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Context: This work presents analytical expressions for ro-vibrational energy models of diatomic molecules by introducing fractional parameters to improve molecular interaction analysis. Thermodynamic models, including Helmholtz free energy, mean thermal energy, entropy, and isochoric heat capacity, are formulated for diatomic molecules such as CO (X 1 ∑ + ), Cs 2 (3 3 ∑ g + ), K 2 (X 1 ∑ g + ), 7 Li 2 (6 1 Π u ), 7 Li 2 (1 3 Δ g ), Na 2 (5 1 Δ g ), Na 2 (C(2) 1 Π u ), and NaK (c 3 ∑ + ). The incorporation of fractional parameters improves predictive accuracy for vibrational energies, as shown by reductions in percentage average absolute deviations from 0.5511 to 0.2185% for CO. Findings indicate a linear decrease in Helmholtz free energy and an initial increase in heat capacity with rising temperature, providing valuable insights for characterizing materials and optimizing molecular processes in chemistry, material science, and chemical engineering. The results obtained show strong agreement with established theoretical predictions and experimental data, validating the robustness and applicability of the proposed models., Methods: The energy equations are derived by solving the radial Schrödinger equation for a variant of the Tietz potential using the generalized fractional Nikiforov-Uvarov (GFNU) method in addition to a Pekeris-type approximation for the centrifugal term. The canonical partition function is derived using the modified Poisson series formula, which serves as a basis for calculating other thermodynamic functions. All computations are carried out using MATLAB programming software., Competing Interests: Declarations. Ethics approval: Not applicable. Consent to participate: Not applicable. Consent for publication: Not applicable. Competing interests: The authors declare no competing interests. Final version of the manuscript: All authors approved the final version of the manuscript submitted., (© 2024. The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.)

The bound-state solution of the radial Klein-Gordon equation has been obtained under the interaction of an exponential-type and Yukawa potential functions. The Greene-Aldrich approximation has been used to overcome the centrifugal barrier and enable the analytical solutions of the energy and wave functions in closed form. The momentum space wave function in D dimensions has been constructed using the Fourier transform. The mean values have been conjectured for the position and momentum spaces using two equivalent equations. The effects of the potential parameters on the expectation values and quantum information measurement have been investigated. For the 1D case, the results obey the Heisenberg uncertainty principle, Fisher, Shannon, Onicescu, and the Rényi entropic inequalities. Other information complexities measures, such as Shannon Power, Fisher-Shannon, and Lopez-Ruiz-Mancini-Calbet, have been verified. For the ground state, the 1D momentum expectation value [Formula: see text] coincides with the 3D [Formula: see text] values, which is an indication of degeneracy. The total energy of a particle in both 1D and 3D space may be degenerate due to the inter-dimensional degeneracy of the quantum numbers. However, in this present result, the degeneracy in 1D and 3D occurred for fixed quantum states at different momentum intervals. Thus, in 1D, a particle may transit an entire space ([Formula: see text] with a certain kinetic energy, which must be equal to its kinetic energy if it moves through the interval [Formula: see text] in 3D space. This may have implications for kinetic energy degeneracy in higher dimensions., Competing Interests: Declarations Competing interests The authors declare no competing interests., (© 2024. The Author(s).)

The inherent association between real quantum systems and their surrounding environment invariably results in decoherence, leading to the loss of entanglement. This diminution in entanglement coincides with a decline in the fidelity of transmitted information using the entangled quantum resource. This study scrutinizes the impact of the squeezed generalized amplitude damping (SGAD) channel on quantum Fisher information (QFI) parameters. The SGAD channel model, a versatile framework, is also employed to simulate other dissipative channels, including amplitude damping (AD) and generalized amplitude damping (GAD). Kraus operators facilitate the modeling of noisy channels. The results reveal that, within the SGAD channel, the QFI remains impervious to the squeezing variables (r and Φ ). In the GAD channel, F θ GAD undergoes enhancement to a constant value with an upswing in temperature (T), while the ϕ parameter in the GAD channel, F ϕ GAD , akin to the SGAD channel, surges around T = 2 before complete loss ensues. Concerning the AD channel, the θ component of the QFI initially experiences decoherence with an augmentation in the AD noise parameter ( λ ). Subsequently, it is restored to its initial value with a further escalation in λ . Conversely, the ϕ component of the QFI in the AD channel experiences decoherence with an elevation in the AD noise parameter ( λ )., (© 2024. The Author(s).)

The approximate solutions of the Dirac equation for spin symmetry and pseudospin symmetry are studied with a coshine Yukawa potential model via the traditional supersymmetric approach (SUSY). To remove the degeneracies in both the spin and pseudospin symmetries, a coshine Yukawa tensor potential is proposed and applied to both the spin symmetry and the pseudospin symmetry. The proposed coshine tensor potential removes the energy degenerate doublets in both the spin symmetry and pseudospin symmetry for a very small value of the tensor strength (H = 0.05). This shows that the coshine Yukawa tensor is more effective than the real Yukawa tensor. The non-relativistic limit of the spin symmetry is obtained by using certain transformations. The results obtained showed that the coshine Yukawa potential and the real Yukawa potential has the same variation with the angular momentum number but the variation of the screening parameter with the energy for the two potential models differs. However, the energy eigenvalues of the coshine Yukawa potential model, are more bounded compared to the energies of the real Yukawa potential model., (© 2024. The Author(s).)

Context: In this paper, the approximate analytical energy equations for the deformed hyperbolic potential have been obtained for arbitrary parameters of the potential. The potential function was transformed to a molecular potential by subjecting it to the Varshni conditions which allows for the determination of the energy levels of diatomic molecules. The molecular vibrational energy spectra for [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] diatomic molecules were obtained and found to match with the results obtained with another analytical approach, potential functions, and experimental data. The noticeable slight differences in the approximate energy spectra obtained in this work and existing literature may be ascribed to the analytical method, computational approach, and the accuracy of the molecular potential functions. The obtained energy equations were used to determine the energy of a particle for arbitrary parameters of the potential function. The obtained energy is bounded and increases with the increase in the quantum numbers. The results conformed to the ones obtained via the path integral approach and numerical solutions obtained via the MATHEMATICA program., Method: The energy spectra equations were obtained via the Nikiforov-Uvarov approach and semi-classical WKB approximation. The Pekeris approximation has been applied to resolve the difficulty in solving the complete energy spectrum of the non-relativistic wave equation for the potential function. The numerical data of the energy spectra was obtained using the MATHEMATICA program., (© 2024. The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.)

In this study, a diatomic molecule interacting potential such as the deformed Schiöberg oscillator (DSO) have been applied to diatomic systems. By solving the Schrödinger equation with the DSO, analytical equations for energy eigenvalues, molar entropy, molar enthalpy, molar Gibbs free energy and constant pressure molar heat capacity are obtained. The obtained equations were used to analyze the physical properties of diatomic molecules. With the aid of the DSO, the percentage average absolute deviation (PAAD) of computed data from the experimental data of the 7 Li 2 (2 3 Π g ), NaBr (X 1 Σ + ), KBr (X 1 Σ + ) and KRb (B 1 Π) molecules are 1.3319%, 0.2108%, 0.2359% and 0.8841%, respectively. The PAAD values obtained by employing the equations of molar entropy, scaled molar enthalpy, scaled molar Gibbs free energy and isobaric molar heat capacity are 1.2919%, 1.5639%, 1.5957% and 2.4041%, respectively, from the experimental data of the KBr (X 1 Σ + ) molecule. The results for the potential energies, bound-state energy spectra, and thermodynamic functions are in good agreement with the literature on diatomic molecules., (© 2023. The Author(s).)

In this paper, the bound state solutions of the radial Schrödinger equation are obtained in closed form under an improved Scarf II potential energy function (ISPEF) constrained by external magnetic and Aharonov-Bohm (AB) flux fields. By constructing a suitable Pekeris-like approximation scheme for the centrifugal barrier, approximate analytical expressions for the bound-states and thermal partition function were obtained. With the aid of the partition function, an explicit equation for magnetization at finite temperatures is developed. The obtained equations were then applied to calculate the energy levels and magnetic properties of 7 Li 2 (2 3 Π g ), K 2 (X 1 Σ g + ), Mg 2 (X 1 Σ g + ) and NaBr (X 1 Σ + ) diatomic molecules. The obtained numerical results of the vibrational energies for these molecules were found to be in good agreement with theoretic and experimental values reported in the existing literature. The results indicated that by turning off the magnetic and AB fields, the energy levels of the diatomic molecules degenerate. The results further revealed that an increase in the temperature of the molecules and the AB field strengths leads to a linear decrease in magnetization., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (©2023PublishedbyElsevierLtd.)

Owing to the devise applications of molecules in industries, the bound state solution of the non-relativistic wave equation with a molecular potential function has been obtained in a closed-form using the Nikiforov-Uvarov method. The solutions of the bound state are then applied to study the information-theoretic measures such as the one-dimensional Shannon and Renyi entropic densities. The expectation values for the position and momentum spaces were obtained to verify the Heisenberg's uncertainty principle. Utilizing the energy spectrum equation, the thermodynamic vibrational partition function is obtained via the Poisson summation. Other thermodynamic function variations with absolute temperature have been obtained numerically for four diatomic molecules (H 2 , N 2 , O 2 , and HF) using Maple 18 software. The Shannon global entropic sum inequality has also been verified. The Renyi sum for constrained index parameters satisfies the global entropic inequality. The thermodynamic properties of the four molecules are similar and conform to works reported in the existing literature. The obtained vibrational energies are in fair agreement with the ones obtained using other forms of potential energy. The result further indicates that the lowest bounds for the Shannon, Renyi, and Heisenberg inequalities are ground states phenomena., (© 2023. The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.)

Thermomagnetic properties, and its effects on Fisher information entropy with Schioberg plus Manning-Rosen potential are studied using NUFA and SUSYQM methods in the presence of the Greene-Aldrich approximation scheme to the centrifugal term. The wave function obtained was used to study Fisher information both in position and momentum spaces for different quantum states by the gamma function and digamma polynomials. The energy equation obtained in a closed form was used to deduce numerical energy spectra, partition function, and other thermomagnetic properties. The results show that with an application of AB and magnetic fields, the numerical energy eigenvalues for different magnetic quantum spins decrease as the quantum state increases and completely removes the degeneracy of the energy spectra. Also, the numerical computation of Fisher information satisfies Fisher information inequality products, indicating that the particles are more localized in the presence of external fields than in their absence, and the trend shows complete localization of quantum mechanical particles in all quantum states. Our potential reduces to Schioberg and Manning-Rosen potentials as special cases. Our potential reduces to Schioberg and Manning-Rosen potentials as special cases. The energy equations obtained from the NUFA and SUSYQM were the same, demonstrating a high level of mathematical precision., (© 2023. The Author(s).)

The solution of the radial Schrödinger equation for the modified shifted Morse potential model is obtained using an approximate supersymmetric approach. The two different formulae for the computation of the centrifugal distortion constant are clearly examined to deduce the formula that gives the result that perfectly aligns with the experimental data. Numerical values for different molecules are computed for the two different values of the centrifugal distortion constant (dissociation energy) obtained from two different equations. The ground state energy spectra for different molecules are obtained using Herzberg's energy level equation as a standard for some molecules. The results of the modified shifted Morse potential are compared with the results from Herzberg's energy level equation and the experimental data. Our study reveals that the results for one of the two centrifugal distortion constants are closer to the standard results and the experimental data in all the molecules studied., Competing Interests: The authors declare no competing interests., (© 2023 The Authors. Published by Elsevier Ltd.)

Since the proposition of the standard form of Morse potential [Formula: see text] model over the years, there has not been much attention on the potential. Its application to different studies such as the thermodynamic properties and information theory are yet to be reported to the best of our understanding. In this study, the solutions of the radial Schrödinger equation for the standard Morse potential is obtained using supersymmetric approach. The effect of the quantum number on the energy eigenvalue for the standard Morse potential is examined numerically for the hydrogen molecule (H 2 ), lithium molecule (Li 2 ), and potassium molecule (K 2 ). Using the energy equation and the wave function obtained, the theoretic measures and thermodynamic properties of hydrogen, lithium, and potassium molecules are calculated via maple program. It has been shown that the energy of the standard Morse potential is fully bounded for the three molecules studied. A higher concentration of electron density corresponds to a strongly localized distribution in the position configuration. The Beckner, Bialynicki-Birula, and Mycieslki (BBM) inequality is satisfied for both the ground state and the first excited state. Finally, the product of uncertainty obtained obeyed the Heisenberg uncertainty relation., (© 2023. The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.)

A shifted Morse potential model is modified to fit the study of the vibrational energies of some molecules. Using a traditional technique/methodology, the vibrational energy and the un-normalized radial wave functions were calculated for the modified shifted Morse potential model. The condition that fits the modified potential for molecular description were deduced together with the expression for the screening parameter. The vibrational energies of SiC, NbO, CP, PH, SiF, NH and Cs 2 molecules were computed by inserting their respective spectroscopic constants into the calculated energy equation. It was shown that the calculated results for all the molecules agreement perfectly with the experimental RKR values. The present potential performs better than Improved Morse and Morse potentials for cesium dimer. Finally, the real Morse potential model was obtained as a special case of the modified shifted potential., (© 2022. The Author(s).)

In this work, we determined an approximate eigen solutions of Modified multi-parameter exponential potential using supersymmetric quantum mechanics approach (SUSY) with improved Greene-Aldrich approximation to the centrifugal term. The energy equation and its corresponding normalised radial wave function were fully obtained. The proposed potential reduces to other useful potentials like Rosen-Morse, Hellmann, Yukawa and Coulomb potential as special cases. The thermodynamic properties like the vibrational mean energy ( U β , V ), Vibrational heat capacity ( C β , V ), vibrational entropy ( S β , V ) and vibrational free energy ( F β , V ) of the interacting potential were studied via partition function ( Z β , V ) obtained from the resulting energy equation. This study was applied to three diatomic molecules: Chromium hydride (CrH), Titanium Hydride (TiH) and Thiocynate (ScN). To ascertain the high degree of our analytical mathematical accuracy, we compared the results of special cases with an existing results. These were found to be in excellent agreement with the existing results., Competing Interests: The authors declare no conflict of interest., (© 2022 The Author(s).)

A molecular potential model is proposed and the solutions of the radial Schrӧdinger equation in the presence of the proposed potential is obtained. The energy equation and its corresponding radial wave function are calculated using the powerful parametric Nikiforov-Uvarov method. The energies of cesium dimer for different quantum states were numerically obtained for both negative and positive values of the deformed and adjustable parameters. The results for sodium dimer and lithium dimer were calculated numerically using their respective spectroscopic parameters. The calculated values for the three molecules are in excellent agreement with the observed values. Finally, we calculated different expectation values and examined the effects of the deformed and adjustable parameters on the expectation values., (© 2021. The Author(s).)

An approximate solution of the Schrӧdinger equation for a molecular attractive potential was obtained using the parametric Nikiforov-Uvarov method. The energy equation and the corresponding radial wave functions were calculated. The effects of the potential parameters on the energy eigenvalues were examined. The thermal properties under the molecular attractive potential were calculated and the behaviour of the thermal properties with the maximum quantum state (λ) and the temperature parameter (β) respectively, were studied. Using the molecular spectroscopic parameters, the Rydberg-Klein-Rees (RKR) of cesium dimer and lithium dimer were both obtained and compared with the experimental values. The RKR values of both cesium dimer and lithium dimer calculated aligned with the observed values. The deviation and average deviation of the RKR for each molecule were also calculated.

The solution of the radial Schrödinger equation was obtained using the methodology of supersymmetric approach with a combination of modified generalized Pöschl-Teller potential and inversely quadratic Yukawa potential model. The non-relativistic ro-vibrational energy spectra and the corresponding wave functions were obtained and numerical results were generated for some states. The variation of energy of the combined potential and the subsets potentials with the screening parameter for various quantum number were graphically studied. The effect of the potential parameters on the energy for different states was also studied numerically. For more usefulness and applications of the work, the vibrational partition function and the various thermal properties like mean energy, Helmholtz energy, heat capacity and entropy were calculated. The behaviour of the thermodynamic properties with respect to temperature change for various quantum number and maximum quantum states were examined in detail. The temperature has positive effect on all the thermal properties except the free energy., Competing Interests: The authors declare no conflict of interest., (© 2021 The Authors.)

An approximate solutions of the radial Schrödinger equation was obtained under a modified Tietz-Hua potential via supersymmetric approach. The effect of the modified parameter and optimization parameter respectively on energy eigenvalues were graphically and numerically examined. The comparison of the energy eigenvalues of modified Tietz-Hua potential and the actual Tietz-Hua potential were examined. The ro-vibrational energy of four molecules were also presented numerically. The thermal properties of the modified Tietz-Hua potential were calculated and the effect of temperature on each of the thermal property were examined under hydrogen fluoride, hydrogen molecule and carbon (ii) oxide. The study reveals that for a very small value of the modified parameter, the energy eigenvalues of the modified Tietz-Hua potential and that of the actual Tietz-Hua potential are equivalent. Finally, the vibrational energies for Cesium molecule was calculated and compared with the observed value. The calculated results were found to be in good agreement with the observed value.

In this research article, the modified approximation to the centrifugal barrier term is applied to solve an approximate bound state solutions of Dirac equation for spin and pseudospin symmetries with hyperbolic Hulthen plus hyperbolic exponential inversely quadratic potential using parametric Nikiforov-Uvarov method. The energy eigen equation and the unnormalised wave function were presented in closed and compact form. The nonrelativistic energy equation was obtain by applying nonrelativistic limit to the relativistic spin energy eigen equation. Numerical bound state energies were obtained for both the spin symmetry, pseudospin symmetry and the non relativistic energy. The screen parameter in the potential affects the solutions of the spin symmetry and non-relativistic energy in the same manner but in a revised form for the pseudospin symmetry energy equation. In order to ascertain the accuracy of the work, the numerical results obtained was compared to research work of existing literature and the results were found to be in excellent agreement to the existing literature. The partition function and other thermodynamic properties were obtained using the compact form of the nonrelativistic energy equation. The proposed potential model reduces to Hulthen and exponential inversely quadratic potential as special cases. All numerical computations were carried out using Maple 10.0 version and Matlab 9.0 version softwares respectively.

An approximate solution of the one-dimensional relativistic Klein-Gordon equation was obtained under the interaction of an improved expression for Wei potential energy function. The solution of the non-relativistic Schrödinger equation was obtained from the solution of the relativistic Klein-Gordon equation by certain mappings. We have calculated Fisher information for position space and momentum space via the computation of expectation values. The effects of some parameters of the Wei potential energy function on the Fisher information were fully examined graphically. We have also examined the effects of the quantum number n and the angular momentum quantum number ℓ on the expectation values and Fisher information respectively for some selected molecules. Our results revealed that the variation of most of the parameters of the Wei potential energy function against the Fisher information does not obey the Heisenberg uncertainty relation for Fisher information while that of the quantum number and angular momentum quantum number on Fisher information obeyed the relation.

The approximate analytical solutions of the three-dimensional radial Schrödinger wave equation with a multiple potential function has been studied using a suitable approximation scheme to the centrifugal term in the framework of parametric Nikiforov-Uvarov method. The energy equation and the wave function were obtained. The calculated wave function was used to study Shannon entropy and variance via expectation values. The behaviour of Shannon entropy and variance respectively with the equilibrium bond length were examined in detail. A special case of the multiple potential (pseudoharmonic-like potential) was equally examined under Shannon entropy and variance. For further application of the study, some diatomic molecules were examined under variance and Shannon entropy. Finally, some variance inequalities were derived using Cramer-Rao uncertainty relation and these were justified by numerical results.

An approximate solution of the radial Schrödinger equation is obtained with a generalized group of potentials in the presence of both magnetic field and potential effect using supersymmetric quantum mechanics and shape invariance methodology. The energy bandgap of the generalized group of potentials was calculated for s - wave cases at the ground state. By varying the numerical values of the potential strengths, the energy band gap of Hellmann's potential and Coulomb-Hulthẻn potential respectively were obtained. It is noted that the inclusion of the potential effect greatly affects the accuracy of the results. Our calculated results are in agreement and better than the existing calculated results. The present results approximately coincide with the standard bandgap of Cu 2 ZnSnS 4 (CZTS)., (© 2020 The Author(s).)

We apply the supersymmetric quantum mechanics and shape invariance approach to obtain the solutions of the 3-dimensional Schrödinger equation in the presence of a newly proposed potential model called hyperbolic-sinus potential function for any ℓ-state. Rotational-vibrational energy eigenvalues of some diatomic molecules are numerically calculated. Special cases of interest of the hyperbolic-sinus potential function are also studied. Our results indicated that the energy eigenvalue is highly sensitive to the potential parameters.

In this paper local and global gradient estimates are obtained for positive solutions to the following nonlinear elliptic equation Δ f u + p ( x ) u + q ( x ) u α = 0 , on complete smooth metric measure spaces ( M N , g , e - f d v ) with ∞-Bakry-Émery Ricci tensor bounded from below, where α is an arbitrary real constant, p ( x ) and q ( x ) are smooth functions. As an application, Liouville-type theorems for various special cases of the equation are recovered. Furthermore, we discuss nonexistence of smooth solution to Yamabe type problem on ( M N , g , e - f d v ) with nonpositive weighted scalar curvature., (© 2019 The Authors.)

The approximate analytical solutions of the non-relativistic Schrӧdinger equation for the Attractive potential model with the centrifugal term are investigated using the elegant methodology of the parametric Nikiforov-Uvarov. The energy equation and the corresponding un-normalized radial wave functions are obtained in a close and compact form after a proper Greene-Aldrich approximation scheme is applied. By changing the numerical values of some potential strengths, special cases of the Attractive potential are investigated in detail. The effects of the potential strengths and potential range respectively on the energy are also studied. The energy is found to be very sensitive to each of the potential parameters. Some theoretic quantities such as information energy, Rẻnyi entropy and Tsallis entropy.

This article describes the data for examining the influence of government expenditure and revenue on Nigerian economic growth. Data were extracted from the World Bank database and Central Bank of Nigeria (CBN) Statistical bulletin. The data are available with this article. The data is related to the research article "Newly proposed estimator for ridge parameter: an application to the Nigerian economy" (Lukman and Arowolo, 2018) but not discussed in detail. This data article will assist economists in identifying factors that will affect the economy of a country, especially in the African region.