This is the second article in which we study the rotating Morse potential model for diatomic molecules using the tridiagonal J-matrix approach. Here, we improve further the accuracy of computing the bound states and resonance energies for this potential model from the poles of the S-matrix for arbitrary angular momentum. The calculation is performed using an infinite square integrable basis that supports a tridiagonal matrix representation for the reference Hamiltonian, which is included in the computations analytically without truncation. Our method has been applied to both the regular and inverted Morse potential with favorable results in comparison with available numerical data. We have also shown that the present method adds few significant digits to the accuracy obtained from the finite dimensional approach (e.g. the complex rotation method). Moreover, it allows us to handle easily both analytic and non-analytic potentials as well as 1/r singular potentials.
This is the first in a series of articles in which we study the rotating Morse potential model for diatomic molecules in the tridiagonal J-matrix representation. Here, we compute the bound states energy spectrum by diagonalizing the finite dimensional Hamiltonian matrix of H2, LiH, HCl and CO molecules for arbitrary angular momentum. The calculation was performed using the J-matrix basis that supports a tridiagonal matrix representation for the reference Hamiltonian. Our results for these diatomic molecules have been compared with available numerical data satisfactorily. The proposed method is…
Application of Genetic Algorithm (GA) for determination of parameters of an analytical representation of diatomic molecule potential is presented. GA can be used for finding potential characteristics of an electronic energy state which can be described by analytical function. GA was tested on two artificially generated datasets which base on potentials with known characteristics and two LIF excitation spectra recorded using transitions in CdKr and CdAr molecules. Tests on generated datasets showed that GA can properly reproduce parameters of the potentials. Tests on experimental spectra indica…
We present modified $\ell$-states of diatomic molecules by solving the radial and angle-dependent parts of the Schrödinger equation for central potentials, such as Morse and Kratzer, plus an exactly solvable angle-dependent potential $V_θ(θ)/r^2$ within the framework of the Nikiforov-Uvarov (NU) method. We emphasize that the contribution which comes from the solution of the Schrödinger equation for the angle-dependent potential modifies the usual angular momentum quantum number $\ell$. We calculate explicitly bound state energies of a number of neutral diatomic molecules composed of a first-ro…
We compute the quantum work distribution for a driven Morse oscillator. To this end, we solve the time-dependent dynamics for a scale-invariant process, from which the exact expressions for the transition probabilities are found. Special emphasis is put on the contributions to the work distribution from discrete (bound) and continuous (scattering) parts of the spectrum. The analysis is concluded by comparing the work distribution for the exact Morse potential and the one resulting from a harmonic approximation.
Due to one of the most representative contributions to the energy in diatomic molecules being the vibrational, we consider the generalized Morse potential (GMP) as one of the typical potential of interaction for one-dimensional microscopic systems, which describes local anharmonic effects. From Eckart potential (EP) model, it is possible to find a connection with the GMP model, as well as obtain the analytical expression for the energy spectrum because it is based on $S\,O\left(2,1\right)$ algebras. In this work we find the macroscopic properties such as vibrational mean energy $U$, specific h…
The vibrational motion equations of both homo and hetero-nuclei diatomic molecules are here derived for the first time. A diatomic molecule is first considered as a one dimensional quantum mechanics oscillator. The second and third-order Hamiltonian operators are then formed by substituting the number operator for the quantum number in the corresponding vibrational energy eigenvalues. The expectation values of relative position and linear momentum operators of two oscillating atoms are calculated by solving Heisenbergs equations of motion. Subsequently, the expectation values of potential and …
We show that a direct connection can be drawn, based on fundamental quantum principles, between the Morse potential, extensively used as an empirical description for the atomic interaction in diatomic molecules, and the harmonic potential. This is conceptually achieved here through a non-additive translation operator, whose action leads to a perfect equivalence between the quantum harmonic oscillator in deformed space and the quantum Morse oscillator in regular space. In this way, our theoretical approach provides a distinctive first principle rationale for anharmonicity, therefore revealing a…
The analytical solution of the Schrödinger equation with exponential coshine screened and Morse potential are presented. The energy eigenvalues and the corresponding wave function are obtained for several values of screening parameters. We also present some numerical results for some selected diatomic molecules which are consistent with the values in the literature
We study the dissociation dynamics of a diatomic molecule, modeled as a Morse oscillator, coupled to an optical cavity. In both classical and quantum dynamics simulations, a marked suppression of the dissociation probability is observed for cavity frequencies significantly below the fundamental transition frequency of the molecule. We show that the suppression in the probability occurs when certain key nonlinear resonances in the classical phase space of the molecule disappear, possible only when the dipole function is nonlinear. This study demonstrates the complexity of cavity-molecule vibrat…
A Generalized Morse Potential (GMP) is an extension of the Morse Potential (MP) with an additional exponential term and an additional parameter that compensate for MP's erroneous behavior in the long range part of the interaction potential. Because of the additional term and parameter, the vibrational levels of the GMP cannot be solved analytically, unlike the case for the MP. We present several numerical approaches for solving the vibrational problem of the GMP based on Galerkin methods, namely the Laguerre Polynomial Method (LPM), the Symmetrized Laguerre Polynomial Method (SLPM) and the Pol…
The shifted Tietz-Wei (sTW) oscillator is as good as traditional Morse potential in simulating the atomic interaction in diatomic molecules. By using the Pekeris-type approximation to deal with the centrifugal term, we obtain the bound-state solutions of the radial Schrödinger equation with this typical molecular model via the exact quantization rule (EQR). The energy spectrum for a set of diatomic molecules ($NO \left(a^4Π_i\right)$, $NO \left(B^2Π_r\right)$, $NO \left(L'^2ϕ\right)$, $NO \left(b^4Σ^{-}\right)$, $ICl\left(X^1Σ_g^{+}\right)$, $ICl\left(A^3Π_1\right)$ and $ICl\left(A'^3Π_2\right…
The trigonometric Pöschl-Teller (PT) potential describes the diatomic molecular vibration. We have obtained the approximate solutions of the radial Schrödinger equation (SE) for the rotating trigonometric PT potential using the Nikiforov-Uvarov (NU) method. The energy eigenvalues and their corresponding eigenfunctions are calculated for arbitrary -states in closed form. In the low screening region, when the screening parameter the potential reduces to Kratzer potential. Further, some numerical results are presented for several diatomic molecules.
Bound-state spectra of shifted Deng-Fan oscillator potential are studied by means of a generalized pseudospectral method. Very accurate results are obtained for \emph{both low as well as high} states by a non-uniform optimal discretization of the radial Schrödinger equation. Excellent agreement with literature data is observed in \emph{both $s$-wave and rotational} states. Detailed variation of energies with respect to potential parameters is discussed. Application is made to the ro-vibrational levels of four representative diatomic molecules (H$_2$, LiH, HCl, CO). Nine states having $\{n,\ell…
We propose an accurate and efficient method to compute vibrational spectra of molecules, based on exact diagonalization of an algebraically calculated matrix based on powers of Morse coordinate. The present work focuses on the 1D potential of diatomic molecules: as typical examples, we apply this method to the standard Lennard-Jones oscillator, and to the ab initio potential of the H2 molecule. Global cm-1 accuracy is exhibited through the H2 spectrum, obtained through the diagonalization of a 30 x 30 matrix. This theory is at the root of a new method to obtain globally accurate vibrational sp…
We propose a method, based on a Generalized Heisenberg Algebra (GHA), to reproduce the anharmonic spectrum of diatomic molecules. The theoretical spectrum generated by GHA allows us to fit the experimental data and to obtain the dissociation energy for the carbon monoxide molecule. Our outcomes are more accurate than the standard models used to study molecular vibrations, namely the Morse and the $q$-oscillator models and comparable to the perturbed Morse model proposed by Huffaker \cite{hf}, for the first experimental levels. The dissociation energy obtained here is more accurate than all pre…
Alchemical harmonic approximation based potential for iso-electronic diatomics: Foundational baseline for $Δ$-machine learning
Technische Universität Berlin（柏林工业大学）；Vector Institute for Artificial Intelligence（向量人工智能研究所）；University of Toronto（多伦多大学）；Berlin Institute for the Foundations of Learning and Data（柏林学习与数据基础研究所）；Acceleration Consortium, University of Toronto（多伦多大学加速联盟）
Explicit algebraic expressions for the expansion of the vibrational matrix elements in series of matrix elements on the wave functions of the ground vibrational state have been obtained for arbitrary sufficiently differentiable functions of the internuclear distance, arbitrary values v and v', and the potential curves whose ladder operators can be constructed. A diagram technique have been developed for it that consists in: 1) the numeration of the matrix elements by points of the 2D diagram with coordinates (l, k), 2) the drawing arrows between points of this diagram corresponding to the acti…
The Deng-Fan-Eckart (DFE) potential is as good as the Morse potential in studying atomic interaction in diatomic molecules. By using the improved Pekeris-type approximation, to deal with the centrifugal term, we obtain the bound-state solutions of the radial Schrödinger equation with this adopted molecular model via the Factorization Method. With the energy equation obtained, the thermodynamic properties of some selected diatomic molecules(H2 , CO , and ScN ) were obtained using Poisson summation method.. The unnormalized wave function is also derived. The energy spectrum for a set of diatomic…
An algebraic model taking into account the influence of the molecular rotation on the wave functions of vibrational-rotational states of the diatomic molecule using the formalism of the ladder operators and an expansion in a small parameter characterizing the vibrational-rotational interaction have been proposed for the potentials whose the creation and annihilation operators can be constructed. Expressions for the expansion of the wave function of the vibrational-rotational states in a set of wave functions of the rotationless vibrational states have been obtained of the second order in the p…
Accurate low and high-lying bound states of Tietz-Hua oscillator potential are presented. The radial Schrödinger equation is solved efficiently by means of the generalized pseudospectral method that enables optimal spatial discretization. Both $\ell=0$ and rotational states are considered. Ro-vibrational levels of six diatomic molecules \emph{viz.}, H$_2$, HF, N$_2$, NO, O$_2$, O$_2^+$ are obtained with good accuracy. Most of the states are reported here for the first time. A detailed analysis of variation of eigenvalues with $n, \ell$ quantum numbers is made. Results are compared with literat…
The computation of vibrational spectra of diatomic molecules through the exact diagonalization of algebraically determined matrixes based on powers of Morse coordinates is made substantially more efficient by choosing a properly adapted quantum-mechanical basis, specifically tuned to the molecular potential. A substantial improvement is achieved while still retaining the full advantage of the simplicity and numerical light-weightedness of an algebraic approach. In the scheme we propose, the basis is parameterized by two quantities which can be adjusted to best suit the molecular potential thro…
