Vibrational Specific Heat Research Articles

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 ℓ-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.

Quantum control of optoelectronic and thermodynamic properties of dopamine molecule in external electric field : A DFT and TD-DFT study

The effect of external electric field on molecular properties of dopamine is investigated using DFT and TD-DFT methods with B3LYP functional and 6-311++G(d,p) basis set. Optoelectronic properties (energy, polarizability, molecular orbitals, HOMO-LUMO gap, optical gap, electron–hole binding energy, UV-visible and electroabsorption spectra), and thermodynamic properties (zero-point vibrational energy, thermal energy, entropy, and specific heat capacity) were investigated. These properties are useful in developing molecular sensing devices, and understanding molecule receptor binding interaction. Our results using TD-DFT for excited states show that when an external electric field of magnitude F = 0.020 a.u. is applied along the x-axis, there is a shift in the absorption spectra of dopamine from the ultraviolet to the visible region which will be useful in optogenetic stimulation using visible light. The tunneling ionization rate increases and tunneling ionization time decreases with increase in electric field. These studies have potential applications in many of the emerging technologies in neurobiology such as deep brain stimulation and iontophoresis.

Quantized momentum eigenstates and thermodynamic properties of the Feinberg–Horodecki equation for the time-dependent Wei-Hua oscillator

In this study, we obtained exact solutions of the Feinberg–Horodecki equation for the time-dependent Wei-Hua potential, which is constructed by the temporal counterpart of the spatial form of this potential. We obtained the quantized momentum eigenvalues and the corresponding wave functions. We obtained the partition function for the system and studied other thermodynamic properties that include vibrational mean momentum ( U), vibrational specific heat capacity ( C), vibrational entropy ( s), and vibrational free momentum ( F) as a signature in the momentum space.

Classical Density Functional Theory Approach to the Vibrational Properties and Lattice Specific Heat of a Quasi-One-Dimensional Antiferromagnet KFeSe2

We have performed ab initio calculations of vibrational properties of KFeSe2 compound utilizing density functional theory. Total and element specific phonon densities of states have been calculated within a direct approach of harmonic approximation. We have used phonon density of states to calculate lattice contribution to the specific heat. The calculated phonon density of states shows numerous of high-frequency vibrational modes of Fe and Se atoms, which strongly restricts application of the Debye model for analysis of the thermodynamic properties of KFeSe2. The results of this work can be used in a further estimation of the magnetic specific heat of KFeSe2.

Energy Eigenvalue and Thermodynamic Properties of q-deformed Hulthen Potential

The objective of this work is to calculate the energy eigenvalue for q-deformed Hulthen potential using a Gaussian hypergeometric function with centrifugal approximation factor and related thermodynamical properties. For this, we develop a mathematical model using the Schrodinger wave equation to find the energy eigenvalue. In addition, the thermodynamic parameters were also calculated for q-deformed Hulthen potential using the partition function. The energy eigenvalue for quantum numbers n=1 to n=5 was observed for screening parameters 0.1, 0.5, and 1. In between, 0.1 to 1 and n=1 to n=2 the energy eigenvalue ranges from -1.048 to -208.572. The energy eigenvalue for considering potential shows that decrease with increasing the quantum number which means electron are loosely bounded nucleus in an atom. Also, the vibrational mean energy, vibrational mean free energy, vibrational specific heat capacity, and vibrational entropy are obtained but due to complexity, the further development of the equation is the limitation of this work. BIBECHANA 19(2022)165-169

Eigensolutions and Thermodynamic Properties of Kratzer Plus Generalized Morse Potential

In this study, we apply the parametric Nikiforov-Uvarov method to obtain the bound state solution of Schrödinger wave equation in the presence of Kratzer plus generalized Morse potential (KPGM). The energy eigen equation and the corresponding normalised wave function were obtained in closed form. The resulting energy eigen equation was used to study partition function and other thermodynamic properties such as vibrational mean energy, vibrational specific heat capacity, vibrational mean free energy and vibrational entropy for the proposed potential as applied to lithium hydride diatomic molecule. The thermodynamic plots obtained were in excellent agreement to work of existing literatures. The wave function and probability density plots for the diatomic molecules were obtained through a well designed and implemented maple programme.

Bound state solutions and thermodynamic properties of modified exponential screened plus Yukawa potential

In this research paper, the approximate bound state solutions and thermodynamic properties of Schrӧdinger equation with modified exponential screened plus Yukawa potential (MESPYP) were obtained with the help Greene–Aldrich approximation to evaluate the centrifugal term. The Nikiforov–Uvarov (NU) method was used to obtain the analytical solutions. The numerical bound state solutions of five selected diatomic molecules, namely mercury hydride (HgH), zinc hydride (ZnH), cadmium hydride (CdH), hydrogen bromide (HBr) and hydrogen fluoride (HF) molecules were also obtained. We obtained the energy eigenvalues for these molecules using the resulting energy eigenequation and total unnormalized wave function expressed in terms of associated Jacobi polynomial. The resulting energy eigenequation was presented in a closed form and applied to study partition function (Z) and other thermodynamic properties of the system such as vibrational mean energy (U), vibrational specific heat capacity (C), vibrational entropy (S) and vibrational free energy (F). The numerical bound state solutions were obtained from the resulting energy eigenequation for some orbital angular quantum number. The results obtained from the thermodynamic properties are in excellent agreement with the existing literature. All numerical computations were carried out using spectroscopic constants of the selected diatomic molecules with the help of MATLAB 10.0 version. The numerical bound state solutions obtained increases with an increase in quantum state.

Approximate Solutions, Thermal Properties, and Superstatistics Solutions to Schrödinger Equation

In this work, we apply the parametric Nikiforov-Uvarov method to obtain eigensolutions and total normalized wave function of Schrödinger equation expressed in terms of Jacobi polynomial using Coulomb plus Screened Exponential Hyperbolic Potential (CPSEHP), where we obtained the probability density plots for the proposed potential for various orbital angular quantum number, as well as some special cases (Hellmann and Yukawa potential). The proposed potential is best suitable for smaller values of the screening parameter α . The resulting energy eigenvalue is presented in a close form and extended to study thermal properties and superstatistics expressed in terms of partition function Z and other thermodynamic properties such as vibrational mean energy U , vibrational specific heat capacity C , vibrational entropy S , and vibrational free energy F . Using the resulting energy equation and with the help of Matlab software, the numerical bound state solutions were obtained for various values of the screening parameter ( α ) as well as different expectation values via Hellmann-Feynman Theorem (HFT). The trend of the partition function and other thermodynamic properties obtained for both thermal properties and superstatistics were in excellent agreement with the existing literatures. Due to the analytical mathematical complexities, the superstatistics and thermal properties were evaluated using Mathematica 10.0 version software. The proposed potential model reduces to Hellmann potential, Yukawa potential, Screened Hyperbolic potential, and Coulomb potential as special cases.

Nonrelativistic solutions of Schrödinger equation and thermodynamic properties with the proposed modified Mobius square plus Eckart potential

We obtain solutions of Schrödinger equation for the modified Mobius square plus Eckart (MMPSE) potential via the formula method. Numerical results are reported. In addition, the partition function Z and other thermodynamic properties such as vibrational free energy, F, vibrational internal energy, U, vibrational entropy, S, and vibrational specific heat, C are presented. We also discuss special cases of this potential. Our result is consistent with previous studies in the literature.

Nonrelativistic treatment of inversely quadratic Hellmann-Kratzer potential and thermodynamic properties

The study presents approximate analytical solutions of the Schrödinger equation with a newly proposed potential model called Inversely Quadratic Hellmann-Kratzer potential (IQHKP). This potential is a superposition of Inversely Quadratic Hellman potential and Kratzer potential. The energy eigenvalues and corresponding wavefunction are calculated via the formula method. We applied our results to evaluate thermodynamic functions such as vibrational free energy, F, vibrational internal energy, U, vibrational entropy, S, and vibrational specific heat, C. We also reported special cases of importance.

Shannon entropy for Feinberg–Horodecki equation and thermal properties of improved Wei potential model

Abstract We solved a one-dimensional time-dependent Feinberg–Horodecki equation for an improved Wei molecular energy potential function using the parametric Nikiforov–Uvarov method. The quantized momentum and the corresponding wave functions were obtained. With the help of the wave functions obtained, we calculated Shannon entropy for both the position space and momentum space. The results were used to study four molecules. The results of Shannon entropy were found to be in excellent agreement with those found in the literature. For more usefulness of these studies, the quantized momentum obtained was transformed into an energy equation with certain transformations. The energy equation was then used to calculate some thermodynamic properties such as vibrational mean energy, vibrational specific heat, vibrational mean free energy, and vibrational entropy via computation of the partition function. The thermodynamic properties studied for CO, NO, CH, and ScH showed that for a certain range of the temperature studied, the molecules exhibited similar features except for the vibrational entropy.

Investigations of mechanical and thermoelectric properties of ‘AlNiP’ novel half-Heusler alloy

Emergent passion for half-Heusler alloys is due to their drastic thermoelectric properties for tremendous applications in spintronics. The detailed physical insight of the elastic, mechanical, and thermodynamical properties of novel half-Heusler alloy ‘AlNiP’ has been investigated under the framework of Density Functional Theory (DFT). The semi-classical Boltzmann transport theory in combination with DFT is implemented to evaluate the thermoelectric properties. The theoretical computation study related to elastic, mechanical, and thermoelectric properties has been made for the first time. The computed elastic constant values satisfy well the Born-Huang stability criteria, which reveal the mechanical stability of the alloy at the ambient conditions of temperature and pressure. The computed value of Poisson's ratio uncovers the metallic bonding of AlNiP. Furthermore, the elastic properties have been evoked in three dimensions (3D) for Young's modulus, linear compressibility, shear modulus, and Poisson's ratio. Thermodynamic calculations showed that vibrational energy, entropy, and specific heat capacity increases remarkably as temperature increases while vibrational free energy decreases steadily with the increase in temperature. Estimates of Seebeck coefficient (~-12 μV/K at 800 K), lattice thermal conductivity (0.06 W/mK at 300 K), and figure of merit (0.03 at 800 K) have also been put forward in this report. The suitable doping substituent or nano-structuring processes can be an alternative to increase the Seebeck coefficient while keeping the other parameters unperturbed. The studied computational appraisal will certainly stimulate further experimental research of novel half-Heusler alloy AlNiP in pioneering applications.

ANALYTICAL SOLUTION OF SCHRÖDINGER EQUATION IN BISPHERICAL COORDINATE FOR MOBIUS SQUARE PLUS MODIFIED YUKAWA POTENTIAL USING NIKIFOROV UVAROV FUNCTIONAL ANALYSIS (NUFA) METHOD WITH ITS THERMODYNAMICS PROPERTIES FOR DIATOMIC MOLECULES

Background: The analytical solution of the Schrödinger equation in bispherical coordinates has attracted a great deal of interest for theoretical physics researchers in the branch of quantum physics. The energy and wave function are solutions of the Schrödinger equation which are very important because it contains all necessary information regarding the behavior of quantum systems. Aim: This study aimed to obtain energy, radial wave functions and thermodynamic properties for diatomic molecules from the radial part of the Schrödinger equation in bispherical coordinates for the modified Mobius square plus Yukawa potential using the Nikiforov Uvarov Functional Analysis (NUFA) method. Methods: The variable separation method was applied to reduce the Schrodinger equation in bispherical coordinates to the radial part and angular part Schrodinger equation. The Schrodinger equation of the radial part in bispherical coordinates was solved using the Nikiforov Uvarov Functional Analysis (NUFA) method to obtain the energy equation and radial wave function. Furthermore, the vibrational partition function 𝑍 was obtained from the energy equation. The vibrational mean energy 𝑈, vibrational specific heat 𝐶, vibrational free energy 𝐹, and vibrational entropy 𝑆 were obtained from the vibrational partition function 𝑍. Results and Discussion: The results showed that the increase of parameters of 𝑛 and 𝛼 caused the decrease of energy, but the increase of parameters of 𝐿 and 𝑚0 caused the increase of energy. The radial quantum number 𝑛 and the potential range 𝛼 had the most effect to the wave functions. The parameters 𝑛𝑚𝑎𝑥, 𝑇, and 𝛼 had effect to the vibrational partition function 𝑍, vibrational mean energy 𝑈, vibrational specific heat 𝐶, vibrational free energy 𝐹, and vibrational entropy 𝑆. Conclusions: From the results of this study, it can be concluded that the energy, radial wave function, and thermodynamic properties for diatomic molecules have been obtained using the Nikiforov Uvarov Functional Analysis (NUFA) method.

Ro‐vibrational energy and thermodynamic properties of molecules subjected to Deng–Fan potential through an improved approximation

AbstractAccurate solution of the Schrödinger equation with Deng–Fan potential is presented by means of Nikiforov–Uvarov method. A modified Pekeris‐type approximation is proposed for the centrifugal term, from a linear combination of the r → 0 and r → re limits. It can potentially offer a series of approximations (depending on an adjustable parameter λ). The existing approximations in the literature can then be recovered in certain special cases. Its efficiency and feasibility is demonstrated by a critical comparison of eigenvalues produced at various λ's for four molecules, namely, H2, LiH, HCl and CO. Analytical expressions are derived for energies, eigenfunctions and the thermodynamic properties such as vibrational mean free energy, vibrational free energy, vibrational entropy and vibrational specific heat. The effect of quantum correction on partition function and thermodynamic properties is discussed by including the correction up to 10th‐order, for H2 and LiH. The effect of λ parameter on these properties is also studied.

Analytical solutions of fractional Schrödinger equation and thermal properties of Morse potential for some diatomic molecules

By employing the concept of conformable fractional Nikiforov–Uvarov (NU) method, we solved the fractional Schrödinger equation with the Morse potential in one dimension. The analytical expressions of the bound state energy eigenvalues and eigenfunctions for the Morse potential were obtained. Numerical results for the energies of Morse potential for the selected diatomic molecules were computed for different fractional parameters chosen arbitrarily. Also, the graphical variation of the bound state energy eigenvalues of the Morse potential for hydrogen dimer with vibrational quantum number and the range of the potential were discussed, with regards to the selected fractional parameters. The vibrational partition function and other thermodynamic properties such as vibrational internal energy, vibrational free energy, vibrational entropy and vibrational specific heat capacity were evaluated in terms of temperature. Our results are new and have not been reported in any literature before.

Planck’s Oscillators at Low Temperatures and Haken’s Perturbation Approach to the Quantum Oscillators Reconsidered

In the first step the extremal values of the vibrational specific heat and entropy represented by the Planck oscillators at the low temperatures could be calculated. The positions of the extrema are defined by the dimensionless ratios between the quanta of the vibrational energy and products of the actual temperature multiplied by the Boltzmann constant. It became evident that position of a local maximum obtained for the Planck’s average energy of a vibration mode and position of a local maximum of entropy are the same. In the next step the Haken’s time-dependent perturbation approach to the pair of quantum non-degenerate Schrödinger eigenstates of energy is re-examined. An averaging process done on the time variable leads to a very simple formula for the coefficients entering the perturbation terms.

ANALYSIS OF ENERGY AND WAVE FUNCTIONS AND THE THERMODYNAMICS PROPERTIES OF THE 6-DIMENSIONAL SCHRODINGER EQUATION UNDER DOUBLE RING-SHAPE OSCILLATOR (DRSO) AND MANNING-ROSEN POTENTIALS USING SUSY QM METHOD

The exact solutions of the Schrodinger equations (SE) in the D-dimensional coordinate system have attracted the attention of many theoretical researchers in branches of quantum physics and quantum chemistry. The energy eigenvalues and the wave function are the solutions of the Schrodinger equation that implicitly represents the behavior of a quantum mechanical system. This study aimed to obtain the eigenvalues, wave functions, and thermodynamic properties of the 6-Dimensional Schrodinger equation under Double Ring-Shaped Oscillator (DRSO) and Manning-Rosen potential. The variable separation method was applied to reduce the one 6-Dimensional Schrodinger equation depending on radial and angular non-central potential into five onedimensional Schrodinger equations: one radial and five angular Schrodinger equations. Each of these onedimensional Schrodinger equations was solved using the SUSY QM method to obtain one eigenvalue and one wave function of the radial part, five eigenvalues, and five angular wave functions angular part. Some thermodynamic properties such, the vibrational mean energy 𝑈, vibrational specific heat 𝐶, vibrational free energy 𝐹, and vibrational entropy 𝑆, were obtained using the radial energy equations. The results showed that except the 𝑛𝑙1, all increment of angular quantum number decreases the energy values. Increments of all potential parameter increase the energy values. Increment of angular quantum number and potentials parameter increases the amplitude and shifts the wave functions to the left. However, the increment of 𝑛𝑙1, 𝛼, 𝜎, and 𝜌 decrease the amplitude and shift wavefunctions to the right. Moreover, the vibrational mean energy 𝑈 and free energy 𝐹 increased as the increasing value of potentials parameters, where the ω parameter has the dominant effect than the other parameters. The vibrational specific heat 𝐶 and entropy 𝑆 affected only by the 𝜔 parameter, where 𝐶 and 𝑆 decreased as the increase of 𝜔.

Vibrational Entropy and Complexity Measures in Modified Pöschl–Teller Plus Woods–Saxon potential

The explicit expression of the vibrational partition function for the modified Poschl–Teller plus Woods–Saxon potential has been presented in a closed-form. The analytical expression for the vibrational mean energy have also been calculated were other thermodynamic functions like the vibrational specific heat, free energy, and the entropy for the gallium nitride wurtzite crystal structure have been determined in details. The dependence of these functions on the potential parameters has also been discussed in detail. By using the ground state energy and the probability density, the behaviours of some theoretic quantities (Shannon entropy and Fisher information entropy) have also been calculated and analyzed graphically as a function of the potential parameters.

Energies Spectra and Thermodynamic Properties of Hyperbolic Pöschl–Teller Potential (HPTP) Model

We solved the Dirac equation using modified factorization method with hyperbolic Poschl–Teller potential model. Relativistic and non-relativistic ro-vibrational energy spectra were obtained and numerical solutions for four diatomic molecules $$\left( {{\text{HCl}},\;{\text{H}}_{2} ,\;{\text{CO}}\;{\text{and}}\;{\text{LiH}}} \right)$$ obtained also. The eigenfunction for this potential has been obtained in terms of hypergeometric function. The energy variations with different potential parameters and quantum numbers of the hyperbolic Poschl–Teller potential are also plotted. In addition, we evaluate the vibrational partition function and other thermodynamic properties such as vibrational internal energy, vibrational free energy, vibrational entropy and vibrational specific heat capacity, in terms of temperature and upper bound vibration quantum number. Our results are consistent with those found in the available literature.

Effect of quantum corrections on thermodynamic properties for dimers

We have calculated the effect of quantum correction (QC) on the vibrational partition function by using improved Rosen-Morse potential for Cs2 molecules. The related thermodynamics properties i.e vibrational mean energy, vibrational specific heat, vibrational free energy and vibrational entropy are derived. The change in the partition function and thermodynamic properties by adding quantum corrections up to 20th order are discussed and then compared with their classical counterpart.