INTERMATHS, VOL. 4, NO. 1 (2023), 9–24

https://doi.org/10.22481/intermaths.v4i1.12921

Article

cb licença creative commons

A methodology to obtain accurate potential ener-

gy Functions for diatomic systems: mathematical

point of view

Uma metodologia para obter funções de energia potencial precisas para

sistemas diatômicos: ponto de vista matemático

Judith de P. Araújo

a,∗

, Maikel Y. Ballester

, Mariana P. Martins

, Rafael P. Silva

, Isadora G. Lugão

Ituen B. Okon

, and Clement A. Onate

Instituto Federal Sudeste de Minas Gerais, Juiz de Fora - MG, Brasil;

Universidade Federal de Juiz de Fora, Juiz de Fora - MG, Brasil;

University of

Uyo, Uyo, Nigéria;

Kogi State University, Anyigba, Nigéria

* Correspondence: judith.araujo@ifsudestemg.edu.br

Abstract: The mathematics used in physical chemistry has changed greatly in the past forty years

and it will certainly continue to change more quickly. Theoretical chemists and physicists must have an

acquaintance with abstract mathematics if they are to keep up with their ﬁeld, as the mathematical

language in which it is expressed changes. Thinking about it, in this article, we want to show some

of the most important concepts of Mathematical Analysis involved in obtaining analytical functions

to represent the potential energy interaction for diatomic systems. A basic guide for the construction

of a potential based on Dunham’s coeﬃcients and an example of a new potential obtained from this

methodology is also presented.

keywords: Mathematical Analysis; Analytical potential energy functions; Born-Oppenheimer Approxi-

mation; Dunham coeﬃcients; Morse-type potential.

Resumo: A matemática usada na físico-química mudou muito nos últimos quarenta anos e certamente

continuará a mudar mais rapidamente. Químicos e físicos teóricos devem ter um conhecimento da

matemática abstrata se quiserem manter-se atualizados em seu campo à medida que a linguagem

matemática na qual ela é expressa muda. Pensando nisso, neste artigo, queremos mostrar alguns dos

conceitos mais importantes da Análise Matemática envolvidos na obtenção de funções analíticas para

representar a interação de energia potencial para sistemas diatômicos. Também é apresentado um guia

básico para a construção de um potencial baseado nos coeﬁcientes de Dunham e um exemplo de um

novo potencial obtido a partir desta metodologia.

Palavras-chave: Análise Matemática; Funções de energia potencial analíticas; Aproximação de

Born-Oppenheimer; coeﬁcientes de Dunham; potenciais tipo-Morse.

Classiﬁcation MSC: 70-11; 81Q05

1 Introduction

The Born-Oppenheimer approximation is a cornerstone for molecular systems in the

study of quantum mechanics. It introduces the concept of the molecular potential

energy surface (PES). The molecular potential energy surface is the potential energy

Received: 30 May 2023 Accepted: 11 June 2023 Available online: 30 June 2023.

that determines the motion of nuclei. In the Born-Oppenheimer Approximation (BOA)

the electrons adjust their positions instantaneously to follow any movement of the nuclei

so that the potential energy surface can be equally thought of as the potential for the

movements of atoms within a molecule or atoms in collision with each other. The motion

with this characteristic is called adiabatic, where the dynamic of the system is associated

with a single potential energy surface [1].

Considering an isolated molecular system composed of electrons and atomic nuclei,

the time-dependent Schrödinger equation is given by:

iℏ

∂

∂t

Φ({r

}, {R

}, t) = H Φ({r

}, {R

}, t) (1)

where H is the Hamiltonian:

H = −

ℏ

∇

−

ℏ

∇

4πϵ

i<j

− r

+ −

4πϵ

I,i

− r

4πϵ

I<J

− R

(2)

for the electronic

}

and nuclear

}

degrees of freedom. In Eq. (2),

and

are

mass and atomic number the

nucleus;

and

−e

are electron mass and charge; and

is the vacuum permittivity. Naming:

n−e

({r

}, {R

}) =

4πϵ

i<j

− r

+ −

4πϵ

I,i

− r

4πϵ

I<J

− R

(3)

and replacing in Eq. (2), we then have:

H = −

ℏ

∇

−

ℏ

∇

+ V

n−e

({r

}, {R

}). (4)

Calling

({r

}, {R

}) = −

ℏ

∇

+ V

n−e

({r

}, {R

}) (5)

we get

H = −

ℏ

∇

+ H

({r

}, {R

}). (6)

Since for each nuclei conﬁguration

(

)

}

, the time-independent Schrödinger equation

for a given k electronic state

({r

}, {R

})Ψ

= E

({R

})Ψ

({r

}, {R

}) (7)

is solving for ﬁxed nuclei in all conﬁgurations

}

, and the adiabatic eigenfunctions are

knowing at all possible nuclear conﬁgurations, the total wave function can be expanded

as:

Φ({r

}, {R

}, t) =

∞

l=0

({r

}, {R

})χ

({R

}, t) (8)

10 | https://doi.org/10.22481/intermaths.v4i1.12921 Judith de P. Araújo et al.

in terms of the complete set of eigenfunctions

{ψ

}

and time dependent nuclear

wave functions {χ

From adiabatic approximation, the wave function Φ(

}, {R

}

;

): can be decoupled

as a single product of an electron and a nuclear wave equation:

Φ({r

}, {R

}; t) ≈ Ψ

({r

}, {R

}) · χ

({R

}; t). (9)

and from Born-Oppenheimer approximation, the Schödinger equation Eq. (1) becomes

−

ℏ

∇

+ E

({R

})

= iℏ

∂

∂t

(10)

that is what will lead to the decoupling of the nuclear and electronic movements, allowing

to calculate them separately.

Now, it is easy to see that nuclei can be approximated to classical point particles. For

this, it is necessary to extract the semi-classical mechanic from the quantum mechanic

(for more details see Ref. [

]) to obtain the diﬀerential equation involving the eﬀective

potential V

(t) = −∇

({R

(t)}). (11)

Thus, within the Born-Oppenheimer approximation, the nuclei move according to

the classical mechanics on an eﬀective potential

, given by the PES

obtained

from solving, for each nuclei conﬁguration

(

)

}

, the time-independent Schrödinger

equation for a given k electronic state.

This potential for time-local interaction of many bodies due to quantum electrons is

a function of the set of all classical nuclear positions at a time t.

We are interested in the case of two bodies, where the potential will depend only on

the internuclear distance

. The construction of an accurate analytical potential energy

function satisfying the Born-Oppenheimer Approximation for diatomic systems from

experimental data is still an important problem in chemistry and molecular physics.

In recent work[

], we have reviewed and compared ﬁfty analytical representations

of potential energy interaction for diatomic systems, proposed from 1920 to 2020.

They can be gathered into two groups: those directly obtained from the spectroscopic

constants, and those depending upon parameters ﬁne-tuned to reproduce ab initio

energies. Functions containing a product of an exponential by a polynomial (with its

variations) are, in general, the best representation of the potential energy functions.

The studied cases have shown that a function that escapes this conﬁguration hardly

provides accurate results [3].

Thus, this work aims to show that functions composed of a product of an exponential

and a polynomial are the main candidates for describing accurate diatomic potentials.

For such, concepts of Mathematical Analysis Theory have been followed.

The conditions of diﬀerentiability, continuity, and convergence of the functions were

the ﬁrst to be veriﬁed, since these are necessary to obtain the spectroscopic parameters

and to evaluate the system dissociation process, respectively. Then, we have shown

some examples of well-known potentials that satisfy these conditions, and others, such

as Dunham and Morse, that produce inaccuracies in some required ranges. Next, a

Judith de P. Araújo et al. INTERMATHS, 4(1), 9–24, June 2023 | 11

step-by-step to build a potential based on the exponential-polynomial model is suggested.

A methodological pathway for obtaining coeﬃcients from experimental data is also

provided.

We expect readers will also be able to build their analytical functions depending only

on experimental data and considering all the necessary physical criteria. Perhaps it is

not a generalized potential, but one that meets the researcher’s needs. In addition, our

proposal may assist in obtaining surfaces for certain diatomic systems that have accurate

data described in the literature, yet without accurate theoretical calculations. For these

cases, a function that depends uniquely on experimental data becomes indispensable.

The manuscript is categories into diﬀerent subsections. Section 2 covers the basic

criteria that every potential energy function satisfying the BOA must satisfy. Section 3

presents deﬁnitions and important results of Mathematical Analysis that will be useful to

support the construction of functions. Section 4 some of the most used potentials in the

literature for their accuracy are presented, highlighting the mathematical characteristics

that they have in common. Section 5, a method to construct correct potential energy

functions is described. Finally, in Section 6 a potential that was obtained using the

described methodology can be seen, followed by the conclusions in Section 7.

2 The choice of functions

To our knowledge, there is not an analytical representation of potential energy capable

to accurately describing all diatomic systems so far studied. Some models describe

satisfactorily the energy of interaction for a reasonable number of systems, mainly in

their ground electronic state. However, for excited states, there are few precise analytical

models and, in general, these can be applied to a minimal number of diatomics (see our

Review [3] where we discussed in detail).

In general, the more accurate and physically well-behaved potentials have some

mathematical characteristics in common: they are sums and/or products of exponential

functions and polynomials (or functional rational) involving spectroscopic constants and

the distance R.

An appropriate non-repulsive potential of Born Oppenheimer

(

) satisﬁes three

criteria:

(i)



R=R

= 0, i. e., V (R) has a minimum at R = R

;

(ii)

V(R) come asymptotically to ﬁnite value as

R → ∞

, in general 0 or

−D

, where

is the depth of the well ;

(iii) If R → 0, then V(R)→ ∞.

How do choose such functions? Although we are talking about a function that

describes a physical problem, we will ﬁnd the answer ﬁrst in mathematics. Let us begin

our discussion with the primordial Dunham potential.

12 | https://doi.org/10.22481/intermaths.v4i1.12921 Judith de P. Araújo et al.

3 Mathematical theory

This section includes mathematical aspects considered as fundamentals for the topics

covered in this paper.

Dunham obtained relationships to calculate the most important spectroscopic pa-

rameters, and these depend on derivatives of potential. Then, the “ideal” potential

energy function must satisfy some mathematical properties related to diﬀerentiability

and continuity. Although derivatives of an order greater than 4 are hardly necessary,

the ideal is to guarantee that the potential energy functions are of class

(at least in

some points), as deﬁned below.

While it is chronologically more obvious to deﬁne continuity before diﬀerentiability,

we are going to reverse the order here. Soon it will be clear why this.

The most signiﬁcant spectroscopic parameters

and

are obtained from

derivatives of the potential at

, the equilibrium distance (for more details see Ref. [

]).

Deﬁnition

Consider

X → R

and

a ∈ X ∩ X

′

, where

′

is the set of accumulation

points of

(for more details see Ref. [

], p.52). The derivative of function

at point

is given by

′

(a) = lim

x→a

V (x) − V (a)

x − a

= lim

h→0

V (a + h) − V (a)

. (12)

Theorem

For the function

X → R

to be diﬀerentiable at point

, it is necessary

and suﬃcient that there is

c ∈ R

so that

h ∈ X ⇒ V

(

) =

(

) +

c · h

(

where lim

h→0

R(h)/h = 0. In this case, c = V

′

(a).

Corollary A function is continuous at points at which it is diﬀerentiable.

This is a relevant result of the Theory of Mathematical Analysis. It is important

to highlight that, the reciprocal is not true, i.e., not all continuous functions are

diﬀerentiable (e.g. the function f(x) = |x|).

Deﬁnition

Consider an open range

and a function

I → R

. Let

be a

non-negative integer. The function

is said to be of class

if it is

times diﬀerentiable

on I, and if all its derivatives are continuous [4].

Then, the ﬁrst mathematical requirement to start building a potential candidate: the

function must be diﬀerentiable

times at

∈ I

. We could demand that the potential

function is of class

for all points in

, ensuring that the function (and its derivatives)

are also continuous at all these points. Although this (the continuity) to be necessary

for every interatomic distance

, the condition of being diﬀerentiable for all of them is

strong.

Thus, the second fundamental characteristic is the continuity of the potential function,

deﬁned below.

Deﬁnition

A function

X → R

, deﬁned in set

X ⊂ R

, is called continuous at point

a ∈ X

, if for all

ϵ >

0 given arbitrary, it is possible to obtain

δ >

0 so that

x ∈ X

and

|x − a| < δ ⇒ |V (x) − V (a)| < ϵ [4].

Judith de P. Araújo et al. INTERMATHS, 4(1), 9–24, June 2023 | 13

Theorem

For the function

X → R

to be continuous at point

, it is necessary

and suﬃcient that, for all sequence of points

∈ X

with

lim x

, implies in

V (x

) = V (a) [4].

Corollary

V, U

X → R

are continuous at point

a ∈ X

, then the functions

U, V · U

X → R

are continuous at same point. Furthermore, if

(

)

= 0, the

function V/U : X → R is continuous at a [4].

This corollary is essential to support the possible combinations with the exponential

functions and polynomial expansions that we will suggest next for the construction of

the potential energy function.

From the results above, we can state:

Statement All polynomial

R → R

is a continuous function. All rational function

(

)

(

) (quotient of two polynomials) is continuous in its domain, which is the set of

points x such that q(x) ̸= 0.

Statement All exponential function

R → R

∗

, where

∗

denotes

R − {

}

, is conti-

nuous and diﬀerentiable for all x ∈ R.

Now, a third (and perhaps one of the most important) characteristic that the potential

function must satisfy is related to convergence. We knew that one of the characteristics

of the BO potential energy function is that

(

) should assume a ﬁnite value as

R → ∞

in general, 0. In contrast, the potential must also satisfy V (R) → ∞, as R → 0.

It is important to note that, if one does not impose correct asymptotic behavior at

inﬁnity the potential will be useless for studying atomic collisions, or even for high-energy

rotation-vibration states of the system [1].

This can be a problem when dealing with inﬁnite expansions in the power series of

some types. However, there are many results of Analysis to ensure the convergence of

such functions, so that it will guide us in choosing the terms of the expansion.

Deﬁnition A power series is a function given by [4]

V (x) =

∞

n=0

(x − x

)

= a

+ a

(x − x

) + · · · + a

(x − x

)

+ · · · . (13)

These functions are considered the most important functions of Analysis and are a

natural generalization of polynomials [

]. The set of values to which this series converges

is a range centered at x

Deﬁnition A series

is absolutely convergent if

| converges.

Theorem

A power series

∞

n=0

(

x − x

)

, or converges only to

= 0, or there is

with 0

< r < ∞

, such that the series converges absolutely in the open range (

−r, r

and diverges outside the closed range [

−r, r

]. At the extremes,

−r

and

, the series can

converge or diverge. The number r is called convergence radius.

14 | https://doi.org/10.22481/intermaths.v4i1.12921 Judith de P. Araújo et al.

Theorem

Suppose that

is the convergence radius of power series

∞

n=0

(

x − x

)

. The

function

: (

−r, r

)

→ R

, deﬁned by

(

) =

∞

n=0

(

x − x

)

, is diﬀerentiable, with

′

(

) =

∞

n=0

n a

(

x − x

)

n−1

, and the power series of

′

(

) still has a convergence radius

equal to r.

This theorem ensures that if the candidate function has a convergence radius

, it will

then be automatically diﬀerentiable of class

∞

. Therefore, the choice of a function

that has a good convergence radius is fundamental, because, consequently, this will

ensure that the other required properties are also satisﬁed.

The following theorem presents a necessary condition, but not enough to investigate

the convergence of a power series.

Theorem The general term of a convergent series has a limit equal to zero.

Note that this theorem ensures that if the general term of a power series is not zero,

then it diverges. However, if the general term is zero, we can´t ensure the convergence

of the power series.

Statement The power series

∞

n=0

, (14)

converges for all

x ∈ R

, then the function

R → R

, deﬁned by

(

) =

∞

n=0

is of

class

∞

. Deriving term by term, we have

′

(

) =

(

). Now, as

(0) = 1, it follows

that V (x) = e

for all x ∈ R, and then

= 1 + x +

+ · · · . (15)

Therefore, the choice of polynomial and exponential functions is not arbitrary, since

in most cases both are expansions in series of powers.

4 Discussion

The Dunham [5] potential given by

= a

[(R − R

)/R

]

(

1 +

∞

i=1

[(R − R

)/R

]

)

, (16)

is a power series expansion, and we can easily verify that this is diﬀerentiable and

therefore continuous for all

R ∈ R

. Although the Dunham potential has correct behavior

in the spectroscopic region (

R ≤ R

), the potential does not converge for very large

values of R.

Thus, in general, a series of powers alone is not enough to provide the appropriate

potential energy for interaction for diatomic systems.

Judith de P. Araújo et al. INTERMATHS, 4(1), 9–24, June 2023 | 15

The same occurs with potentials that involve only exponentials. The well known

Morse [6] potential

MOR

(R) = D

−2a(R−R

)

− 2D

−a(R−R

)

(17)

is an example of this.

The functional form to describe diatomic potentials is quite adequate to represent

atoms forming a chemical bond, providing greater precision in the region of the minimum

potential. Besides, this function is diﬀerentiable and continuous for all

. However, note

that when

R →

MOR

(

) assumes the ﬁnite value

(

2aR

−

), and then does

not satisfy the criterion (iii). Furthermore, the Morse potential does not have a correct

asymptotic behavior for many systems, where his function is too negative at large R.

We chose the potential of Dunham and Morse as a reference because they are the

most widely known diatomic potentials. Furthermore, they have the characteristics of

potentials that we want to unite: one is composed only by exponential and the other by

a series of powers.

The history [

] and recent comparative studies [

] have shown that, in general, accurate

analytical potential energy functions are obtained by joining both functions. We can

list the following functions as good examples of accurate analytical potentials:

() Extended Rydberg [8, 9]

(R) = D

(1 + a

(R − R

) + a

(R − R

)

+ a

(R − R

)

−γ(R−R

)

. (18)

() Varshni III [10]

V AR

III

(R) = D



1 −

exp [−β(R

− R

)]



(19)

() Levine [11]

LEV

(R) = D



1 −

exp [−a(R

− R

)]



. (20)

The Extended Rydberg is still considered one of the most accurate analytical potenti-

als [3]. The Levine function can be considered a modiﬁed version of V

V AR

III

The Hulburt-Hirschfelder [12] potential

(R) = D

[(1 − e

−x

)

+ (1 + bx)cx

−2x

] (21)

does not appear in this list, in turn, it corresponds to the type of function we are

searching for. This potential is considered a Morse modiﬁed function, being the repulsive

branch of the potential multiplying by a polynomial in (

R −R

). However, the attractive

branch is not modiﬁed and therefore does not produce signiﬁcant improvements over

the Morse potential.

Now, we can also list some functions that are sums and/or product of exponential by

polynomials, and in this case, they have adjustable parameters:

16 | https://doi.org/10.22481/intermaths.v4i1.12921 Judith de P. Araújo et al.

(i) EHFACE2U [13]

EHF ACE2U

= V

EHF

+ V

(22)

where

EHF

(R) = −DR

1 +

i=1

exp(−γr), (23)

and

= −

n=6,8,10,···

(R)R

−n

(24)

(ii) Aguado and Paniagua [14]

(2)

) = V

short

(2)

+ V

long

(2)

(25)

where

short

(2)

−α

(26)

and

long

(2)

i=1

(27)

with

= R

−β

(2)

, β

(2)

> 0. (28)

Both potentials satisfy all the criteria described in Section 3. These are two of the

most well-known and used functions for ﬁtting potential energy curves to ab initio points.

Very ﬂexible, these functions can be used for many diﬀerent diatomic systems in their

fundamental and excited electronic states (see more details in Ref [3]).

5 Proposed Methodology

In this section, we will describe a methodology for building potential, based on

Dunham’s function.

First, two functions that satisfy all the criteria described in Section 3 must be

chosen, one being a polynomial expansion and the other an exponential one;

Make the product of the chosen functions. Remember that one must satisfy the

short-range while the other satisﬁes

large. To verify this, do the test with

R →

and R → ∞ using the complete function;

3. The function can be given, for example, by:

V (R) = b

(R)F

(R)

1 +

i=1

(R)

(29)

or even by

V (R) = b

(R)

1 +

i=1

(R)F

(R)

. (30)

Judith de P. Araújo et al. INTERMATHS, 4(1), 9–24, June 2023 | 17

where

(

) is a exponential-type function involving powers of

and

, such as

those presented in Section 4,

(

) is a polynomial term (in general (

R − R

) and

its variations) and N must be truncated in some satisfactory value;

In the region of its convergence, the Dunham potential converges to the RKR [

–

] potential derived from the energy levels (see Ref. [

]). Then, for the corres-

ponding property to hold for the new expansion

(

), it must be equal to the

Dunham expansion in the region where both series converge:

(R)F

(R)

1 +

i=1

(R)

= a

[(R − R

)/R

]



1 +

∞

i=1

[(R − R

)/R

]



(31)

(R)

1 +

i=1

(R)F

(R)

= a

[(R − R

)/R

]



1 +

∞

i=1

[(R − R

)/R

]



(32)

by considering Eq. Eq. (30).

The coeﬃcients

can be straightforwardly obtained, taking the derivatives of

both sides of Eqn. 32 regarding to R and equated them at R = Re;

Then, a series of expressions relating the new potential coeﬃcients

and the

Dunham coeﬃcients a

is obtained, providing the full potential.

This method, although simple and illustrative, can be used to obtain potential energy

surfaces. One of the diﬃculties that may arise concerns the Dunham’s coeﬃcients

These are not widely available in the literature and in general, for

i >

6, they are quite

inaccurate and diﬃcult to obtain (some of them can be obtained from Refs. [

]).

Thus, to use this method, the function must not have a degree greater than 8.

6 Results

The following function was obtained from the methodology described in the previous

section, and this new potential is published in Ref. [21], and is given by:

V (R) =











n=2



1 + e

−2β

(

R−R

)





R−R





, R ≤ R

(1−e

−2α(R−R

)

(1+e

−γα(R−R

)

, R > R

(33)

where β =

α with α the Morse [6] parameter, γ is a ﬁne-tuning parameter 1 ≤ γ ≤ 3,

to be ﬁxed by direct comparison with RKR data. The

= 2

, · · · ,

8 are coeﬃcients

related with the Dunham [5] coeﬃcients, as can be seen below.

18 | https://doi.org/10.22481/intermaths.v4i1.12921 Judith de P. Araújo et al.

The potential Eq. (33) presents accuracy in the spectroscopic region and the correct

asymptotic behavior, being given by two distinct functions. The function chosen to

represent the short-range is Dunham type. Although the Dunham potential [

] is

accurate around the minimum, it does not have adequate asymptotic behavior.

Thus, the Morse-type function

(1−e

−2α(R−R

)

(1+e

−γα(R−R

)

, at

R > R

was introduced, ensu-

ring by construction, that the potential has an asymptotically correct form.

Note that as we commented in Section 4, Dunham-type functions describe the

spectroscopic region well, but in general, do not describe the asymptotic region well.

On the other hand, Morse-type functions describe the asymptotic region well, but fail

in the short range. Thus, to obtain the ”ideal” potential, it was necessary to join the

two potentials to guarantee convergence, without forgetting also to verify the continuity

in the coupling of the two functions in R

The coeﬃcients

have been obtained using items 4 and 5 described in Section 5,

from the following procedure:

Calculating the second derivative of

(

) (at

R ≤ R

) and of Dunham’s poten-

tial Eq. (16) at R = R

, we obtain:

i=2



1 + e

−2β

(

R−R

)



R − R



R=Re

8 · c

(34)

[(R − R

)/R

]

(

1 +

∞

i=1

[(R − R

)/R

]

R=Re

2 · a

(35)

Then,

8 · c

2 · a

(36)

⇒ c

Calculating the third derivative of

(

) (at

R ≤ R

) and of Dunham’s poten-

tial Eq. (16) at R = R

, we obtain:

i=2



1 + e

−2β

(

R−R

)



R − R



R=Re

48[c

− c

(1 + β)]

(37)

[(R − R

)/R

]

(

1 +

∞

i=1

[(R − R

)/R

]

R=Re

6 · a

· a

(38)

Then,

48[c

− c

(1 + β)]

6 · a

· a

(39)

⇒ c

+ 8(1 + β)c

] .

Judith de P. Araújo et al. INTERMATHS, 4(1), 9–24, June 2023 | 19

Calculating the fourth derivative of

(

) (at

R ≤ R

) and of Dunham’s poten-

tial Eq. (16) at R = R

, we obtain:

i=2



1 + e

−2β

(

R−R

)



R − R



R=Re

96[4c

− 6c

(1 + β) + c

(3β

+ 4β + 3)]

(40)

[(R − R

)/R

]

(

1 +

∞

i=1

[(R − R

)/R

]

R=Re

24 · a

· a

(41)

Then,

96[4c

− 6c

(1 + β) + c

(3β

+ 4β + 3)]

24 · a

· a

(42)

⇒ c

+ 24(1 + β)c

− 4(3β

+ 4β + 3)c

Continuing with this procedure until the eighth derivative, we obtained the relations

for the function’s coeﬃcients c

, · · · c

, in terms of the Dunham coeﬃcients a

, · · · , a

; (43)

+ 8(1 + β)c

] (44)

+ 24(1 + β)c

− 4(3β

+ 4β + 3)c

; (45)

+ 64(1 + β)c

− 4(12β

+ 18β + 12)c



+ 6β

+ 6β + 4



;

(46)

+ 160(1 + β)c

− 4(40β

+ 64β + 40)c

+4 (18β

+ 36β

+ 36β + 20) c

−4



3β

+ 9β

+ 8β + 5



;

(47)

128

+ 384(1 + β)c

− 4(120β

+ 200β + 120)c



224

+ 160β

+ 160β + 80



−4(22β

+ 54β

+ 72β

+ 60β + 30)c



+ 6β

+ 10β

+ 12β

+ 10β + 6



] ;

(48)

20 | https://doi.org/10.22481/intermaths.v4i1.12921 Judith de P. Araújo et al.

256

+ 896(1 + β)c

− 4(336β

+ 576β + 336)c



800

+ 600β

+ 600β + 280



−4



340

896

+ 400β

+ 320β + 140





114

+ 66β

+ 108β

+ 120β

+ 90β + 42



−4



+ 9β

+ 15β

+ 12β + 7



] .

(49)

The potential Eq. (33) fulﬁll the necessary continuity conditions in R = R

(a) Note that

lim

R→R

−

i=2



1 + e

−2β

(

R−R

)



R − R



= 0

lim

R→R

(1 − e

−2α(R−R

)

(1 + e

−γα(R−R

)

= 0.

(50)

(b) The same occurred with the ﬁrst order derivatives,

lim

R→R

−

i=2



1 + e

−2β

(

R−R

)



R − R



= 0

lim

R→R





(1 − e

−2α(R−R

)

(1 + e

−γα(R−R

)





= 0.

(51)

Furthermore, the new potential Eq. (33) satisﬁes the following necessary criteria [

(i)



R=R

= 0, i. e., V(R) has a minimum at R = R

;

(ii)

V(R) come asymptotically to ﬁnite value as

R → ∞

, and in this case

(

∞

) =

;

(iii) If R → 0, then V(R)→ ∞.

We have also added the condition,

(

) = 0, which simply shifts the zero of potential,

without physically aﬀecting its properties.

This potential function, obtained within the methodology here introduced, proved

to be accurate for 22 diatomic systems. Among them, for the ground electronic state

of the ion

, for CN and

in their excited electronic states (

) and (

)

respectively. Furthermore, diatomic systems formed by heavier atoms and/or with many

electrons, such as

, BiI,

were also well described by this function (For

more details see Ref. [21]).

Judith de P. Araújo et al. INTERMATHS, 4(1), 9–24, June 2023 | 21

7 Conclusions

We have described all the mathematical details to obtain good potentials. The search

for a correct functional form is not so simple, it also requires a lot of physical knowledge

to deﬁne the parameters that will compose the functional form and in which positions.

These diﬃculties were conﬁrmed when we built our potential from the methodology

described here.

The ﬁve listed functions, with adjustable or not adjustable parameters, besides

involving exponential and polynomials, have another characteristic in common: the

correct asymptotic behavior of the potential for dissociation into atoms. This is a

necessary condition to obtain a potential that is satisfactory overall accessible values of

R [1].

Signiﬁcants points were also observed to obtain a correct potential energy curve.

Firstly, we proposed only the function deﬁned at

R ≤ R

as being a function for all

. However, we found that the behavior of this function at

R → ∞

was not correct.

It providing non-zero values for such R. Then, we searched for functions that could

correct this asymptotic limit. Again, we obtained a function in terms of the exponential

function, as Morse-type potential.

Thus, to obtain correct potential energy functions, it is necessary to ensure convergence

and all other requirements described in Section 3 of this paper. In general, functions not

satisfying the aforementioned conditions will hardly represent the inter atomic potentials

accurately.

Acknowledgement(s)

J.P.A. thanks IF Sudeste MG-Campus Juiz de Fora, Brazil, for the leave of absence during

her Ph.D. studies. Financial support from Conselho Nacional de Desenvolvimento Cientíﬁco e

Tecnológico (CNPq) is acknowledged.

ORCID

Judith de Paula Araújo https://orcid.org/0000-0002-4608-492X

Maikel Y. Ballester https://orcid.org/0000-0002-5475-8808

Mariana P. Martins https://orcid.org/0009-0004-5340-0689

Rafael P. Silva https://orcid.org/0009-0000-9506-3159

Isadora G. Lugão https://orcid.org/0009-0003-4430-7055

Ituen B. Okon https://orcid.org/0000-0002-8172-7249

Clement A. Onate https://orcid.org/0000-0002-9909-4718

Disclosure statement. No potential conﬂict of interest was reported by the author(s).

Funding. Financial support from Conselho Nacional de Desenvolvimento Cientíﬁco e

Tecnológico (CNPq) is acknowledged.

22 | https://doi.org/10.22481/intermaths.v4i1.12921 Judith de P. Araújo et al.

Referências

J. N. Murrell and S. Carter and S. C. Farantos and P. Huxley and A. J. C. Varandas,

Molecular Potential Energy Functions. New York: Wiley, 1977.

D. Marx and J. Hutter. Ab Initio Molecular Dynamics. New York,: Cambridge University

Press, 2009.

J. P. Araújo and M. Y. Ballester, “A comparative review of 50 analytical representation of po-

tential energy interaction for diatomic systems: 100 years of history”, International Journal

of Quantum Chemistry, vol. 121, pp.e26808 1–102, 2021. https://doi.org/10.1002/qua.26808

4. E. L. Lima, Análise Real: Funções de uma variável, Rio de Janeiro: IMPA, 2008.

J. L. Dunham, “The Energy Levels of a Rotating Vibrator ”, Physical Review, vol. 41,

pp.721–731, 1932. https://doi.org/10.1103/PhysRev.41.721

P. M. Morse, “Diatomic Molecules According to the Wave Mechanics. II. Vibrational

Levels”, Physical Review, vol. 34, pp.57–64, 1929. https://doi.org/10.1103/PhysRev.34.57

J. P. Araújo and M. D. Alves and R. S. da Silva and M. Y. Ballester, “A compara-

tive study of analytic representations of potential energy curves for O

, N

, and SO in

their ground electronic states”, Journal of Molecular Modeling, vol. 25, pp.1–17,2019.

https://doi.org/10.1007/s00894-019-4079-3

J. N. Murrell and K. S. Sorbie, “New Analytic Form for the Potential Energy Curves of

Stable Diatomic States”, Journal of the Chemical Society, Faraday Transactions 2, vol.70,

pp.1552–1556, 1974. https://doi.org/10.1039/F29747001552

P. Huxley and J. N. Murrell, “Ground-state Diatomic Potentials”,Journal of

the Chemical Society, Faraday Transactions 2, vol. 79, pp.323–328, 1983. DOI

https://doi.org/10.1039/F29837900323

10.

Y. P. Varshni, “Comparative Study of Potential Energy Functions for Di-

atomic Systems”, Reviews of Modern Physics, vol. 29, pp.664–682, 1957.

https://doi.org/10.1103/RevModPhys.29.664

11.

I. N. Levine, “Accurate Potential Energy Function for Diatomic Molecules”, The Journal

of Chemical Physics, vol.45, pp.827–828, 1966. https://doi.org/10.1063/1.1727689

12.

H. M. Hulburt and J. O. Hirschfelder,

‘Potential Energy Functions for Diato-

mic Molecules ”textitThe Journal of Chemical Physics, vol. 9, pp. 61–69, 1941.

https://doi.org/10.1063/1.1750827

13.

A. J. C. Varandas and J. D. da Silva, “Potential model for diatomic molecu-

les including the united-atom limit and its use in a multiproperty ﬁt for ar-

gon”,Journal of the Chemical Society, Faraday Transactions, vol. 88, pp. 941–954, 1992.

https://doi.org/10.1039/FT9928800941

14.

A. Aguado and M. Paniagua, “A new functional form to obtain analytical potentials of

triatomic molecules”, The Journal of Chemical Physics, vol. 96, pp. 1265–1275, 1992.

https://doi.org/10.1063/1.462163

15.

R. Rydberg, “Graphische Darstellung einiger bandenspektroskopischer Ergebnisse”, Zeits-

chrift für Physik, vol. 73, pp. 376–385, 1932. https://doi.org/10.1007/BF01341146

16.

O. Z. Klein, “Zur Berechnung von Potentialkurven für zweiatomige Moleküle mit

Hilfe von Spektraltermen”, Zeitschrift für Physik, vol. 76, pp. 226–2355, 1932.

https://doi.org/10.1007/BF01341814

17.

A. L. G. Rees, “The calculation of potential-energy curves from band-spectroscopic

data”, Proceedings of the Physical Society of London A, vol. 59, pp. 998–1008,1947.

https://doi.org/10.1088/0959-5309/59/6/310

18.

A. C. Hurley, “Equivalence of RydbergKleinRees and Simpliﬁed Dunham

Potentials”, The Journal of Chemical Physics, vol.36, pp.1117–1118, 1962.

http://dx.doi.org/10.1063/1.1732678

19.

J. F. Ogilvie and D. Koo, “Dunham potential energy coeﬃcients of the hydrogen halides

Judith de P. Araújo et al. INTERMATHS, 4(1), 9–24, June 2023 | 23

and carbon monoxide”,Journal of Molecular Spectroscopy, vol. 61, pp. 332–336, 1976.

https://doi.org/10.1016/0022-2852(76)90323-4

20.

J. F. Ogilvie, “Dunham energy parameters of isotopic carbon monoxide, hydrogen halide,

and hydroxyl radical molecules”, Journal of Molecular Spectroscopy, vol. 69, pp.169–172,

1978. https://doi.org/10.1016/0022-2852(78)90056-5

21.

J. P. Araújo and M. Y. Ballester, “New generalized potential energy function for diatomic

systems”, Physica Scripta, vol. 96, pp.125407 1-10, 2021. https://doi.org/10.1088/1402-

4896/ac3150

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications

are solely those of the individual author(s) and contributor(s) and not of Edições UESB and/or the

editor(s). Edições UESB and/or the editor(s) disclaim responsibility for any injury to people or property

resulting from any ideas, methods, instructions or products referred to in the content.

CC BY-NC 4.0

24 | https://doi.org/10.22481/intermaths.v4i1.12921 Judith de P. Araújo et al.