The knowledge of internuclear interaction potential energy between two atoms as a function of their relative positions is of fundamental importance in a wide variety of fields. There has been a growing interest in obtaining an available closed-form representation governing the interaction of two atoms for diatomic molecules in chemistry and physics. Numerous contributions have been made to build analytical internuclear potential energy functions.^[1–15] In recent years, by utilizing the dissociation energy and equilibrium bond length for a diatomic molecule as explicit parameters, some authors^[16–21] constructed improved versions with more convenient applications for some empirical potential energy functions, including the well-known Rosen-Morse, Tietz, Frost-Musulin, Manning-Rosen, and Pöschl-Teller potentials. The improved molecular potentials have received considerable attentions due to their real applications in many fields, including simulations of molecular potential curves, predictions of thermochemical quantities of gaseous substances, and calculations of molecular vibrational energies.^[21–50] There has been an increasing recognition of the importance of the Tietz potential as a typical potential energy model. Hassanabadi et al.^[51–53] studied the solutions of the Dirac equation and Klein-Gordon equation with the Tietz potential in terms of the supersymmetry quantum mechanics approach. Ikot et al.^[54] investigated the bound state solutions of the Duffin-Kemmer-Petiau equation with the Tietz potential by employing the Nikiforov-Uvarov method. Onate et al.^[55] computed the Shannon entropy and information energy under the modified Tietz-Hua potential. Khordad and Mirhosseini^[56] studied the optical properties of spherical quantum dots by using the Tietz potential.

In 2001, the authors in the literature^[12] proposed a five-parameter exponential-type potential model, and obtained a general energy spectrum formula by solving the Schrödinger equation with that in terms of the supersymmetry shape invariance approach. By choosing suitable forms of the parameters, the five-parameter exponential-type potential can turn to some well-known potential functions, including the Hulthén potential, Eckart potential, and improved Manning-Rosen potential. It is unfortunate that the model parameters in the five-parameter exponential-type potential lack explicit physical definitions. This fault has greatly affected its effective applications. In this work, we attempt to establish an improved five-parameter exponential-type potential energy model for diatomic molecules, and explore the relationship between it and the well-known Tietz potential in the presence of a diatomic molecule. We also apply the improved five-parameter exponential-type potential to model the internuclear interaction potential curve for the ground electronic state of the carbon monoxide (CO) molecule. A great deal of CO is released from tail gases, including coke oven gas, carbon black manufacturing tail gas, etc. CO is one of the most prominent gaseous environment pollutants, and a highly toxic gas for humans as well as for animals. Due to its high harmful effect on human health, the adsorption processes of CO in various adsorbent materials have aroused considerable attention.^[57–61]

The five-parameter exponential-type potential energy model can be represented as Eq. (1),^[12] 1
where P₁, P₂, P₃, q, and α are five adjustable parameters, and .

For a diatomic molecular potential energy function U(r), it is directly related to the three molecular constants and satisfies the following three relationships:


where D_e denotes the dissociation energy for a diatomic molecule, r_e is the equilibrium bond length, ω_e represents the equilibrium harmonic vibrational frequency, μ is the reduced mass of the molecule, and c is the speed of light. Taking the first order derivate of U(r) with respect to the r variable and substituting it into Eq. (2), we obtain
From Eq. (5), we have the following expression,
With the help of expression (6), we obtain the following relationship from Eq. (3),
Solving Eqs. (6) and (7) for parameters P₂ and P₃ yields the following two expressions,

Substituting expressions (8) and (9) into Eq. (1) and making simple algebraic manipulations, we can rewrite potential function (1) in the following form
In order to achieve a choice that the minimum value of the potential energy is zero, i.e. , an additional term of is added to the right hand of expression (10). This manipulation does not produce any change in the physical properties of the original potential energy function. The five-parameter exponential-type potential function can be further represented as the following form
here we have replaced α by α/2 for the sake of simplification. The improved five-parameter exponential-type potential given in the above expression is just the improved Tietz potential.^[16] Therefore, the five-parameter exponential-type potential is identical to the Tietz potential for diatomic molecules. Employing the improved Tietz potential to represent the internal vibration of a molecule, the molar entropy values have been satisfactorily predicted for gaseous substances, including the gases CO, HCl, HF, DF, NO, and gaseous BBr.^[30] Choosing q = 0, the improved five-parameter exponential-type potential becomes the standard Morse potential.^[1] When q = 1 and q = −1, the improved five-parameter exponential-type potential turns to the standard improved Rosen-Morse potential^[16] and improved Manning-Rosen potential.^[18]

We calculate the second order derivation of U(r) given in expression (11) with respect to r, and substitute it into Eq. (4). In the presence of given parameter q, we solve Eq. (5) for parameter α and achieve the following expression,
in which W represents the Lambert W function, which satisfies .^[62]

By choosing an available value of parameter q and putting the experimental values of the molecular constants D_e, r_e, and ω_e as inputs, we reproduce the potential energy curve for the ground electronic state of CO in terms of the improved five-parameter exponential-type potential. The experimental values of three molecular constants are obtained by referring to the literature:^[13] , , and . Substituting these values into expression (12) and taking q = −0.55, we obtain the value of parameter α as α=2.191 89 Å⁻¹. The variation of internuclear interaction potential with internuclear distance is shown in Fig. 1, in which the green solid line stands for the predicted values and red solid circles represent the experimental Rydberg-Klein-Rees (RKR )^[63–65] data points from the literature.^[66] Figure 1 shows that the predicted potential energy values are in excellent agreement with the experimental RKR potential data points over the entire interaction distance range except at much large r.

Fig. 1.

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	Fig. 1. Experimental RKR data points and the improved five-parameter exponential-type potential energy curve for the ground electronic state of CO.

In order to evaluate quantificationally the accuracy of the proposed potential, we consider the average absolute deviation of the improved five-parameter exponential-type potential versus the experimental RKR potential. The average absolute deviation is represented as , here N_p denotes the number of experimental data points considered in the RKR potential, U_RKR(r and U_calc(r) represent the experimental RKR potential and the empirical analytical potential, respectively. The average deviation of the improved five-parameter exponential-type potential versus the RKR potential reported by Kirschner, and Watson^[66] is 0.333% of D_e. This average absolute deviation satisfies satisfactorily the Lippincottʼs error criterion for an available empirical closed-form potential energy representation. The Lippincottʼs error criterion implies that an average absolute deviation should be less than 1% of the dissociation energy for a successful analytical energy function.^[10]

Through the introduction of the dissociation energy and equilibrium bond length as explicit parameters, we establish the improved five-parameter exponential-type potential model for diatomic molecules. We show exactly and simply that the five-parameter exponential-type potential is identical to the Tietz potential in the presence of a diatomic molecule. Using the experimental values of the dissociation energy, equilibrium bond length and equilibrium harmonic vibrational frequency, we model satisfactorily the potential energy curve for the ground electronic state of CO. By the calculation of average absolute deviation of the proposed potential versus the experimental RKR potential, we show quantitatively that the improved five-parameter exponential-type potential is an available model to represent the internuclear interaction potential for the ground electronic state of the CO molecule.

We would like to thank the kind referee for positive and invaluable suggestions which have greatly improved the manuscript.