFF [13], DREIDING [14], and ESFF [15]. In 1933, Manning and Rosen [6] proposed a potential function for diatomic molecules,
U MR ðrÞ ¼
h
"
2 2
8lp2 b
bðb 1Þe2r=b ð1 er=b Þ2
# Aer=b ; 1 er=b
ð2Þ
⇑ Corresponding author. Fax: +86 28 8303 2901. E-mail addresses: [email protected] (P.-Q. Wang), [email protected] (C.-S. Jia). 0022-2852/$ - see front matter Ó 2012 Published by Elsevier Inc. http://dx.doi.org/10.1016/j.jms.2012.03.005
in which b and A are two dimensionless parameters, parameter b is related to the range of the potential and has dimension of length. The Manning–Rosen potential can be used to describe the diatomic molecular vibrations [16–18]. In 1957, Deng and Fan [11] proposed a simple potential model for diatomic molecules,
2 eare 1 U DF ðrÞ ¼ De 1 ar ; e 1
ð3Þ
in which De, re, and a denote the dissociation energy, equilibrium bond length and the range of the potential, respectively. The potential (2) is also called a generalized Morse potential [19]. Deng and Fan [11] suggested that the potential function (3) is better than the Morse potential function (1) in representing diatomic interactions for vibration of diatomic molecules. By employing the Deng–Fan potential, some authors investigated the F–H stretching motion and rotational transitions for HF molecule [20,21], and studied the statistical mechanics of the quasi-one-dimensional system of DNA [22]. In 1986, Schiöberg [12] proposed a new potential function for diatomic molecules,
U S ðrÞ ¼ Dð1 r coth arÞ2 ;
ð4Þ
in which D, r, and a are three adjustable positive parameters, r < 1. Schiöberg [12] suggested that the potential function (4) is more accurate than the Morse potential function in fitting the experimental Rydberg–Klein–Rees (RKR) potential energy curves and the rotational-vibrating levels for some diatomic molecules. The Schiöberg potential has been used to describe diatomic interactions [23–27]. In the present work, we investigate the equivalence of the Manning–Rosen potential, Schiöberg potential and Deng–Fan
6
P.-Q. Wang et al. / Journal of Molecular Spectroscopy 274 (2012) 5–8
potential. We also compare the Morse potential with the Manning– Rosen potential, Deng–Fan potential and Schiöberg potential according to quantitative tests on 16 molecules with diverse values of essential parameters. 2. Equivalence of the three empirical potential functions The empirical potential energy function U(r) should satisfy the following conditions:
dUðrÞ ¼ 0; dr r¼re
ð5Þ
Uð1Þ Uðre Þ ¼ De ;
ð6Þ
2 d UðrÞ 2 dr
ð7Þ
¼ ke ¼ 4p2 lc2 x2e ;
r¼re
A bðb 1Þ ¼ ere =b 1 ; 2 2bðb 1Þ þ1 : r e ¼ b ln A
ð8Þ
ð9Þ
Employing the condition (6) and expression (9), we have the following relationship: 2
32lp
A2 ¼ De : bðb 1Þ
2 b2
ð10Þ
Using Eqs. (8) and (10), we get
A¼
16lp2 b h
2
2
ðere =b 1ÞDe :
ð11Þ
Substituting the expressions (8) and (11) into expression (2) and making some algebraic simplifications, we can rewrite the Manning–Rosen potential in the following form:
2 ere =b 1 U MR ðrÞ ¼ De 1 r=b : e 1
ð12Þ
Here, we have added one term De to the original Manning–Rosen potential energy function given in Eq. (2). This change only yields an energy of zero at the potential minimum, i.e. UMR(re) = 0, and does not affect the physical properties of the potential function. We can rewrite the Schiöberg potential function (4) in the following form:
U S ðrÞ ¼ D 1 r
2r e2ar 1
2
e2are 1 : e2are þ 1
ð14Þ
By using the condition (6), we obtain the following relation
Dð1 rÞ D 1 r 2
2r e2are 1
lim U MR ðrÞ ¼ limU DF ðrÞ ¼ limU S ðrÞ ¼ De a!0
2 De 2are þ1 : e 4
a!0
r r 2 e : r
ð18Þ
The Kratzer–Fues molecular potential has been extensively used for investigating the properties of diatomic molecules [30,31]. 3. Evaluation of spectroscopic parameters ae and xexe On the basis of wave mechanics, Dunham [32] showed that the molecular parameters can be expressed in terms of the derivatives of the potential function. The vibrational rotational coupling parameter ae and anharmonicity parameter xexe are given by
ae ¼
r e f3 ; 1þ xe 3f 2
6B2e
ð19Þ
0 !2 1 Be @ r 2e f4 xe ae A ; xe xe ¼ þ 15 1 þ 8 f2 6B2e
ð20Þ
in which Be ¼ 8p2hclr2 , h is the Planck constant, fi (i = 2, 3, 4) denotes e the derivatives of the potential energy function through
2 d UðrÞ f2 ¼ 2 dr
; r¼r e
3 d UðrÞ f3 ¼ 3 dr
4 d UðrÞ and f 4 ¼ : 4 dr r¼re
r¼r e
ð21Þ
By employing the expressions (19) and (20), we obtain the following expressions for spectroscopic parameters ae and xexe corresponding to the Morse potential and Manning–Rosen potential, respectively,
6B2e
1
a 3 De r e ; 2p2 lc2 x2e
ð22Þ
0 !2 1 Be @ 7a4 De r 2e xe ae ðMÞ A ; xe xe ðMÞ ¼ 2 2 2 þ 15 1 þ 8 2p l c xe 6B2e
ð23Þ
ae ðMÞ ¼
xe
ae ðMRÞ ¼
r e f3 ; 1þ xe 3ke
6B2e
ð24Þ
0 !2 1 Be @ r2e f4 xe ae ðMRÞ A ; xe xe ðMRÞ ¼ þ 15 1 þ 8 ke 6B2e
ð25Þ
where
2 ¼ De :
ð15Þ
Be ¼
Substituting Eq. (14) into Eq. (15), and solving it for D, we obtain
D¼
ð17Þ
Here, we replace 2a by 1/b, this expression becomes exactly the expression (12). By comparing expressions (3) and (17) with expression (12), we can conclude that the Deng–Fan potential and Schiöberg potential are the same empirical diatomic potential function, and they are just the Manning–Rosen potential. When the parameter b in Eq. (2) becomes infinity, and the parameter a in Eqs. (3) and (4) goes to zero, the limits of the Manning–Rosen potential, Deng–Fan potential and Schiöberg potential become the Kratzer–Fues molecular potential [28,29], namely,
ð13Þ
:
Substituting the expression (13) into the condition (5), we have
r¼
2 e2are 1 U S ðrÞ ¼ De 1 2ar : e 1
b!1
in which l is the reduced mass of the diatomic molecule, c is the speed of light, and xe denotes the vibrational frequency. Substituting the Manning–Rosen potential function (2) into the condition (5), we obtain
h
Substituting the expressions (14) and (16) into the potential function (13), we obtain the new expression for the Schiöberg potential function,
ð16Þ
f3 ¼
h ; 8p2 clr2e
ke ¼ 4p2 lc2 x2e ;
12De e3re =b b
3
ðere =b
1Þ
3
þ
6De e2re =b 3
2
b ðere =b 1Þ
ð26Þ
;
ð27Þ
7
P.-Q. Wang et al. / Journal of Molecular Spectroscopy 274 (2012) 5–8 Table 1 Comparison of experimental values of ae with calculated values obtained by using the Morse potential and Manning–Rosen potential. All quantities are in 103 cm1. Molecule
State
12 16
1
+
ae(exp)
ae(M)
Dae (%)
ae(MR)
17.50406 306.807 793.4125 302.203 0.8683
16.42597 310.743 658.5493 251.679 1.4569
6.16 1.28 16.998 16.72 67.79
18.99192 377.316 909.783 348.126 1.6777
Dae (%)
C O H35Cl HF DF Na2
X R X1R+ X1R+ X1R+
7
X1 Rþ g
7.017
7
LiH 39 7 K Li Cs2
X1R+ X1R+
216.391 1.9078 0.022012
193.021 3.15753 0.043499
10.80 65.5 97.6
309.501 3.60793 0.046466
43.03 89.1 111.09
RbCs Na39K AgH MgH NaH 16 O2
X1R+ X1R+ X1R+ X2R+ X1R+
0.03661 0.4485 201.93 177.2981 137.09097 15.910
0.07054 0.7737 212.31 263.6870 135.19571 17.493
92.7 72.5 5.14 48.72 1.38 9.95
0.07583 0.8681 250.44 289.3235 181.25006 18.693
107.1 93.5 24.02 63.18 3.22 17.5
35
X1 Rþ g
1.9027
25.5
X1 Rþ g
Li2
X1 Rþ g
X3 R g
Cl2
10.12
1.5163
43.7
Average
12.54
1.9543
36.4
8.50 22.98 14.667 15.20 93.21 77.7
28.9 50.8
Table 2 Comparison of experimental values of xexe with calculated values obtained by using the Morse potential and Manning–Rosen potential. All quantities are in cm1. Molecule
State
xexe(exp)
xexe(M)
Dxexe
xexe(MR)
Dxexe
12 16
X1 R+ X1 R+ X1 R + X1 R +
13.2880 52.746 89.94511 47.2712 0.7214
12.9814 60.047 86.73872 45.5989 1.0569
0.3066 7.301 3.56 3.54 46.5
14.3466 68.436 106.82988 56.2016 1.1651
1.0586 15.690 18.77 18.89 61.5
C O H35Cl HF DF Na2
X1 Rþ g
7
X1 Rþ g
2.5779
3.5894
39.2
4.1431
60.7
7
LiH 39 7 K Li Cs2
X1 R+ X1 R+
23.1679 1.26617 0.08182
24.3446 1.83123 0.12095
5.08 44.63 47.8
32.6174 2.00857 0.12689
40.79 58.63 55.1
RbCs Na39K AgH MgH NaH 16 O2
X1 R + X1 R+ X1 R+ X2 R + X1 R +
0.10953 0.4827 34.180 29.84682 19.52352 11.981
0.16301 0.7285 40.226 50.16752 21.58835 14.846
48.8 50.9 17.68 68.08 10.58 23.9
0.17177 0.7897 45.023 53.58953 26.12419 15.580
56.8 63.6 31.72 79.55 33.81 30.04
35
X1 Rþ g
Li2
X1 Rþ g
X3 R g
Cl2
2.694271
3.863106
43.38
Average
f4 ¼
28.8
72De e4re =b 4
4
b ðere =b 1Þ
72De e3re =b 4
b ðere =b 1Þ
3
þ
14De e2re =b 4
b ðere =b 1Þ2
:
ð28Þ
According to condition (7), the following relations are satisfied by the potential parameters a and b in the Morse potential and Manning–Rosen potential, respectively,
sffiffiffiffiffiffiffi 2l ; a ¼ pcxe De e2re =b b
2
ðere =b
2
1Þ
¼
ð29Þ 2p2 lc2 x2e : De
ð30Þ
The experimental data of De, re, xe, ae and xexe are taken from the recent literature [2]. Choosing the experimental values of De, re and xe as inputs, we determine the potential parameters a and b from Eqs. (29) and (30). By employing expressions (22)–(25), we calculate the values of vibrational rotational coupling parameter ae and anharmonicity parameter xexe for 16 molecules. The corresponding calculated values and experimental data are listed in Tables 1 and 2, respectively. Average absolute relative deviations from the experimental data for ae are 36.4% and 50.8%, respectively, and are 28.8% and 42.1%, respectively, for xexe. The absolute relative deviation is defined by
Dae ¼
jae ðexpÞ ae ðcalcÞj ; ae ðexpÞ
Dxe xe ¼
jxe xe ðexpÞ xe xe ðcalcÞj ; xe xe ðexpÞ
3.943132
46.35 42.1
ð31Þ
ð32Þ
where ae(exp) and xexe(exp) correspond to the experimental values, and ae(calc) and xexe(calc) correspond to the calculated values. Tables 1 and 2 tell us that the Morse potential is better than the Manning–Rosen potential to model diatomic interactions. In view of the equivalence of Manning–Rosen potential, Deng–Fan potential and Schiöberg potential, we suggest that the three potential functions are not better than the Morse potential function in representing diatomic interactions. By comparing 21 empirical potential functions in terms of fitting experimental RKR data for some diatomic molecule, Royappa et al. [33] also suggested that the Morse potential is better than the Deng–Fan potential. 4. Conclusions In this work, we have shown that the Manning–Rosen potential, Deng–Fan potential and Schiöberg potential are the same empirical potential energy function for diatomic molecules. By choosing the experimental values for the molecular parameters De, re and xe as
8
P.-Q. Wang et al. / Journal of Molecular Spectroscopy 274 (2012) 5–8
inputs, we calculate the spectroscopic parameters ae and xexe. The results show that the Manning–Rosen potential, Deng–Fan potential and Schiöberg potential are not better than the traditional Morse potential in simulating the atomic interaction for diatomic molecules. Acknowledgments This work was supported by the National Science Foundation for Distinguished Scholars of China under Grant No. 51125019, the National Natural Science Foundation of China under Grant No. 51174169, the Science Foundation of Southwest Petroleum University under Grant No. 2010XJZ198 and the Science Foundation of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation of China under Grant No. PLN-ZL001. References [1] [2] [3] [4] [5] [6]
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