Thermal Responses and the Energy Spectral of Diatomic Molecules Using Nikiforov–Uvarov Methodology

: The parametric Nikiforov–Uvarov approach and the Greene–Aldrich approximation scheme were used to achieve approximate analytical solutions to the Schrödinger equation, involving an interaction of the modiﬁed deformed Hylleraas potential mixed linearly with the improved Frost–Musulin diatomic molecular potential. For each (cid:96) -state, the energy spectra and normalized wave functions were generated from the hypergeometric function in the closed form. The thermal properties of such a system, including the vibrational partition function, vibrational mean energy, vibrational mean free energy, vibrational speciﬁc heat capacity, and vibrational entropy, were then calculated for the selected diatomic molecules using their experimental spectroscopic parameters. Furthermore, the peculiar conditions of this potential were evaluated, and their energy eigenvalues were calculated for the purpose of comparison. The acquired results were found to be in reasonable agreement with those reported in the literature.


Introduction
In recent years, several studies have focused on the examination of quantum mechanical concerns with diverse physical potentials because they include all of the necessary features required to appropriately define physical quantum models [1][2][3]. Describing the Schrödinger wave equation for particles interacted via the physical potential terms of interest in quantum mechanics has prompted substantially more thought in several researchers, as well as its significant influence on quantum mechanics in diverse areas of physics and chemistry [4,5]. It leads to a precise explanation of the particles' behavior in quantum mechanics and may serve as a prologue to the examination of other features such as thermodynamics [6,7], mass spectra of quarkonia systems [8][9][10], and the structure of diatomic molecules [11][12][13][14].
It is also recognized that the explicit viable actual possibilities through the Schrödinger equations are uncommon, with the exception of several notable solvable quantum systems, such as the hydrogen atom [15] and harmonic oscillator [16,17], whereas the case of states with arbitrary angular momentum, which do not reveal exact solutions, is known analytically or by approximation methods [18,19]. These solutions have vital uses in many domains of physics and chemistry for assessing and enhancing the models established to analyze quantum mechanical systems, as well as for developing numerical approaches. As a consequence, several quantum modes should indeed be treated through approximation schemes if the solutions of the wave equation are to be obtained under such a system; the choice of these applicable approximation layouts is based on their adaptability to the required potential [20,21]. Because of their diverse uses, solutions to the relativistic and non-relativistic wave equations have been used in various quantum potential interactions employing various methodologies [22,23]. These approaches include the 1/N shifted expansion procedure [24], the Nikiforov-Uvarov approach [25][26][27], the asymptotic iteration method [28], the factorization method [29,30], the formula technique [31], the supersymmetric approach [32,33], the ansatz methodology [34], the Laplace transform approach [35,36], the functional analysis approach [37,38], the appropriate quantization rule [39], and others [40,41]. For many solvable quantum frameworks, the hypergeometric Nikiforov-Uvarov technique has demonstrated its ability to determine the exact energy levels of bound states [42]. The parametric adaptation of this method is used in this study to obtain the bound state solutions of the radial Schrödinger equation. Its satisfactory execution is backed by comparisons with numerous methodologies as well as a straightforward, precise, and user-friendly approach.
Various quantum mechanical wave equations have been widely explored using exponential-type potentials [43,44]. It is argued that these potentials are far more efficient than their counterparts, which exist in Coulomb or inverse forms [45] and play important roles in solid-state physics, nuclear physics, and other intriguing fields [46,47]. An appropriate form of a composite exponential-type potential is assumed in the study of potentials with hypergeometric wavefunctions, allowing for the generation of varied exponential potentials and a broader variety of applications. Such potentials may be extensively used as interaction models of diatomic molecules, which motivated the current investigation. The desired combined potential encompasses the modified deformed Hylleraas potential with an improved Frost-Musulin diatomic molecular potential function, which is given by where D e , σ and r e are the dissociation energy, the screening parameter, and the equilibrium bond length, respectively, where σ < 1 and V 0 , V 1 and V 2 are the depths of the potential well, and a along with b are two adjustable parameters representing the properties of the potential. Because it includes the significant potential models as special instances, this combination might reflect the effective potential of a many-body system, Figure 1. This paradigm may be applied to atomic, solid-state, plasma, and molecular physics. As a result, in some circumstances, this potential provides superior therapy to other exponential potentials and has the ability to affect the efficiency of other potentials. As a result, the investigation of quantum systems associated with these types of potentials appears to be significant in suggesting numerous frameworks for quantum chemistry and molecular physics. Despite the fact that there has been little research into this linear combination of diatomic molecule potential energy functions, the current inquiry was driven by the concept that this form of potential might be used as a mathematical example to strikingly illustrate intramolecular and intermolecular interactions and atomic pair correlations, as well as to check the accuracy of the models and approximations in computational physics and chemistry as well as in quantum mechanics. As a result, the success of quantum mechanics in depicting the physics of atomic and subatomic particles cannot be emphasized [22,46,48,49]. Various differential problems may be solved using the Nikiforov-Uvarov technique in mathematical physics, which is founded on masking a second-order linear differential equation to an extended equation of a hypergeometric type [25]. Several breakthroughs in utilizing this methodology to successfully generate genuine solutions for different wave equations in a quantum system have been described [50,51]. This technique's parametric form was designed to have a practical deployment [27] and can effectively solve certain systems of equations of type  Various differential problems may be solved using the Nikiforov-Uvarov technique in mathematical physics, which is founded on masking a second-order linear differential equation to an extended equation of a hypergeometric type [25]. Several breakthroughs in utilizing this methodology to successfully generate genuine solutions for different wave equations in a quantum system have been described [50,51]. This technique's parametric form was designed to have a practical deployment [27] and can effectively solve certain systems of equations of type The eigenfunctions are represented as follows in this approach, where ( , ) ( ) and ,ℓ are the orthogonal Jacobi polynomial [52] and the normalization constant, respectively. The following condition yields the parametric energy eigenvalues as In this sense, the requirements for this paper are as follows: Section 2 applies the fundamental notion of the parametric Nikiforov-Uvarov method via the Greene-Aldrich approximation technique to derive ℓ -wave approximate analytical solutions of the radial The eigenfunctions are represented as follows in this approach, where P (α,β) n (x) and N n, are the orthogonal Jacobi polynomial [52] and the normalization constant, respectively. The following condition yields the parametric energy eigenvalues as where In this sense, the requirements for this paper are as follows: Section 2 applies the fundamental notion of the parametric Nikiforov-Uvarov method via the Greene-Aldrich approximation technique to derive -wave approximate analytical solutions of the radial Schrödinger equation and the normalized radial wavefunctions in terms of the generalized hypergeometric functions for the appropriate potential. There is a great deal of interest in using various potential energy functions and experimentally established spectroscopic data to solve a variety of problems, including thermodynamic quantity calculations [53,54]. To the state of the art, the energy spectra and corresponding thermodynamic properties for the linear combination of the modified deformed Hylleraas potential with the improved Frost-Musulin diatomic molecular potential function have not been obtained, motivating the author to investigate the nonrelativistic vibrational energy and corresponding thermal responses of such a system within the Schrödinger equation in Section 3. Section 4 is devoted to a brief discussion, followed by a summary of the conclusions.

Any -State Solutions through the Greene-Aldrich Approximation and Parametric Nikiforov-Uvarov Approach
For an empirical potential, the radial Schrödinger equation is stated as [55] where , µ and E n, are the reduced Planck's constant, the reduced mass of the particle, and the rotational-vibrational energy of the diatomic molecules, respectively. n and represent the radial and orbital angular momentum quantum numbers. This equation is a nonrelativistic wave equation for diatomic molecules with an effective potential specified as V(r) + ( + 1) 2 /2µr 2 where the boundary condition of U n, (r) vanishes at the points r = 0 and r → ∞ . Substituting Equation (1) into Equation (6) and rearranging, the following second-order Schrödinger equation may be acquired as which cannot be solved analytically even in the s-wave case. The following Greene-Aldrich approximation approach might be utilized [19,56], for σ 1 as to solve Equation (7) for any given -state while additionally considering the columbic characteristic (1/r) of the suggested potential. Figure 2 compares the trend of 1/r 2 with the proposed approximation term. This approximation appeared to be most appropriate for the low values of σ that were investigated in this research. As a result, this approximation was employed throughout this inquiry. By substituting Equation (8) into Equation (7) and applying the coordinate transformation of the form = , the modified hypergeometric-type equation was given as By substituting Equation (8) into Equation (7) and applying the coordinate transformation of the form P = e −σr , the modified hypergeometric-type equation was given as where The corresponding parametric coefficients might well be derived by analogizing Equation (9) with Equation (2), as By substituting Equation (13) into Equation (4), the relevant nonrelativistic energy spectra for the modified deformed Hylleraas potential coupled directly with the improved Frost-Musulin diatomic molecular potential model could be obtained precisely as Furthermore, by replacing Equation (13) with Equation (3), the appropriate wavefunction for the suggested potential was achieved, where and P (χ,ς) n refers to the Jacobi polynomials, which are widely applied in mathematical analysis and practical applications. The orthogonality feature of these polynomials [52] may be used to represent the normalizing coefficient, Mathematics 2023, 11, 3338 6 of 18 In this manner, the normalization coefficient N n can be expressed as

The Thermal Behaviour of the Schrödinger Equation with MDH-IFM Potential
Thermal properties are considered as characteristic features of a system capable of determining its state. To investigate the thermodynamic features of the proposed potential model in which only pure vibrational states were addressed, we recast Equation (14) in the form where The primary objective for investigating a system's thermodynamic characteristics is to evaluate its vibrational partition function, which is commonly regarded as the cornerstone of statistical thermodynamics.

Partition Function
The calculation of thermodynamic functions [56], whose applications are frequently employed in statistical mechanics and molecular physics [55,57], is aided by the vibrational partition function for any given potential system. This intriguing function expressly relies on temperature and is commonly referred to as the Boltzmann distribution function in statistical mechanics, which was initially introduced by Boltzmann in 1870 [58]. For certain potential models, the partition function may be determined simply by adding all the possible rotational-vibrational energy levels to the system. The partition function Q(β) of the modified deformed Hylleraas potential with the improved Frost-Musulin diatomic molecular potential at a finite temperature T may be calculated by using the energy spectra of Equation (19) as where β = 1/κT with κ is the Boltzmann constant. Hence, The error function (a normalized variant of the Gaussian function), commonly abbreviated as erf (w), is a comprehensive function defined as: The various thermodynamic variables of the current system, such as vibrational mean energy, vibrational mean free energy, vibrational specific heat capacity, vibrational entropy, and others, may be easily determined from the canonical partition function as shown below.

The Vibrational Mean Energy
Evaluating the thermodynamic value of the total energy can also be used to illustrate the partition function's utility. The expected value, or ensemble average, for the energy is just the sum of the microstate energies weighted by their probabilities. The vibrational mean free energy of the modified deformed Hylleraas potential with improved Frost-Musulin diatomic molecular potential is calculated as follows:

The Vibrational Mean Free Energy
The Helmholtz free energy is a thermodynamic potential in statistical mechanics and thermodynamics that evaluates the useful work achievable from a closed thermodynamic system with constant temperature, volume, and particle number. The Helmholtz free energy is reduced for such an equilibrium system and may be used to calculate all other thermodynamic parameters of the material. As a result, the vibrational mean free energy, through its compact form, is directly represented by

The Vibrational Entropy
Entropy is an essential quantity in physics, chemistry, and biology because of its numerous applications, such as dissolution [59], fluorescence microscopy [60], adsorption [61], material creation [62], and protein activity [63]. Determining the analytical expression of entropy for diatomic molecules is still a tough proposition in science. Entropy, which may be defined and quantified in many domains other than thermodynamics, serves to assess the degree of order and disorder, or chaos. As a result, the vibrational entropy is more specifically expressed as

The Vibrational Specific Heat
The specific heat capacity of materials and components is critical in their ultimate functionality, which includes thermal storage in building elements or transitory heat flow. The associated vibrational specific heat can be described further as

Discussion
In order to understand the nature of the chemical bond and the behavior of molecules for equal internuclear distance and equilibrium bond length, Figure 1 displays the variation of the modified deformed Hylleraas potential mixed directly with the improved Frost-Musulin diatomic molecular potential in terms of the internuclear separation for the different diatomic molecules. It should be noted that the value of the proposed potential rapidly reduced to a minimum at approximately certain r after which it steadily grew with increasing r. Figure 2 depicts the treatment of 1/r 2 and the preferred approximation with various screening parameters, σ. For various values of the screening parameters, the pattern tended to accumulate asymptotically, providing a decent approximation of the system. It can be observed from Figure 3, that the energy was purely attractive in the sense that for the lower values of r e , the energy was less appealing and less bounded, and grew vigorously as the equilibrium bond length increased.  It is also noted that at the lower , the energies for the selected diatomic molecules were almost the same, but as the equilibrium bond length increased, the energy spectra diverged significantly. Figure 4 depicts the variation of the energy eigenvalue with respect to the dissociation energy , where the fluctuations of the energy spectra with , illustrated a parabolic-like trend. The nonrelativistic energies increased to a peak value and later decreased as the dissociation energies increased. H and HF show the monotonic drop in the energy as the dissociation energy increased, which was related to the bond length that increased as the atom's size grew, and the bond dissociation energy decreased, leading to a steady decline in the bond strength. It is also noted that at the lower r e , the energies for the selected diatomic molecules were almost the same, but as the equilibrium bond length increased, the energy spectra diverged significantly. Figure 4 depicts the variation of the energy eigenvalue with respect to the dissociation energy D e , where the fluctuations of the energy spectra with D e , illustrated a parabolic-like trend. The nonrelativistic energies increased to a peak value and later decreased as the dissociation energies increased. H 2 and HF show the monotonic drop in the energy as the dissociation energy increased, which was related to the bond length that increased as the atom's size grew, and the bond dissociation energy decreased, leading to a steady decline in the bond strength. Figure 5 indicates the approximate energy E nl , as far as the screening parameter, σ, within the proposed potential. It can be seen that as the screening parameter expanded, the energy spectra increased negatively for the selected diatomic molecules.
These diatomic substances were being evaluated for this study due to their importance in atomic, molecular, optical, and chemical physics.
In Figure 6, the behavior of the partition function was plotted as a function of the inverse temperature β. It can be concluded that there was a monotonic decrease in the partition function with increasing β for the chosen diatomic molecules.
It is also noted that at the lower , the energies for the selected diatomic molecules were almost the same, but as the equilibrium bond length increased, the energy spectra diverged significantly. Figure 4 depicts the variation of the energy eigenvalue with respect to the dissociation energy , where the fluctuations of the energy spectra with , illustrated a parabolic-like trend. The nonrelativistic energies increased to a peak value and later decreased as the dissociation energies increased. H and HF show the monotonic drop in the energy as the dissociation energy increased, which was related to the bond length that increased as the atom's size grew, and the bond dissociation energy decreased, leading to a steady decline in the bond strength.   Figure 5 indicates the approximate energy , as far as the screening parameter, , within the proposed potential. It can be seen that as the screening parameter expanded, the energy spectra increased negatively for the selected diatomic molecules. These diatomic substances were being evaluated for this study due to their importance in atomic, molecular, optical, and chemical physics.
In Figure 6, the behavior of the partition function was plotted as a function of the inverse temperature . It can be concluded that there was a monotonic decrease in the partition function with increasing for the chosen diatomic molecules.  These diatomic substances were being evaluated for this study due to their importance in atomic, molecular, optical, and chemical physics.
In Figure 6, the behavior of the partition function was plotted as a function of the inverse temperature . It can be concluded that there was a monotonic decrease in the partition function with increasing for the chosen diatomic molecules.  It can be deduced from Figure 7 that the internal energy of the diatomic molecules exhibited an exponential characteristic that decreased monotonically with increasing β. It can be deduced from Figure 7 that the internal energy of the diatomic molecules exhibited an exponential characteristic that decreased monotonically with increasing . The variations of the Helmholtz free energy , with regard to the varied inverse temperature , are plotted in Figure 8 for the selected diatomic molecules. The variations of the Helmholtz free energy F, with regard to the varied inverse temperature β, are plotted in Figure 8 for the selected diatomic molecules.
It can be seen that there is an early increase in the Helmholtz free energy as the parameter β grows and tends to converge in high β. The plots of the entropy S in terms of the inverse temperature β are displayed in Figure 9, where the curves of the entropy decayed asymptotically in the region of low temperatures. The variations of the Helmholtz free energy , with regard to the varied inverse temperature , are plotted in Figure 8 for the selected diatomic molecules. It can be seen that there is an early increase in the Helmholtz free energy as the parameter grows and tends to converge in high . The plots of the entropy in terms of the inverse temperature are displayed in Figure 9, where the curves of the entropy decayed asymptotically in the region of low temperatures.       In the low inverse temperature , one can characterize an increment in the capacity , where beyond this region, there was a descending tendency with increasing for the selected diatomic molecules. The spectroscopic constants of the chosen diatomic molecules discussed in this research can be obtained via Table 1. In the low inverse temperature β, one can characterize an increment in the capacity C, where beyond this region, there was a descending tendency with increasing β for the selected diatomic molecules. The spectroscopic constants of the chosen diatomic molecules discussed in this research can be obtained via Table 1. The unique cases of Equation (1) generated by choosing the proper controlling parameters might easily propose distinct interaction potential patterns. Based on this, it is possible to see that when D e = 0, and V 1 = V 2 = 0, Equation (1) reduces to the deformed Hylleraas potential, which is a subset of the multi-parameter exponential-type potential and was first explored for diatomic molecules [64], This form of potential, on the other hand, is a subset of the Morse potential, which has sparked a lot of attention over the years and is one of the most useful potential models for describing the interaction of two atoms in a diatomic molecule [28]. Because of its capacity to simulate the interaction of two atoms in these systems, this is considered as a persuasive model with applications in several branches of condensed matter and nuclear physics [65]. As a result, the approximate analytical solutions to the -state Schrödinger equation are derived as (DHP) n, However, when D e = 0 and assigning V 0 = V 1 = 0, the Pöschl-Teller potential [66] is obtained as (1 − e −σr ) 2 , whose energy spectra, in accordance with Equation (14), are given by Furthermore, the results correspond well with those presented in [67] in the nonrelativistic limit for the -wave case. Setting V 2 = 0 and a = 0 at D e = 0 in Equation (1), yields the Hulthén potential, which is particularly essential in the atomic and molecular fields. It is a short-range potential that has been used to investigate the bound state energies of diatomic molecules [68], the optical characteristics of quantum dots [69], and other topics. Thus, given this condition, the energy eigen-spectrum may be calculated as E (HP) n, In particular, the results have good agreement with those obtained in [70] as a special case. As the internuclear distance r approaches 0, it reduces to a screened coulomb potential [71,72], whereas for large values of r, it becomes a diminishing exponential potential [73]. It is demonstrated that the findings produced by this approach are compatible with the ones obtained in [71][72][73]. Using D e = 0, as well as V 0 = 0, or equivalently b → ∞ , in Equation (1), one may produce the Eckart potential, which was established in 1930 [74] and is one of the most significant exponential-type potentials in chemical physics [75].
Applying D e = 0 in addition to V 1 = V 2 = 0, and substituting a = −1 and b = 1, results in the Rosen-Morse potential [76], expressed as The corresponding energy eigenvalues of this potential are calculated as which agrees well with that obtained in [77]. The Frost-Musulin diatomic molecular potential has been used in a variety of fields of physics [78,79]. In the case when V 0 = V 1 = V 2 = 0 in Equation (1) This potential model has also been shown to suit experimental data for the study of diatomic molecules [80].

Conclusions
Because of the intriguing characteristic of the modified deformed Hylleraas potential with the improved Frost-Musulin diatomic molecular potential (which is a more generic instance of the exponential-type ones commonly seen in many physical fields), the approximate -state solutions for the Schrödinger equation with this potential function were studied in this research utilizing the Nikiforov-Uvarov technique and the Greene-Aldrich approximation scheme. For several selected diatomic molecules, the treatment of the corresponding energy eigenvalues as a function of various potential parameters was explored. It was readily apparent that the energy spectra for the chosen diatomic molecules grew adversely when the screening parameter was raised. The partition function, as well as other thermodynamic functions such as entropy, mean free energy, and specific heat capacity, were determined in this manner. The graphs of the thermodynamic characteristics as well as β indicated that for certain diatomic molecules, there was a monotonous decline in the partition function owing to the rising B. The internal energy and the entropy dropped gradually as β boosts but Helmholtz free energy increased and tended to converge as β increased. The capacity advanced at the low inverse temperature B, with an ongoing decline as β increased. Specific samples of the potential and their analogous energy relations were determined, which were consistent with those reported in the literature. Furthermore, it is important to highlight that, to the best of our knowledge, the findings of this study have not been investigated in the prior literature, which could pave the way for additional studies in other fields of theoretical chemistry, such as mathematical chemistry, molecular mechanics, molecular dynamics, and chemical thermodynamics, to explain the geometric and electronic structures of molecular assemblies, polymers, and chemical processes.
Funding: This research received no external funding.

Data Availability Statement:
The calculated and acquired data employed to support the findings of this study are included in the article and are mentioned at appropriate positions within the manuscript as references.

Acknowledgments:
The author gratefully thanks the kind referees for their constructive suggestions and invaluable comments, which definitely helped to improve the readability and quality of the manuscript.

Conflicts of Interest:
The author declares that there are no conflicts of interest regarding the publication of this paper.