Vol.2, No.3, 184-189 (2010) Natural Science
http://dx.doi.org/10.4236/ns.2010.23028
A novel analytic potential function applied to neutral
diatomic molecules and charged lons
Chang-Feng Yu, Chang-Jun Zhu, Chong-Hui Zhang, Li-Xun Song, Qiu-Pin Wang
Department of physics, School of Science, Xi’an Polytechnic University, Xi’an, China;
Received 24 November 2009; revised 8 January 2010; accepted 30 January 2010.
ABSTRACT
In this paper, a new method on constructing
analytical potential energy functions is pre-
sented, and from this a analytical potential en-
ergy function applied to both neutral diatomic
molecules and charged diatomic molecular ions
is obtained. This potential energy function in-
cludes three dimensionless undetermined pa-
rameters which can be determined uniquely by
solving linear equations with the experimental
spectroscopic parameters of molecules. The
solutions of the dimensionless undetermined
parameters are real numbers rather than com-
plex numbers, this ensures that the analytical
potential energy function has extensive uni-
versality. Finally, the potential energy function is
examined with four kinds of diatomic molecules
or ions—homonuclear neutral diatomic mole-
cule 

1
2g
H(X) ,

1
2u
K(B) and

1
2u
Li (B),
homonuclear charged diatomic molecular ion


2
2u
He(X) ,  
2
2g
N(X ) and 
2
2g
O(X) , heter-
nuclear neutral diatomic Molecule

1
AlBr(A) ,
)PuO(X g
1
 and 

1
g
NaLi
(
X
)
, heternuclear ch-
arged diatomic Molecular ion 

3
BC (X),


1
MgH(X) and )(XHCl i
2 ,as a conseque-
nce, good results are obtained.
Keywords: Diatomic Molecules And Ions; Potential
Energy Function; Force Constants; Spectroscopic
Parameters; Phase Factor
1. INTRODUCTION
Analytical potential energy functions are of great sig-
nificance in the study of material science, molecular
spectrum, reaction dynamics of atoms and molecules,
vibrational and rotational energy-level structures of
molecules, interactions between laser and matter,
photoionization etc. [1-3] Due to the importance and
extensive applications of the potential energy function,
the corresponding research works have been carried on
all along [4-6]. So far, the representative analytical po-
tential energy function proposed have Morse potential
[7], Rydberg potential [8], Murrell-Sorbie potential (M-S)
[9] and Huxley-Murrell-Sorbie potential (HMS) [10] etc.
Recently, Sun Weiguo et al have proposed an energy
consistent method (ECM) and constructed a new phy-
sically well behaved analytical potential function of a
diatomic system called ECM potential [11]. These po-
tential functions above have merits and defects respec-
tively, they are valid in describing the behaviors of some
individual or classificatory diatoms and molecules. But
none of them can describe both neutral diatomic mole-
cules and charged diatomic molecular ions and describe
precisely the behaviors of potential energy function over
the whole range of internuclear distance. Seen from ex-
pressional forms, most of these potential energy func-
tions adopt the forms of ploynomial and exponential. In
this paper, a cosine function with a phase factor is used
as basic potential energy function and, through renor-
malization to the phase factor, a universal potential en-
ergy function applied to four kinds of diatomic mole-
cules or ions — homonuclear neutral diatomic molecules,
homonuclear charged diatomic molecular ions, heternu-
clear neutral diatomic molecules and heternuclear charged
diatomic molecular ions is given. Finally, the potential
energy function is examined with twelve different kinds
of diatomic molecules and ions etc., as a consequence,
good results are obtained.
2. FUNDAMENTAL SUPPOSITIONS,
AND DERIVATION OF A UNIVERSAL
ANALYTIC POTENTIAL FUNCTION
Suppose that the potential function of diatomic molecu-
lar satisfies the following relation
()cos ()VrAr B


 (1)
where ()arccos(/ )rr



 (2)
where, BA, are undetermined constants, )(r

is a

C. F. Yu et al. / Natural Science 2 (2010) 184-189
185
phase factor related to

,

and the internuclear dis-
tance
r
, here

is equivalent phase difference be-
tween two interacting atoms,

is equilibrium internu-
clear distance. Substituting Eq.2 into Eq.1 , yields

( )cosarccos(/)Vr ArB   

22
(/)cos1/ sin
A
rrB   
(3)
Eq.3 is a basic analytical potential energy function. In
order to obtain the universal analytical potential function
of diatomic molecules and ions, renormalization should
be needed for the term 22
1r   in Eq.3, so as to
ensure that the derivatives of each order of the Eq.3 are
continuous and finite at equilibrium distance


r
.
Thus we can expand the term into binomial series
i
i
ir
ii
i
r2
0
2
22 )(
)12()!(4
)!2(
1





 (4)
Here, Eq.4 is a infinite series, it need to be truncateed
into finite terms and its following infinite terms should
be absorbed into three undetermined coefficients
,,abc , so from Eq.4, we have











 


n
i
n
ir
a
r
ii
i
ri
0
22
2
22 )()(
)12()!(4
)!2(
12







 


n
i
nnni
r
c
r
b
r
a
r
iH
0
6242222 )()()())((

(5)
where
)12()!(4
)!2(
)( 2 ii
i
iH i (6)
Generally, the potential energy function satisfies
asymptotic condition lim()0
rVr 

,so from Eq.3
we have

sinAB  (7)
Substituting Eq.5 and Eq.7 into Eq.3, and notice
1)0( H, yields













624222
1
2
)()()(
))((sincos)(
nnn
n
i
i
r
c
r
b
r
a
r
iHA
r
A
rV




(8)
In Eq.8, the undetermined constant
A
can be deter-
mined according to the properties of potential energy
function. At the equilibrium distance


r
, the potential
value is equal to the negative value of dissociation en-
ergy e D, i.e. e
D
V
)(

, and the first derivatives of
)
(
r
V with respect to
r
is zero.
So from Eq.8, we obtain
e
n
i
DcbaiHAAV 










 
1
)(sincos)( 
(9)


 






n
i
iiH
AA
r
V
r1
)2)(({sincos
d
d





 0)62()42()22( 




ncnbna (10)
From Eq.9 and Eq.1 0, the solutions of
A
and

cos can be given as follows
1
1
)52()32()12()12)((
sin








 
n
i
encnbnaiiH
D
A

(11)
)22()2()({sincos
1
 

naiiH
n
i

)}62()42( 


ncnb (12)
Substituting Eq.11 and Eq.12 into Eq.8, yields

)(rV 4222
1
2)/()/()/)(( 






nn
n
i
i
erbrariHD

62
)(  n
r
c



 

n
i
nbnaiiH
1
)42()22()2)((




 r nc

)62( 





n
i
naiiH
1
)12()12)((

1
)52()32(   ncnb (13)
Eq.13 is the universal analytical potential energy
function that is required. The undetermined parameters
cba ,, can be determined with the experimental
spectroscopic parameters (eeeee B  ,,) of mole-
cules or fitting method using singlepoint potential en-
ergy scanning. When 3,2,1n, from Eq.13,we have

24
68
21
() 16 10142
1468,(1)
e
D
Vr a
abc rr
bc abcn
rr r

 

  


 

  





 
 

 
 

(14)
24
6810
811
() 74056 7228
36810
2
e
D
Vr abc rr
abcabc
rrr r


 

  


 

 





 
 

 
 


(2)n (15)

C. F. Yu et al. / Natural Science 2 (2010) 184-189
186

 )3(12
1088/15
16
1
8
1
2
1
17614411219
16
)(
121086
42



















 





 































n
r
c
ba
r
c
r
b
r
a
r
rrcba
D
rV e



(16)
3. USING EXPERIMENTAL
SPECTROSCOPIC PARAMETERS TO
DETERMINE a,b,c
The undetermined parameters ,,abc can be deter-
mined with the experimental spectroscopic parameters
(eeeee B
 ,, ) of diatomic molecules or ions. The
principle of this method is, according to the relationship
between undetermined parameters and force constants,
to obtain cba ,, by solving linear equations. From
Eq.13, the general expression of force constants at the
equilibrium internuclear distance can be given as follows
1
0
10
(1)
d[() (2)]
d
(21)!(2 3)!(2 5)!
(2 1)!(23)!(25)!
m mm
n
mmm
ij
r
V
V
f
Hiij
r
nmn mn m
ab c
nnn





 
 
  
 
  








 

!])62()42()22()2)(([
1
mncnbnaiiH
n
i
(4,3,2m) (17)
where



n
i
e
ncnbnaiiH
D
V
1
0
)52()32()12()12)((
(18)
From Eq.17 and Eq.18, when 3,2,1n, the fol-
lowing linear equations can be obtained
2
2
3
3
4
4
1123056
1610 142
2 32100224
1610146
362240 644
1 6101424
(1,and 62254320)
e
e
e
f
abc
X
abc D
f
abc
Y
abc D
f
abc
Z
abc D
nXYZ



 
 



 

 

 



(19)
2
2
3
3
4
4
10 120224360
74056 722
24 4008961680
74056 726
43 96025765640
74056 7224
(2,and 98314800)
e
e
e
f abc
X
abc D
f ab c
Y
abc D
f abc
Z
abc D
nXYZ



 






 





(20)
2
2
3
3
4
4
35 4487201056
19 1121441762
98 179233605632
191121441766
206 51521128021648
19 11214417624
(3,and1423741600)
e
e
e
f
ab c
X
abc D
f
abc
Y
abc D
f
abc
Z
abc D
nXYZ



 






 





(21)
In Eqs.19-21, the relationships between force con-
stants and spectroscopic parameters are as follows
222
24cf e
  (22)
)
6
1(
3
2
2
3
e
ee
B
f
f


 (23)







B
B
f
fee
e
ee


8
)
6
1(15 22
2
4 (24)
The Eqs.19-21 above are all linear equations, when
the conditions of 03242562

ZYX , 98 31
X
Y

4800Z


and 160437142 

ZYX 0 are satisfied
with respect to Eqs.19-21 , they have unique real number
solutions for the undetermined parameters cba,, .
Calculations show that the conditions above are always
tenable in general. This ensures that the analytical poten-
tial function Eq. 13 has extensive universality, which can
describe any of diatomic molecules and ions especially
the behaviors of molecules near equilibrium internuclear
distance. So far, the most extensively used analytical
potential energy function is Murrel-Sorbie (M-S) poten-
tial. The undetermined parameters in Murrel-sorbie po-
tential which are determined by experimental spectro-
scopic parameters have no unique solutions and contain
complex number solutions. Thus, the M-S potential is
extremely limited in applications to some diatomic
molecules and ions. [12]
4. APPLIED EXAMPLES OF THE
UNIVERSAL ANALYTICAL
POTENTIAL ENERGY FUNCTION
For examining potential energy function Eq.13, fifty
kinds of neutral diatomic molecules and charged diatomic

C. F. Yu et al. / Natural Science 2 (2010) 184-189
187
Table 1. Experimental spectroscopic parameters of diatomic molecules and ions.
states eV/nm/cm/cm/cm/cm/ 1111 eeeeeeDB 
 Refs.
)(XH g
1
2
 4401.21 121.34 60.809 3.062 0.0741 4.747 [13]
)B(K u
1
2 75.00 0.3876 0.04824 0.000235 0.4235 0.514 [14]
)B(Li u
1
2 270.7 2.9530 0.5770 0.0083 0.2936 0.3700 [15]
)(XHe u
2
2  1698.52 35.30 7.211 0.2240 0.1080 2.475 [16]
)X(Ng
2
2

 2207.20 16.1360 1.9320 0.0200 0.1116 6.341 [14]
)X(O g
2
2
 1905.30 16.304 1.6905 0.0189 0.1117 6.7792 [17]
)AlBr(A1 297.2 6.400 0.1555 0.00216 0.2322 2.400 [14]
)X(PuO g
1
 822.28 2.500 0.3365 0.00146 0.1830 7.3372 [18]
)X(NaLi g
1
 256.80 1.610 0.3960 0.0036 0.2810 0.8570 [13]
)(XBC 3
 1301.4 9.820 1.418 0.0155 0.1445 5.588 [19]
)X(MgH 1  1226.60 16.300 3.321 0.0640 0.16530 2.100 [14]
)X(HCli
2
 2675.4 53.50 9.9463 0.3183 0.13152 4.480 [14]
Table 2. Potential parameters and force constants of diatomic molecules and ions.
states 44
4
33
3
22
2nmaJ10/nmaJ10/nmaJ10/.nm/eV/   fffcbanDe

)(XH g
1
2
 4.747 0.0741 1 –0.4615 0.2008 –0.0367 5.752 –37.43 238.7
)B(K u
1
2 0.514 0.4235 3 –0.64605 0.47875 –0.10628 0.0646 –0.1035 0.045
)B(Li u
1
2 0.370 0.2936 3 –0.69431 0.57753 –0.14667 0.1463 –0.3177 0.4546
)(He 2
2 u
X 2.475 0.1080 3 –0.8073 0.6766 –0.1729 3.401 –20.97 101.23
)X(N g
2
2

 6.341 0.1116 3 –0.5889 0.44335 –0.10575 20.11 –160.61 1059.1
)X(O g
2
2
 6.7792 0.1117 3 –0.55136 0.37972 –0.0829 17.09 –142.33 918.04
)AlBr(A1 2.400 0.2322 1 –0.4874 0.2759 –0.0577 1.049 –7.357 21.837
)X(PuO g
1
 7.3372 0.183 3 –0.64234 0.49584 –0.11975 5.959 –27.03 98.61
)X(NaLi g
1
 0.857 0.281 3 –0.78369 0.67093 –0.17435 0.2095 –0.4434 0.7012
)X(BC 3
 5.588 0.1445 3 –0.6900 0.5404 –0.1317 5.677 –31.49 140.50
)X(MgH1  2.10 0.1653 3 –0.74505 0.62333 –0.15926 1.6468 –6.5343 19.547
)X(HCli
2
 4.480 0.13152 3 –0.80085 0.66178 –0.16814 4.3433 –25.61 138.1
Table 3. Potential parameters of Murrel-Sorbie potential of diatomic molecules and ions.
states eV/nm/nm/nm/nm/ 3
3
2
2
1
1e
Daaa 


)(XHg
1
2
 39.601 405.91 3577.1 0.0741 4.747
)B(Ku
1
2 1.227 –38.457 161.65 0.4235 0.514
)B(Li u
1
2 28.79 291.03 1317.4 0.2936 0.370
)(He2
2u
X 32.363 94.792 584.08 0.108 2.475
)X(N
g
2
2

 70.966 1528.3 15675 0.1116 6.341
)X(Og
2
2
 1.376 68.968 1085.1 0.1117 6.7792
)AlBr(A1 9.1044 –95.068 2072.2 0.2322 2.400
)X(PuO g
1
 30.377 207.88 804.25 0.183 7.3372
)X(NaLi
g
1
 19.865 121.01 329.24 0.281 0.857
)
(
XBC 3
 27.880 71.540 634.30 0.1445 5.588
)X(MgH1  33.953 331.66 1450.8 0.1653 2.10
)X(HCl i
2
 29.618 150.74 1133.1 0.13152 4.48

C. F. Yu et al. / Natural Science 2 (2010) 184-189
188
Figure 1. Potential curve of   g
1
2XH.
Figure 2. Potential curve of )(He 2
2
 u
X
Figure 3. Potential curve of  1 AAlBr .
molecular ions have ever been investigated and good results
are obtained. Calculations show that two common poten-
tial energy curves, i.e. steadystate and metastable state
Figure 4. Potential curve of )(XBC 3
.
can be given by using the potential energy function de-
termined with experimental spectroscopic parameters.
The experimental spectroscopic parameters of 1
2g
H(X ),


2
2u
He(X) ,

)
1
AlBr(A and )(XBC 3
 etc. are listed
in Table 1. According to Eqs.22-24, the corresponding
force constants can be obtained by using the experimen-
tal spectroscopic parameters above, and substituting
these force constants into Eq.19 or Eq.2 1, then the un-
determined parameters ,, ba c can be calculated by
solving the linear equations. The calculation values are
listed in Table 2. The potential energy curves (to be cal-
culated and plotted by using Eq.14 and Eq.16 directly
with Origin 7.0 software) plotted by Eq.14 and Eq.16 of
1
2g
H(X ),

 2
2u
1
He(X), AlBr(A)


 and )(XBC 3

are illustrated in Figures 1-4. As comparison, in the
Figs., the dot lines are the potential curves which are
plotted by using the most extensively used Mur-
rel-Sorbie Potential. The M-S potential expression is as
follows

 
2
12
3
31
() 1
exp
e
VrDa rar
ar ar




 


 





(25)
The relationships between undetermined parameters
of M-S potential and force constants are as follows
22
2
1)2( faaDe (26)
3
3
1321 )33(2faaaaDe (27)
4312
2
1
4
1)24123(faaaaaDe (28)
5．CONCLUSIONS
In this paper, we first introduce the phase concept to the
studies of analytical potential energy functions and get

C. F. Yu et al. / Natural Science 2 (2010) 184-189
189
good results. This shows that the method of constructing
analytical potential energy function by means of phase is
effective and reliable. Compared with other potential
energy functions, the potential energy function given in
this paper has two merits: 1) The undetermined parame-
ter equations determined by experimental spectroscopic
parameters are linear equations. Because these linear
equations have unique real number solutions, so this
potential energy function has a extensive universality; 2)
This potential energy function can describe four different
kinds of diatomic molecules or ions—homonuclear neu-
tral diatomic molecules, homonuclear charged diatomic
molecular ions, heternuclear neutral diatomic molecules
and heternuclear charged diatomic molecular ions; In
addition, This potential energy function can also de-
scribe accurately the behaviors of potential curves over a
fairly wide range of internuclear distance.
Potential energy functions of diatomic molecules are
the basis to the studies of multi-atomic molecules, ions
and clusters, which have extremely significances and
applied values in the study of material science, molecu-
lar spectrum chemical reaction etc. Chemical reaction,
molecular collision and many other problems need pre-
cise analytical potential energy functions. Thus, the stud-
ies of analytical potential energy function will still be
important subject in atomic and molecular physics.
REFERENCES
[1] Yiannopoulou, A., Jeung, G.-H., Su, J.P., et al. (1999)
Potential-energy curves for highly excited electronic
states in diatomic molecules related to the atomic orbital
undulations [J]. Physical Review A, 59(2), 1178-1186
[2] Liu, G.Y., Sun, W.G. and Feng, H. (2004) Studies on the
analytical potential energy function of diatomic molecu-
lar Ion XY+ using variational method [J]. Science in
China (Series G), 47(2), 154-164.
[3] Maniero, M. and Acioli, P.H. Full configuration interac-
tion pseudopotential determination of the ground-state
potential energy curves of Li2 and LiH [J]. International
Journal of Quantum Chemistry, 103(5), 711-717.
[4] Xie, R.H. and Gong, J.B. (2005) A Simple Three- pa-
rameter model potential for diatomic systems: From
weakly and strong to metastable molecular ions [J].
Physical Review Letters, 95, 263202.
[5] Yu, C.f., Yan, K. and Liu, D.Z. (2006) A universal ana-
lytic potential-energy function based on a phase factor [J].
Acta Metallurgica Sinca (English Letters), 19(6), 455-468.
[6] Esteves, C.S., de Oliveira, H.C.B., Ribeiro, L., et al.
(2006) Modeling diatomic potential energy curves through
the generalized exponential function [J]. Chemical Phys-
ics Letters, 427(1), 10-13.
[7] Morse, P.M. (1929) Diatomic molecules according to the
wave mechanics. Ⅱ. Vibrational levels [J]. Physical Re-
view Letters, 34: 57-64.
[8] Rydberg, R. (1931) Graphische Darstellung einiger Ban-
denspe-ktroskopischer Ergebnisse [J]. Z Physics, 73:
376-385.
[9] Murrel, J.N. and Sorbie, K.S. (1974) New analytic from
for the potential energy curves of stable diatomic state [J].
Journal of the Chemical Society, Faraday Trans, II, 70,
1552-1557.
[10] Huxley, P. and Murrel, J.N. (1983) Ground-state diatomic
potential [J]. Journal of the Chemical Society, Faraday
Trans
Ⅱ , 79, 323-328.
[11] Sun, W.G. and Feng, H. (1999) An energy-consistent
method for potential energy curves of diatomic mole-
cules [J]. Journal of Physics B: Atomic, Molecular and
Optical Physics, 32(21), 5109-5121.
[12] Zhu, Z.H. and Yu, H.G. (1997) Molecular structure and
molecular potential energy function [M]. Science Press,
Beijing, 38-123.
[13] Geng, Z.D., Fan, X.W. and Zhang, Y.S. (2006) Structure
and potential energy function of the ground state of XY
(H, Li, Na).[J] Acta Physica Sinica, 55(05), 2175-2179.
[14] Herzberg, G. (1983) Molecular spectra and molecular
structure (i. spectra of diatomic molecules) [M]. Science
Press, Beijing, 397-454
[15] Yu, B.H., Shi, D.H., Sun, J.F., Zhu, Z.L., Liu, Y.F. and
Yang, X.D. (2007) Ab initio Calculation on Accurate
Analytic Potential Energy Functions and Harmonic Fre-
quencies of 3
g
c


and 1
u
B States of Dimer 7Li2 [J].
Chinese Physics, 16(8), 2371-2377.
[16] Haberland, H., Issendorff, B.V., Frochtenicht, R., et al.
(1995) Absorption Spectroscopy and Photodissociation
Dynamics of Small Helium Cluster Ions [J] Journal of
Chemical Physics, 102(22), 8773-8779.
[17] Sarvpreet, K. and Mahajan, C.G. (1999) Wei hua’s
four-parameter potential comments and computation of
molecular constants_ e

and ee

 [J]. Pramana jour-
nal of physics, 52(4), 409-420.
[18] Gao, T., Wang, H.Y., Yi, Y.-G., Tan, M.-L., Zhu, Z.-H.,
Sun, Y., Wang, X.-L. and Fu, Y.-B. (1999) Ab intio cal-
culation of the potential energy function and thermody-
namic functions for ground state 5
X

 of PuO [J].
Acta. Physica Sinica, 48(12), 2222-2227.
[19] Tzeli, D. and Mavridis, A. (2001) Continuing our study
on the electronic structure of the carbides BC and AlC [J].
Journal of Physical Chemistry, A105, 1175-1184.