Morse Oscillator Research Articles

Vibrational spectrum of a 1D oscillator: The quantum, the Wigner, and the classical ways

AbstractInfrared spectra have been used in many chemical applications, and theoretical calculations have been useful for analyzing these experimental results. While quantum mechanics is used for calculating the spectra for small molecules, classical mechanics is used for larger systems. However, a systematic understanding of the similarities and differences between the two approaches is not clear. Previous studies focused on peak position and relative intensities of the spectra obtained by various quantum and classical methods, but here, we included “absolute” intensities in the evaluation. The infrared spectrum of a one‐dimensional (1D) harmonic oscillator (HO) and Morse oscillator were examined using four treatments: quantum, Wigner, truncated Wigner, and classical microcanonical treatments. For a 1D HO with a linear dipole moment function (DMF), the quantum and Wigner treatments give nearly the same spectra. On the other hand, the truncated Wigner underestimates the fundamental transition's intensity by half. In the case of cubic DMF, the truncated Wigner and classical methods fail to reproduce the relative intensity between the fundamental and second overtone transitions. Unfortunately, all the Wigner and classical methods fail to agree with the quantum results for a Morse oscillator with just 1% anharmonicity.

Algebraic vibrational description of the symmetric isotopologues of CO2: 13C16O2, 12C18O2 and 12C17O2

• A polyad preserving Hamiltonian involving interactions up to sixth order is used to describe the vibrational degrees of freedom of the symmetric carbon dioxide isotopologues 13 C 16 O 2 , 12 C 17 O 2 and 12 C 18 O 2 . • Different polyad schemes are explored to consider the more relevant interactions. • The connection between the algebraic description and the configuration space is outlined. • Predictions of energy levels of 13 C 16 O 2 , 12 C 17 O 2 and 12 C 18 O 2 are provided up to ∼ 24000 cm −1 . We present a polyad-conserving local model applied to the vibrational excitations of symmetric isotopologues of carbon dioxide. The description is carried out in the framework of local internal coordinates where the stretching degrees of freedom are modeled with Morse oscillators while for the bending modes the algebraic U ( 3 ) model is introduced. Three isotopologues are considered, namely 13 C 16 O 2 , 12 C 18 O 2 and 12 C 17 O 2 in their electronic ground states. In all cases the most extensive experimental dataset of vibrational energies were taken into account. For the three isotopologues a Hamiltonian with interactions up to sixth order involving 16 parameters was considered. The description of the isotopologue 13 C 16 O 2 involved 110 experimental energies with a rms = 0.06 cm −1 . On the other hand, for the isotopologue 12 C 18 O 2 , only 42 experimental energies were available and a rms = 0.05 cm −1 was obtained with 15 parameters. For the isotopologue 12 C 17 O 2 , considering 28 experimental energy terms a rms = 0.17 cm −1 was obtained. In addition, as a consequence of the normal mode behavior of these molecules, alternative polyad schemes were explored.

Morse oscillator equation of state: An integral equation theory based with virial expansion and compressibility terms

A number of interaction energy types are employed in the vibrations studies, especially in the spectroscopic analysis, such as the harmonic oscillator and Morse oscillator. In this research, a derivation of an analytical formula of equation of state of Morse oscillator is considered by employing the approximations used in the simple fluids theory. The compressibility formula of the pressure and the virial expansion formula of the pressure using the solutions of the main equation of the simple fluids theory with one of the approximations of the theory are employed for the purpose of the derivation. The virial coefficients of the total Morse oscillator pressure (the first order one, and the second order one) are found for Morse oscillator with respect to the fractional volume of the components, where we conclude that the first order term is proportional to the absolute temperature directly and depends on the diameter of the particles, while we concluded that the second order coefficient term is more complicated than the first order one with temperature, and also, depends on the three Morse oscillator parameters and the diameter of the particles. Besides, we conclude that the total pressure of Morse oscillator, generally, depends on the minimum energy of the well of Morse oscillator, the width parameter of Morse oscillator, and the equilibrium bond distance of the oscillator, in addition to their dependence on the absolute temperature of the components, and the diameter of the particles. The formula of the Morse oscillator equation of state which is found in this research can be applied to multiple materials described using Morse oscillator such as lots of dimers in the vibrations spectroscopy.

On the exact revival of Morse oscillator wave packets

On the exact revival of Morse oscillator wave packets

An alternative method to obtain analytical solutions of the time-independent Schrödinger equation

ABSTRACT An alternative method for obtaining analytical solutions of the time-independent Schrödinger equation (TISE) is developed and applied to 1-dimensional model quantum systems of interest in atomic and molecular physics. The method is based on proposing a wave function that is a product of two functions, one of them an integrating factor. The differential equation resulting from operating the TISE on the product wave function is compared with a differential equation with a known solution. This comparison produces a system of differential equations for the integrating factor. This system of equations is easily solved for model quantum systems such as the harmonic oscillator, the hydrogen atom and the Morse oscillator. The method presented simplifies some approaches shown in textbooks, based on asymptotic analyses of the TISE, and power series methods with recurrence relations. In addition, the method presented here facilitates the understanding of the relationship between the ordinary differential equations of the mathematical physics and the TISE of physical models as the harmonic oscillator, the Hydrogen atom, and the Morse oscillator.

ExoMol molecular line lists – XLIII. Rovibronic transitions corresponding to the close-lying X 2Π and A 2Σ+ states of NaO

ABSTRACT The sodium monoxide radical (NaO) is observed in night-glow in the Earth’s mesosphere and likely has astronomical importance. This study concerns the optical transitions within the ground X 2Π state and to the very low-lying (Te ≈ 2000 cm−1) excited A 2Σ+ state. A line list consisting of rovibronic term values, allowed electric dipole transitions, Einstein coefficients, and partition functions for varying temperature are produced using a variational solution of the coupled-channel Schrödinger equations using the program duo. multi-reference configuration interaction (MRCI) ab initio calculations characterizing the potential energy curves of the two states, spin-orbit and L-uncoupling non-adiabatic matrix elements, as well as permanent and transition dipole moments were integral in the formation of the final deperturbation model. Ab initio potential energy curves are represented in the analytical Extended Morse Oscillator form and refined, along with the spin-orbit and L-uncoupling functions, by least-squares fitting to the available spectroscopic data. The input experimental data consisted of pure rotational transitions within the fine-structure components of the X 2Π state for v″ ∈ [0, 3] vibrational levels as well as the rovibronic A 2Σ+(v′ = 0) ← X 2Π(v″ = 0) transitions, both with limited coverage over rotational excitation. The lack of data detailing the vibrational structure of the X and A states points to the need for further experimental study of higher excited levels, which would provide a more robust spectroscopic model. The NaO NaOUCMe line list is available via www.exomol.com and the CDS data base.

A flexible approach to vibrational perturbation theory using sparse matrix methods.

A sparse linear algebra based implementation of Rayleigh-Schrödinger vibrational perturbation theory is presented. This implementation allows for flexibility in the coordinates used to expand the vibrational Hamiltonian as well as the order to which the perturbation theory is performed. It also provides a powerful tool for investigating the origin of spectral intensity and transition frequencies. Specifically, this flexibility allows for the analysis of which terms in the expansions of the Hamiltonian and dipole surface lead to the largest corrections to the energies and transition intensities, and how these conclusions depend on the coordinates used for these expansions. Comparisons of corrections to transition frequencies are reported for the Morse oscillator when the potential is expanded in Δr and Morse coordinates as well as for water, water dimer, and peroxynitrous acid when the molecular Hamiltonians and dipole surfaces are expanded in Cartesian displacement coordinates and in the displacements of the bond-angle-dihedral internal coordinates. Further comparisons of the corrections to the transitions moments are made for H2O and (H2O)2. It is found that while the transition frequencies and intensities are independent of coordinate choice, a good choice of coordinates leads to a cleaner interpretation of the origins of the anharmonicities in these systems.

Compacting the density matrix in quantum dynamics: Singular value decomposition of the surprisal and the dominant constraints for anharmonic systems.

We introduce a practical method for compacting the time evolution of the quantum state of a closed physical system. The density matrix is specified as a function of a few time-independent observables where their coefficients are time-dependent. The key mathematical step is the vectorization of the surprisal, the logarithm of the density matrix, at each time pointof interest. The time span used depends on the required spectral resolution. The entire course of the system evolution is represented as a matrix where each column is the vectorized surprisal at the given time point. Using the singular value decomposition (SVD) of this matrix, we generate realistic approximations for the time-independent observables and their respective time-dependent coefficients. This allows for a simplification of the algebraic procedure for determining the dominant constraints (the time-independent observables) in the sense of the maximal entropy approach. A non-stationary coherent initial state of a Morse oscillator is used to introduce the approach. We derive the analytical exact expression for the surprisal as a function of time, and this offers a benchmark for comparison with the accurate but approximate SVD results. We discuss two examples of a Morse potential of different anharmonicities, H2 and I2 molecules. We further demonstrate the approach for a two-coupled electronic state problem, the well-studied non-radiative decay of pyrazine from its bright state. Five constraints are found to be enough to capture the ultrafast electronic population exchange and to recover the dynamics of the wave packet in both electronic states.

Supersymmetry of 𝒫𝒯-symmetric tridiagonal Hamiltonians

We extend the study of supersymmetric tridiagonal Hamiltonians to the case of non-Hermitian Hamiltonians with real or complex conjugate eigenvalues. We find the relation between matrix elements of the non-Hermitian Hamiltonian [Formula: see text] and its supersymmetric partner [Formula: see text] in a given basis. Moreover, the orthogonal polynomials in the eigenstate expansion problem attached to [Formula: see text] can be recovered from those polynomials arising from the same problem for [Formula: see text] with the help of kernel polynomials. Besides its generality, the developed formalism in this work is a natural home for using the numerically powerful Gauss quadrature techniques in probing the nature of some physical quantities such as the energy spectrum of [Formula: see text]-symmetric complex potentials. Finally, we solve the shifted [Formula: see text]-symmetric Morse oscillator exactly in the tridiagonal representation.

Optimal series expansion method for bound states of quantum systems

An optimal series expansion method is developed to improve variational wave functions for the bound state calculation of quantum systems. With one variational parameter residing in the exponential decay factor, the finite power series expansion of the remaining factor in the trial wave function is determined with the criterion that the Schrödinger equation be satisfied as well as possible. In this way, more and more accurate trial wave functions can be obtained without undesirably increasing the number of variational parameters. Thus, instead of the standard variational principle of minimizing the energy expectation value, we utilize an alternative variational principle with which one minimizes the deviation of the trial wave function from being a true eigenstate of the Hamiltonian. The variational principle applies to both the ground state and the excited states. We then employ the optimal series expansion method to calculate the bound states of several quantum systems, including the one-dimensional double well potential, the Morse oscillator, the two-dimensional double well potential, and the vibrational states in the lowest electronic surface of methyl iodide. Computational results indicate that this method can yield excellent results for both energy eigenvalues and eigenfunctions. Therefore, the optimal series expansion method provides an alternative approach to the bound state calculation of quantum systems.

Vibrational Hamiltonian of tetrachloro-, tetrafluoro-, and mono- silanes using U(2) Lie algebras

In this paper, we have applied the symmetry adapted one-dimensional framework of the U(2) Lie algebras to estimate the vibrational frequencies of tetrachloro-, tetrafluoro-, and mono- silanes in the gas phase having the spectroscopic interest of terrestrial volcanic plumes and other planetary atmospheres. A vibrational Hamiltonian that preserves the Td point group symmetry of each of these silane molecules is devised using ten interacting Morse oscillator bound state spectra. The calculated vibron numbers and locality parameters indicate that the vibrational motion is highly anharmonic in SiH4 (nearest to local mode), moderately anharmonic in SiF4 (mixed mode) and the least anharmonic in SiCl4 (near to local mode). RMS deviations of the derived vibrational frequencies [0.41 cm−1 (SiH4), 0.83 cm−1 (SiCl4), and 0.63 cm−1 (SiF4)], with reference to their experimental counterparts, assert that the U(2) Lie algebraic Hamiltonian is successful in deriving all the fundamental vibrations, their higher overtones and combination bands up to the fifth excitation, of each of the three silane molecules at the sub-cm−1 level of accuracy, possibly at a much lower computational cost as compared to other theoretical methods.

More about properties of Morse oscillator

Using Birge-Sponer extrapolation we have analyzed the approximation of the potential of a real diatomic molecule by the Morse model, which implies a constant value of anharmonicity ωx. The real values of ωx*(v) for each vibrational level are estimated from transition frequencies between neighboring levels. The dependence of ωx* on the vibrational quantum number v up to dissociation is calculated from the literature data for the ground electronic state of H2, O2, Be2, Li2, ArXe, Xe2, Kr2 and the excited state of Li2. Characteristic features of deviations of the anharmonicity parameter x* - x from the Morse model are described.

Enhanced optical nonlinearities under collective strong light-matter coupling

Optical microcavities and metallic nanostructures have been shown to significantly modulate the dynamics and spectroscopic response of molecular systems. We present a study of the nonlinear optics of a model consisting of $N$ anharmonic multilevel systems (e.g., Morse oscillators) undergoing collective strong coupling with a resonant infrared microcavity. We find that, under experimentally accessible conditions, molecular systems in microcavities may have nonlinear phenomena significantly intensified due to the high quality of polariton resonances and the enhanced microcavity electromagnetic energy density relative to free space. Particularly large enhancement of multiphoton absorption happens when multipolariton states are resonant with bare molecule multiphoton transitions. In particular, our model predicts two-photon absorption cross section enhancements by several orders of magnitude relative to free space when the Rabi splitting $\Omega_R$ is approximately equal to the molecular anharmonic shift $2\Delta$. Our results provide rough upper bounds to resonant nonlinear response enhancement factors as relaxation to dark states is treated phenomenologically. Notably, ensembles of two-level systems undergoing strong coupling with a cavity (described by the Tavis-Cummings model) show no such optical nonlinearity enhancements, highlighting the rich phenomenology afforded by multilevel anharmonic systems. Similar conclusions are expected to hold for excitonic systems that share features with our model (e.g., molecular dyes with accessible S_0 -> S_1 -> S_2 transitions) and strongly interact with a UV-visible cavity.

Two-dimensional electron gas in a non-Euclidean space

A charged particle in the presence of a perpendicular magnetic field is studied within the position-dependent translation operator formalism. We show that the solution of the resulting Schrödinger-like equation is mapped on the Morse oscillator, instead of the expected quantum harmonic oscillator. The anharmonicity that shows up naturally from the theory is analogous to corrections introduced by relativistic effects. Due to the non-Euclidean space, it is demonstrated that: (i) the degeneracy of the spin up and spin down levels is lifted and (ii) the Fermi energy and the total ground state energy of the two-dimensional electron gas is also modified.

Vibrational Hamiltonian of Naphthalene (C10H8) Using Dynamical U(2) Lie Algebras

This article is the first to demonstrate the application of dynamical U(2) Lie algebras to model a vibrational Hamiltonian of naphthalene (C10H8), a prototype polycyclic aromatic hydrocarbon molecule containing a fused pair of benzene rings, in the gas phase having considerable astrophysical interest. A vibrational Hamiltonian that preserves the D2h symmetry of naphthalene is modeled using 19 coupled one-dimensional Morse oscillators to describe the eight C-H single bonds, the eleven C-C aromatic bonds, and their dynamic interactions, making use of the Casimir and Majorana operators. Results indicate that the dynamical U(2) Lie algebraic Hamiltonian has successfully reproduced the vibrational frequencies at the 48 fundamental excitations near to the level of spectroscopic accuracy, as well as predicted their first three overtones. The observed degeneracy of the stretching, bending, and combination vibrations at the fundamental and lower overtone excitations are expected to modify the shape of the two major bands (centered at 782 cm−1 and 3065 cm−1), and several minor bands in the infrared absorption spectrum of naphthalene in the gas phase.

Accuracy of Morse and Morse-like oscillators for diatomic molecular interaction: A comparative study

This study aims at comprehensively evaluating and comparing the goodness-of-fit of several potential energy curves (PECs), which are Morse, Manning-Rosen, Rosen-Morse and Modified-Rosen-Morse oscillators, to the corresponding experimental Rydberg-Klein-Rees (RKR) or RKR-style data of diatomic molecules. In this work, we also determined vibrational energies of the diatomics based on Bohr-Sommerfeld quantization rule. The goodness-of-fit to RKR data of 34 electronic states of 17 diatomics and the accuracy of the calculated vibrational energies were compared for the four oscillators and reported simultaneously for the first time in this study. These results showed that some oscillators demonstrated better goodness-of-fit than the others. Overall, the (Modified)-Rosen-Morse oscillators generated more accurate vibrational levels for most molecules studied, while Manning-Rosen oscillator was generally better for molecules in their excited electronic states. Although all oscillators performed well for more classical molecules, the Morse oscillator was superior to the other three in describing the ground states of 12C16O and 28Si32S. Particularly, it reproduced all 38 observed experimental ground state vibrational levels of 12C16O with error below 0.79% and even accurately predicted four states unobserved experimentally.

On the Method of Correct Determination of Eigenvalues of a Truncated Hamiltonian Matrix on the Example of a Morse Oscillator

A method of precise determination of the eigenvalues of energies of the molecular Hamiltonian based on high-order perturbation theory has been developed and implemented for a diatomic molecule. The proposed method makes it possible not only to obtain the energy values, but also to estimate the accuracy of their prediction and the applicability limits for the employed model. Numerical calculations have been performed for the extended Morse oscillator with corrections for the potential functions up to the sixth power of the Morse coordinate. The results obtained are compared with the results of calculation for the model of the truncated Hamiltonian matrix. The possibilities of application of the method are analyzed compared to other approaches to the determination of the potential functions for polyatomic molecules.

Eigensolutions and expectation values of shifted-rotating Möbius squared oscillator

In this paper we have employed exact quantization rule and the Hellman–Feynman theorem to obtain bound states eigensolutions and expectation values of the shifted-rotating Mobius squared oscillator, and we have used the Pekeris-like approximation recipe to effect solution for bound states energy eigenspectra, and in solving the Riccati equation for the eigenfunctions, cases of Q ≠ 0 and Q = 0 were considered in our solutions. We have also applied our derived formulas to compute numerical values to the spectroscopic parameters of four diatomic molecules: CO, LiH, H2 and ICl. Our analysis showed that numerical results we obtained for potential parameters, bound states energy eigenspectra and expectation values are consistent with available data in the literature and that molecular interactions in CO, LiH, H2 and ICl are best described by improved Wei oscillator than by Morse oscillator.

Genetic Algorithm for quick finding of diatomic molecule potential parameters

ABSTRACT 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 an analytical function. GA was tested on two artificially generated datasets which base on potentials with known characteristics and two laser induced fluorescence 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 indicated that changing the potential model from a Morse, which is frequently used as a starting potential in the inverse perturbation approach method, to an expanded Morse oscillator leads to noticeable improvement of agreement between simulated and experimental data.

The shape of the electric dipole function determines the sub-picosecond dynamics of anharmonic vibrational polaritons

Vibrational strong coupling has emerged as a promising route for manipulating the reactivity of molecules inside infrared cavities. We develop a full-quantum methodology to study the unitary dynamics of a single anharmonic vibrational mode interacting with a quantized infrared cavity field. By comparing multi-configurational time-dependent Hartree simulations for an intracavity Morse oscillator with an equivalent formulation of the problem in Hilbert space, we describe for the first time the essential role of permanent dipole moments in the femtosecond dynamics of vibrational polariton wavepackets. We classify molecules into three general families according to the shape of their electric dipole function de(q) along the vibrational mode coordinate q. For polar species with a positive slope of the dipole function at equilibrium, an initial diabatic light-matter product state without vibrational or cavity excitations evolves into a polariton wavepacket with a large number of intracavity photons for interaction strengths at the conventional onset of ultrastrong coupling. This buildup of the cavity photon amplitude is accompanied by an effective lengthening of the vibrational mode that is comparable with a laser-induced vibrational excitation in free space. In contrast, polar molecules with a negative slope of the dipole function experience an effective mode shortening, under equivalent coupling conditions. We validate our predictions using realistic ab initio ground state potentials and dipole functions for HF and CO2 molecules. We also propose a non-adiabatic state preparation scheme to generate vibrational polaritons with molecules near infrared nanoantennas for the spontaneous radiation of infrared quantum light.