Acta Chim. Slov. 2005, 52, 149-152
149
Scientific Paper
Approximate Relationships Between the Generalized Morse and the Extended-Rydberg Potential Energy
Functions
Teik-Cheng Lim
Faculty of Engineering, Nanoscience & Nanotechnology Initiative,National University of Singapore, S 117576, Republic of
Singapore. E-mail: alan_tc_lim@yahoo.com
Received 11-02-2005
Abstract

Parameters of the Generalized Morse and the Extended-Rydberg potential functions are connected herein. Due to the existence of a polynomial portion in the latter, we apply the Maclaurin series expansion to the former to allow comparison of terms that lead to the parametric relationships. Two schemes for the parametric connections were developed in order to cater to two broad categories of the Extended-Rydberg parameters on the basis of the sign of a2. In either čase, it was shown that the curvatures at the minimum well-depth are equal, thereby indicating validity of the parametric connections for small distortion. Theoretical plots also reveal that, in the čase of large interatomic distortion, the Generalized Morse approximation gives slight over-estimation and under-estimation to the Extended-Rydberg potential for a2<0 and a2>0 respectiveh/.
Keywords: interatomic potentials, Maclaurin series, parametric connections

Introduction
Mathematical functions have been exploited in recent years to relate interatomic potential functions, as well as to observe any discrepancies upon equating them.1"4 A result of these works led to the develop-ment of a prototype interatomic potential function converter,5'6 which relates various potential functions vvithin the categories of bond-stretching, bond-bending, bond-torsion and van der Waals interaction. Although connections of potential functions have been extensively established for various combinations of pair potentials, these have been largeh/ confined to simple potentials such as the harmonic,7 Lennard-Jones,8 Morse,9 Rydberg10 and Buckingham11 potential functions (see ref. 12, 13). Potential functions with greater flexibiliry - which consist of more parameters such as the Generalized Morse potential (adopted as the 2-body portion of the Biswas-Hamann potential)14 and the difference of two Gaussians (adopted as the 2-body portion of Kaxiras-Pandey)15 - were also related to simple potential functions by equating derivatives of these potentials at the equilibrium bond length and by introducing scaling functions.16"23 This paper takes advantage of the Maclaurin series24'25 to relate the parameters of the Generalized Morse and the Extended-Rydberg potential functions. The reason for so doing is discussed with reference to the mathematical forms of these two potentials.
It should be stated herein that interatomic po-
tential energy functions - like intramolecular and intermolecular functions - have no exact analytical expressions. Given that the results of the quantum chemical calculations and the potentials derived from experimental data are furnished in tabular form, the approximate functionals discussed in this paper are nonetheless of great practical importance. It has been appreciated that “the simple functions such as Morse, Rydberg, Born-Mayer, etc, give a qualitatively correct description of the potential and only modest extensions are needed to obtain functions which stand up to the most stringent experimental test”.26
Analysis
The Generalized Morse potential14
UGM=_^JAiexp(-Air)
(1)
where subscripts 1 and 2 refer to the repulsive and at-tractive terms respectiveh/, and the Extended-Rydberg potential27
i>„/?"]exp(-ai/?)          (2)
uER=
D
J
in which
p = r-R,

are the more generalized forms of the Morse
potential9
!=1
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150
Acta Chim. Slov. 2005, 52, 149–152
UM
D[exp(-2ap) - 2 exp(-ap)] and that of the Rydberg potential10
URyd =
-D[l + aip]exp(-aip)
(4)
(5)
respectively. Here, we refer to the Generalized Morse potential as that applicable for the diatom molecule. For the čase of multiatom molecules, the generalization of Morse potentials can be expressed by the Empirical Valence Bond (EVB) approach.28
Reduction of the generalized forms to the original forms of these potential functions can be made by the following substitution
X,
A, exp(-XtR) = D; (i, j = 1,2) (6)
and
X1 = 2X2 = 2a                                            (7)
for the čase of Generalized Morse to original Morse, and
0
(8)
for the čase of Extended-Rydberg to original Rydberg function. A brief examination of the Extended-Ryd-berg’s parameters (Table 1 of reference 27) reveals that D, R, a1 and a3 are positive while the parameter a2 is negative for some ground-states, although most are positive. As such, we develop two sets of relationship betvveen the Generalized Morse and the Extended-Ry-dberg parameters: (i) for the čase where a2<0, and (ii) for the čase where a2>0. For both cases, we begin with the Generalized Morse potential written in another, but equivalent, form as
UGM=D
X,

X1-X2
xx-x2
exp(-X1p) exp(-X2p)
(9)
where D is defined from Equation (6). We next proceed with the analysis, bearing in mind the following bounds for the Morse indices: 4 >0
X2>0
(X1-X2)>0
For a2<0 Here we express Equation (9) as
4 l
(10)
(11)
(12)
uGM

xx-x2
exp[-(X1-X2)p]
(13)
exp(-X2p)
X1-X2\           2
vvhereupon substituting the first four terms of the Maclaurin series,
oxp[-(Xi-X2)ph±[  ^ >)T>,(
n\
Equation (13) becomes
UGM=
D
1
1 + X2p — X2(X1 - X2 )p
2
+ -X2(X1-X2)2p3
(15)
exp(-X2p)
Comparing the terms of the Extended-Rydberg potential, Equation (2), with those of Equation (15) gives the following relationships
a1=X2
a0 =

-X2(X1-X2)
a3--^l{\-^2)

(17)
(18)
With reference to equations (11) and (12), Equa-tion (17) implies that a2 is negative, hence indicating that equations (16) to (18) apply for the following five ground-states: Al-O, Be-S, F-Mg, Mg-O and Mg-S, as specified by the negative values of their a2 parameters (in Table 1 of reference 27).
For a2>0 For this čase, we express Equation (9) as
UGM=D
X,
X

/tj     A2      /tj     /t2
Qxp[(X1-X2)p\

(19)
exp(-/tj/7)
which, upon substituting the following first four terms of the Maclaurin series
exP[(Xl-X2)ph±^lp".
n\
(20)
gives
uGM=
D
1
l + /l1/7 + -/l1(/l1-/l2)/72
1
2
2      3
+-(/L -x„y p
6
.(21)
exp(-/l1/7)
Comparison of terms in this equation with those of Equation (2) leads to the following connections
^   =  Xy
a2 =
-Xi(X1-X2f
(22) (23)
(24)
n=0
a
a

2
3
n=0
"3
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Acta Chim. Slov. 2005, 52, 149–152
151
By virtue of Equations (10) and (12), Equations (22) to (24) are applicable for 66 other ground-states with positive a2 (as listed in Table 1 of reference 27).
Discussion
The reason for using the Maclaurin Series for relating the two generalized forms of interatomic po-tentials can be seen in the polynomial portion of the Extended-Rydberg potential function. This polynomial portion can be easih/ reproduced from Equations (13) and (19), thereby allowing comparison of terms to be made. Since it is well known that both potential func-tions adhere to sharp rise, minimum well-depth and zero at bond compression, at equilibrium and at bond dissoci-ation respectiveh/, then the only theoretical verification required for small distortion about equilibrium is equal curvature at the equilibrium interatomic distance, r=R. Taking double derivatives for the Generalized Morse and the Extended-Rydberg potential functions, we have
-\2tj
GM
dP2
S p=0
and
d2U
ER
dP2
\\D
 [at — 2a2 jD
(25)
(26)
/ p=0
respectively at equilibrium. Substituting either equa-tions (16) and (17), or equations (22) and (23), into Equation (26), we recover the result shown in Equa-tion (25). Equal minimum well-depth and curvature at equilibrium therefore proves the validity of the parametric connections establish herein at and near the minimum well-depth. Further to this, plots of Be-S and H-Na potentials, as examples of a2<0 and a2>0 respectiveh/, were plotted using the Extended-Ryd-berg parameters given by Huxley and Murrell.27 Using equations (16) and (17), the Generalized Morse parameters were obtained for Be-S potential (with a2<0) as
K

2^
and
/Ul
(27)
(28)
From Equations (22) and (23), the Generalized Morse parameters of H-Na (i.e. a2>0) were obtained as
Al=a1                                                 (29)
and
/Ui

2^
(30)
Figures 1 and 2 depict the potentials of Be-S and H-Na respectiveh/, using the Extended-Rydberg and Generalized Morse potential functions, whereby the latter was plotted on the basis of the former’s
1 0 -1
>"2
=> -3
-4
-5
-6
	.Generalized					
	IV	orse N				
		1,		nded-Rydberg		
l		1	Exte			
		1				
\		/	Be-!	i Poter tial (a: <0)		
						
-1
o
12          3          4          5
r-R (Angstrom)
Figure 1. Generalized Morse approximation to Extended-Ryd-berg curve for Be-S potential.
0.5
0
-0.5
>
3L    -1
-1.5
-2
-2.5
-10         1          2         3         4         5
r-R (Angstrom)
Figure 2. Generalized Morse approximation to Extended-Ryd-berg curve for H-Na potential.
parameters. See Table 1 for the potential functions’ parameters. The plotted potential curves reveal that: Good correlation is seen at and near equilibrium, as the theoretical analysis indicates.
Table 1. Potential function parameters for Be-S and H-Na based on Extended-Rydberg parameters27 and Generalized Morse parameters calculated in this paper.
	Extended				
	Rydberg v    ___			eralizec 1orse	
			u en l\		
					
		H-Ne	i Poter tial (a ,>0)		
					
ŠT^v,*^
^
i
A2
Be-S H-Na
1.7415   5.007 1.8874   1.952
2.128 2.154
-0.308 1.071
0.220 0.365
4.27086   2.128 2.154   1.13159
Generalized Morse function over-estimates the Extended-Rydberg potential at long range for a2<0.
Generalized Morse function under-estimates the Extended-Rydberg potential at long range for a2>0.
For completeness’ sake, we extend the present parametric connection to those of the original Morse potential. Substituting Equation (7) into Equations (16) to (18) gives
a


-2
-]
-i
a

a
a
2
a

a
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152
Acta Chim. Slov. 2005, 52, 149–152
Table 2. Parametric conversion between Generalized Morse (GM) and Extended-Rydberg (ER) potential function.
From GM to ER
FromERtoGM
Remarks
*fa-xy
n\
in = 1,2,3)
M-
u,
 = JU±
UiJ
Fara2<0,=>/ = 2,y = l. For a2 >0, =>i = l,7' = 2.
(-1)'

;
in = 1,2,3)
(31)
for a2<0, whilst substitution into equations (22) to (24) yields
r/n
—     ;   (n = 1,2,3)                      (32)
n! J
for a2>0. In any čase, substituting Equation (7) into Equation (25) gives the same curvature as substituting either equations (31) or (32) into Equation (26), thereby not violating the equality of potential curve at and near the minimum well-depth.
Conclusion and Recommendation
By relating the parameters of the Generalized Morse and the Extended-Rydberg potential func-tions, we have included as well the parameters of the original Morse and Rydberg potential functions. Parametric connections betvveen Generalized Morse and the Extended-Rydberg potential functions have been attained by taking advantage of the Maclaurin series that elegantly fits into the polynomial portion of the Extended-Rydberg potential. A summary of parametric conversion is furnished in Table 2. Since the number of parameters for Extended-Rydberg exceeds those of Generalized Morse by 1, therefore only a1 and a2 (a3 being redundant) are required for converting the Extended-Rydberg Parameters into Generalized Morse parameters but both the Morse indices are needed to obtain ax, a2 and av
Without doubt, vibrational analysis is extremely sensitive to the details of the potential. It would hence be a challenge for the future to solve vibrational 1-D Schrödinger equation for both functional forms (see also ref. 29 and 30).
References
1.   T. C. Lim, /. Math. Chem. 2002, 31, 421-428.
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7.   R. Hooke, The Trne Theory of Elasticity and Springiness 1676.
8.   J. E. Lennard-Jones, Proč. Roy Soc. Lond. A 1924, 106, 463-477.
9.   P. M. Morse, Phys. Rev. 1929, 34, 57-64.
10.   R. Rydberg, Z. Phys. 1931, 73, 376-385.
11.   R. A. Buckingham, Proč. Roy. Soc. Lond. A 1938, 169, 264-283.
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14.   R. Biswas, D.R. Hamann, Phys. Rev. Lett. 1985, 55, 2001-2004.
15.   E. Kaxiras, K.C. Pandey, Phys. Rev. B 1988, 38, 12736-12739.
16.   T. C. Lim, Z. Naturforsch. A 2004, 59, 116-118.
17.   T. C. Lim, Czech. J. Phys. 2004, 54, 553-559.
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26.   J. N. Murrell, S. Carter, S.C. Farantos, P. Huxley, A.J.C. Varandas, Molecular Potential Energy Functions, Wiley: Chichester, 1984, pp.182-185.
27.   P. Huxley, J. N. Murrell, / Chem. Soc. Faraday Trans II 1983, 79, 323-328.
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Povzetek
Raziskovali smo povezavo med posplošeno Morsejevo in razširjeno Rydbergovo potencialno funkcijo. Ker Ryd-bergov zapis vsebuje razvoj v potenčno vrsto, smo v Morsejevi funkciji uporabili Mac Laurinov razvoj, kar nam je omogočilo primerjavo med parametri obeh funkcijskih zapisov. Razvili smo dve shemi povezav med parametri z namenom dobiti dve kategoriji parametrov razširjene Rydbergove funkcije glede na predznak parametra a2. V obeh primerih se je pokazalo, da sta ukrivljenosti krivulj v bližini minimuma enaki, kar kaže na uspešnost opisa potenciala pri majhnih odmikih. Za velike odmike pa se je pokazalo, da posplošeni Morsejev približek da nekoliko večje vrednosti od Rydbergovega pri negativnih a2, medtem ko je pri pozitivnih vrednostih za a2 učinek obraten.


an =
a
Lim    Potential Energy Functions