Exact quantization rule to the Kratzer-type potentials: an application to the diatomic molecules (original) (raw)
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In this study, the Schrödinger equation with the Hulthén plus screened Kratzer potentials (HSKP) are solved via the Nikiforov-Uvarov (NU) and the series expansion methods. We obtained the energy equation and the wave function in closed form with Greene-Aldrich approximation via the NU method. The series expansion method was also used to obtain the energy equation of HSKP. Three distinct cases were obtained from the combined potentials. The energy eigenvalues of HSKP for HCl, LiH, H2, and NO diatomic molecules were computed for various quantum states. To test the accuracy of our results, we computed the bound states energy of HCl and LiH, for a special case of Kratzer and screened Kratzer potentials, which are in excellent agreement with the report of other researchers.
Energy States of Some Diatomaic Molecules: The Exact Quantisation Rule Approach
Zeitschrift für Naturforschung A, 2015
In this study, we obtain the approximate analytical solutions of the radial Schrödinger equation for the Deng–Fan diatomic molecular potential by using the exact quantisation rule approach. The wave functions were expressed by hypergeometric functions via the functional analysis approach. An extension to the rotational–vibrational energy eigenvalues of some diatomic molecules is also presented. It is shown that the calculated energy levels are in good agreement with those obtained previously (E nℓ –D; shifted Deng–Fan).
Spectroscopic study of some diatomic molecules via the proper quantization rule
Journal of Mathematical Chemistry, 2015
Spectroscopic techniques are very essential tools in studying electronic structures, spectroscopic constants and energetic properties of diatomic molecules. These techniques are also required for parametrization of new method based on theoretical analysis and computational calculations. In this research, we apply the proper quantization rule in spectroscopic study of some diatomic molecules by solving the Schrödinger equation with two solvable quantum molecular systems-Tietz-Wei and shifted Deng-Fan potential models for their approximate nonrelativistic energy states via an appropriate approximation to the centrifugal term. We show that the energy levels can be determined from its ground state energy. The beauty and simplicity of the method applied in this study is that, it can be applied to any exactly as well as approximately solvable models. The validity and accuracy of the method is tested with previous techniques via numerical computation for H 2 and CO diatomic molecules. The result also include energy spectrum of 5 different electronic states of NO and 2 different electronic state of ICl.
Analytical Solutions of the Molecular Kratzer-Feus potential by means of the Nikiforov-Uvarov Method
Journal of Mathematical Chemistry
The analytical methods for solving Schrödinger equation are essential and effective tools with which we can investigate the spectroscopic properties, the electronic structure, and the energetic properties of the diatomic molecules (DMs). Accordingly, in this work, we used the Nikiforov-Uvarov (NU) method to solve the three-dimensional nonrelativistic Schrödinger equation with the molecular Kratzer-Feus (KF) potential and obtain the exact analytical bound state energy eigenvalues as well as their corresponding normalized eigenfunctions. The effective KF diatomic molecular potential well is investigated and represented graphically for several well-known DMs. The bound state energy levels are tabulated numerically for arbitrary values of the vibrational and rotational quantum numbers. The results obtained in this work are found to be in excellent agreement with the already-existing results in the literature.
2009
The Schrodinger equation for the rotational-vibrational (ro-vibrational) motion of a diatomic molecule with empirical potential functions is solved approximately by means of the Nikiforov-Uvarov method. The approximate ro-vibratinal energy spectra and the corresponding normalized total wavefunctions are calculated in closed form and expressed in terms of the hypergeometric functions or Jacobi polynomials P_n^(μ,ν)(x), where μ>-1, ν>-1 and x included in [-1,+1]. The s-waves analytic solution is obtained. The numerical energy eigenvalues for selected H_2 and Ar_2 molecules are also calculated and compared with the previous models and experiments.
Journal of Mathematical Chemistry, 2012
For arbitrary values n and quantum numbers, we present the solutions of the 3-dimensional Schrödinger wave equation with the pseudoharmonic potential via the SU (1, 1) Spectrum Generating Algebra (SGA) approach. The explicit bound state energies and eigenfunctions are obtained. The matrix elements r 2 and r d dr are obtained (in a closed form) directly from the creation and annihilation operators. In addition, by applying the Hellmann-Feynman theorem, the expectation values of r 2 and p 2 are obtained. The energy states, the expectation values of r 2 and p 2 and the Heisenberg uncertainty products (HUP) for set of diatomic molecules (CO, NO, O 2 , N 2 , CH, H 2 , ScH) for arbitrary values of n and quantum numbers are obtained. The results obtained are in excellent agreement with the available results in the literature. It is also shown that the HUP is obeyed for all diatomic molecules considered.
Bound state solutions to the Schrödinger equation for some diatomic molecules
Pramana, 2018
The bound state solutions to the radial Schrödinger equation are obtained in three-dimensional space using the series expansion method within the framework of a general interaction potential. The energy eigenvalues of the pseudoharmonic and Kratzer potentials are given as special cases. The obtained analytical results are applied to several diatomic molecules, i.e. N 2 , CO, NO and CH. In order to check the accuracy of the present method, a comparison is made with similar results obtained in the literature by using other techniques.
A numerical procedure to obtain accurate potential energy curves for diatomic molecules
Journal of Molecular Structure: THEOCHEM, 1992
The potential energy curves for the X'Z+ state of W60 and 'Li'H were obtained by fitting the Rydberg-Klein-Rees potential in the Chebyshev sense (minimum-maximum approximation) to a simple functional form very similar to a perturbed-Morse-oscillator potential but with a minor number of parameters. Using Hermite orthogonal functions (eigenfunctions of the harmonic oscillator) as the basis set we solved variationally the radial Schriidinger equation to obtain the vibrational energies E, and the rotational constants B,. Using the potential thus determined and the proposed basis set, the matrix elements of the Hamiltonian may be calculated analytically, presenting only round-off errors. The agreement between the calculated and experimentally determined E, and B, values is good, the self-consistency of our potentials being very similar to that obtained by other authors.
Systematic Approach to Compute the Vibrational Energy Levels of Diatomic Molecules
Journal of Applied Mathematics and Physics, 2020
In continuation of our previous paper of the anharmonic potentials analysis through the Floquet representation, we performed in this work a systematic calculation of the diatomic vibrational energy levels as well as the corresponding wave functions. The solution of Schrödinger equation according to Morse potential, which is a suitable model to describe the diatomic vibrational spectra, has been introduced; thus the explicit formulas to the second order have been established. As an illustration, the dissociation energies of some molecules species (i.e. ScN, LiH, Cl 2 and NO) have been computed, as well as the wave functions and the corresponding probability densities, relating to the (ScN) molecule have been represented. Comparisons of our results with those of literature have been made.