Morse Oscillator Research Articles

Connecting the Dunham Expansion to the Dissociation Limit for Interatomic Potentials: Application to Lennard-Jones m- n Potentials.

Dunham generated the expansion for energy levels of a rotating, vibrating diatomic molecule from an expansion of the potential about the equilibrium position. For partition functions, however, the energy levels are needed all the way to dissociation. Analytic Morse oscillator energies are not very useful because the exponential decay of the Morse potential is much too short-ranged for any physical system. The longer-range Lennard-Jones 12-6 potential could be used, but quantum energies have not previously been conveniently fit. I show how Dunham coefficients begin a set of asymptotic functions for any interaction potential, one function arising from each successive term in the WKB expansion. I apply this to the family of Lennard-Jones m- n (LJ m- n) potentials with an R-m repulsive term and R-n attractive term ( m > n) and demonstrate how m can be used as a parameter to adjust either the equilibrium distance or harmonic frequency. I present an empirical parametrization of LJ m- n vibrotor energies starting with Dunham coefficients generated from four terms in the WKB expansion. This information is combined with data from numerically solved energies and asymptotic limits to fit the functions all the way to dissociation. One can also treat exp-6 and similar model potentials with different repulsive parts using the same method because the expansion form is controlled by the long-range part of the potential.

Infrared Spectral Assignment of Pyrimidine and Pyrazine in the C[sbnd]H Stretching Region by an Effective Spectroscopic Hamiltonian

Mid-Infrared spectra of pyrimidine (PM) and pyrazine (PZ) were recorded in the gas phase using a multi-pass long path gas cell. The IR band structure of these compounds above and below 3000 cm−1 is very broad and contains many humps and shoulders. These humps and shoulders are due to various higher quantum excitation of low-frequency vibrational modes, which participate in Fermi resonance with the nearby CH stretch fundamentals and appears in this region. We constructed an Effective Spectroscopic Hamiltonian (ESH) in a mixed local mode (LM) normal mode (NM) basis to assign the various overtone and combination bands in the CH stretching region of these compounds. The CH stretching vibrations of both PZ and PM were treated as symmetrized anharmonic Morse oscillators in local coordinates and the in-plane deformations down to 1000 cm−1 were treated as normal coordinates. The ESHs were diagonalized and the resulting eigenvalues were subsequently fitted in a given parameter space with the experimentally observed bands. The eigenvalues of the converged Hamiltonian are the anharmonic frequencies and the transition intensities were obtained by summing the squared eigenvector components. The overtone and combination transitions near 3000 cm−1 of both PM and PZ were identified and assigned from the eigenvector coefficients of the ESH matrix. The wavefunctions of a pure CH stretch, overtone of the HCC in-plane bend and due to Type 1 Fermi coupling (resonance between a fundamental with an overtones of a low frequency mode, in this case resonance between the CH strech and the overtone of HCC in-plane bending modes) has been demonstrated pictorially.

Relative Fisher information for Morse potential and isotropic quantum oscillators

The relative Fisher information associated with quantum-mechanical states for two prototypical models is presented; the radial density functions that characterize the Morse oscillator and the d-dimensional isotropic quantum oscillators are analyzed in terms of this relative information by specifying the ground state as the reference. We find that the information increases as the eigenvalue (i.e., the degree of the associated Laguerre polynomials) increases for the Morse density function, but for the isotropic quantum harmonic models, it decreases with the dimension and increases almost linearly with the increasing quantum numbers. We present also the explicit expression for the information associated with the angular density as a function of the orbital quantum number and the magnetic quantum number for the three-dimensional isotropic quantum oscillator.

Line list for the ground state of CaF

The molecular potential energy function and electronic dipole moment function for the ground state of CaF were studied with MRCI, ACPF, and RCCSD(T) ab initio calculations. The RCCSD(T) potential function reproduces the experimental vibrational intervals to within ∼2 cm−1. The RCCSD(T) dipole moment at the equilibrium internuclear separation agrees well with the experimental value. Over a wide range of internuclear separations, far beyond the range associated with the observed spectra, the ab initio dipole moment functions are similar and highly linear. An extended Morse oscillator (EMO) potential function was also obtained by fitting the observed lines of the laboratory vibration-rotation and pure rotation spectra of the 40CaF X2Σ+ ground state. The fitted potential reproduces the observed transitions (v ≤ 8, N ≤ 121, Δv = 0, 1) within their experimental uncertainties. With this EMO potential and the RCCSD(T) dipole moment function, line lists for 40CaF, 42CaF, 43CaF, 44CaF, 46CaF, and 48CaF were computed for v ≤ 10, N ≤ 121, Δv = 0–10. The calculated emission spectra are in good agreement with an observed laboratory spectrum of CaF at a sample temperature of 1873 K.

Morse oscillator propagator in the high temperature limit II: Quantum dynamics and spectroscopy

This paper is a continuation of Paper I (Toutounji, 2017) of which motivation was testing the applicability of Morse oscillator propagator whose analytical form was derived by Duru (1983). This is because the Morse oscillator propagator was reported (Duru, 1983) in a triple-integral form of a functional of modified Bessel function of the first kind, which considerably limits its applicability. For this reason, I was prompted to find a regime under which Morse oscillator propagator may be simplified and hence be expressed in a closed-form. This was well accomplished in Paper I. Because Morse oscillator is of central importance and widely used in modelling vibrations, its propagator applicability will be extended to applications in quantum dynamics and spectroscopy as will be reported in this paper using the off-diagonal propagator of Morse oscillator whose analytical form is derived.

Application of the mixed time-averaging semiclassical initial value representation method to complex molecular spectra.

The recently introduced mixed time-averaging semiclassical initial value representation of the molecular dynamics method for spectroscopic calculations [M. Buchholz, F. Grossmann, and M. Ceotto, J. Chem. Phys. 144, 094102 (2016)] is applied to systems with up to 61 dimensions, ruled by a condensed phase Caldeira-Leggett model potential. By calculating the ground state as well as the first few excited states of the system Morse oscillator, changes of both the harmonic frequency and the anharmonicity are determined. The method faithfully reproduces blueshift and redshift effects and the importance of the counter term, as previously suggested by other methods. Different from previous methods, the present semiclassical method does not take advantage of the specific form of the potential and it can represent a practical tool that opens the route to direct ab initio semiclassical simulation of condensed phase systems.

A tunable mechanism to control photo-dissociation with invariant tori with variable energies

In this work, we investigate the classical dynamics of a polar diatomic molecule under the action of a space and time-dependent laser field. In the absence of laser fields, the phase space consists of a bound and an unbound region divided by a separatrix. When an interacting laser field is present, some surviving invariant tori are deformed. Therefore, the initial conditions in the bound region which are over the surviving tori may visit the unbound region but still undergoing a bounded motion. Based on this analysis, we propose a mechanism to control the photo-dissociation process using the driven Morse oscillator as a model.

Polyad breaking phenomenon associated with a local-to-normal mode transition and suitability to estimate force constants

ABSTRACTIn a system of two interacting harmonic oscillators a local-to-normal mode transition is manifested as a polyad breaking phenomenon. This phenomenon is associated with the suitability to estimate zeroth-order force constants in the framework of a local mode description. This transition is also exhibited in two interacting Morse oscillators. To study this case, an appropriate parameterisation going from a molecule with local mode behaviour (H2O) to a molecule presenting a normal mode behaviour (CO2) is introduced. Concepts from quantum mechanics like fidelity, entropy and probability density, as well from nonlinear classical mechanics like Poincaré sections are used to detect the transition region. It is found that fidelity and entropy are sensitive complementary properties to detect the local-to-normal transition. Poincaré sections allow the local-to-normal transition to be detected through the appearance of chaos as a consequence of the polyad breaking phenomenon. In addition, two kinds of avoided energy crossings are identified in accordance with the different regions of the spectrum.

Line list for the MgF ground state

An extended Morse oscillator (EMO) potential function was obtained by fitting the observed laboratory vibration-rotation and pure rotational spectra of the 24MgF X2Σ+ ground state. The fitted potential reproduces the observed transitions within the observation uncertainties. With this EMO potential and an analytic dipole moment function in the form of a Padé approximant fitted using ab initio dipole moment data, line lists for 24MgF, 25MgF and 26MgF were computed for v≤8, N≤100, Δv=0–8. It was discovered that directly using the ab initio dipole moment points with cubic spline interpolation to calculate line intensities worked for Δv<3, but failed for higher Δv values. A simple solution was found by fitting the ab initio dipole points with a suitable analytical dipole moment function. The calculated emission spectra are in good agreement with an observed laboratory spectrum with the MgF sample at 1823K.

Q-Deformed Morse and Oscillator Potential

We studied the q-deformed Morse and harmonic oscillator systems with appropriate canonical commutation algebra. The analytic solutions for eigenfunctions and energy eigenvalues are worked out using time-independent Schrödinger equation and it is also noted that these wave functions are sensitive to variation in the parameters involved.

Coexistence of Multiple Attractors, Hysteresis, and Vibrational Resonance in the Classical Morse Oscillator Driven by an Amplitude Modulated Signa

Coexistence of Multiple Attractors, Hysteresis, and Vibrational Resonance in the Classical Morse Oscillator Driven by an Amplitude Modulated Signa

NEW QUANTUM APPROACH TO DETERMINATION OF THE MOLECULAR SPECTRAL CONSTANTS AND PROBABILITIES FOR COOPERATIVE VIBRATIONROTATION-NUCLEAR TRANSITIONS IN SPECTRA OF DIATOMICS AND THE HADRONIC MOLECULES

It is proposed a new approach to construction of the potential function of diatomic molecules as a sum of the known perturbed Morse oscillator function, the Simons-Parr-Finlan molecular potential in the middle of the potential curve, function of the -Cn/Rn type at the large internuclear distances. Within this approach it is presented a precise scheme for computing the molecular spectral parameters, namely, vibrational, rotational, centrifugal constants for the electronic states of diatomics. As application it was carried out calculation of the of molecular constants (cm-1) for the X1Σ+ B1Π states of the KRb dimer and rubidium dimer and performed further comparison with experimental data. Within consistent approach to calculation of the electron-nuclear γ transition spectra (set of vibrationrotational satellites in molecule) of molecule there are obtained the estimates for vibration-rotationnuclear transition probabilities in a case of the emission and absorption spectrum of nucleus 127I(E(0)g = 203 keV) in the molecule of H127I for different approximations of the for potential curves: the hadmonic oscillator, the Dunham model and presented approach.

Morse oscillator propagator in the high temperature limit I: Theory

In an earlier work of the author the time evolution of Morse oscillator was studied analytically and exactly at low temperatures whereupon optical correlation functions were calculated using Morse oscillator coherent states were employed. Morse oscillator propagator in the high temperature limit is derived and a closed form of its corresponding canonical partition function is obtained. Both diagonal and off-diagonal forms of Morse oscillator propagator are derived in the high temperature limit. Partition functions of diatomic molecules are calculated.

Supersymmetry of the Morse Oscillator

While dealing in [1] with the supersymmetry of a tridiagonal Hamiltonian H, we have proved that its partner Hamiltonian H(+) also have a tridiagonal matrix representation in the same basis and that the polynomials associated with the eigenstates expansion of H(+) are precisely the kernel polynomials of those associated with H. This formalism is here applied to the case of the Morse oscillator which may have a finite discrete energy spectrum in addition to a continuous one. This completes the treatment of tridiagonal Hamiltonians with pure continuous energy spectrum, a pure discrete one, or a spectrum of mixed discrete and continous parts.

Forward Period Analysis Method of the Periodic Hamiltonian System

Using the forward period analysis (FPA), we obtain the period of a Morse oscillator and mathematical pendulum system, with the accuracy of 100 significant digits. From these results, the long-term [0, 1060] (time unit) solutions, ranging from the Planck time to the age of the universe, are computed reliably and quickly with a parallel multiple-precision Taylor series (PMT) scheme. The application of FPA to periodic systems can greatly reduce the computation time of long-term reliable simulations. This scheme provides an efficient way to generate reference solutions, against which long-term simulations using other schemes can be tested.

A study of vibrational excitations of ozone in the framework of a polyad preserving model of interacting Morse oscillators

The vibrational spectroscopic description of the ozone molecule 16O3 in its electronic ground state X1A1 is presented in the framework of a simple local model, where Morse potentials are associated with both stretching and bending modes. The Hamiltonian is written in terms of internal coordinates considering the local mode character of the ozone molecule. Later on an algebraic representation in terms of Morse ladder operators is introduced through a linear approximation in the expansion of the coordinates and momenta. Three polyads are considered in our study: P11=ν1+ν3+ν2, P21=2(ν1+ν3)+ν2, and P32=3(ν1+ν3)+2ν2, as suggested by resonances derived from the fundamentals as well as from previous variational analysis. The best description is provided by the P11 polyad scheme, yielding an rms deviation of 1.85cm−1 for a fit involving 121 energy levels. Considering the other two polyads the description is less accurate: rms=2.78cm−1 for polyad P21 and rms=2.63cm−1 for polyad P32, considering 99 and 100 energy levels, respectively. These fits represent the best descriptions in the framework of an algebraic approach. In addition, since our algebraic model keeps the connection with configuration space, the force constants derived from the three fits have been estimated. We have found that all the available experimental energies may be assigned at least to one of the three fits. As the energy increases the eigenstates obtained from different polyad schemes differ. This fact paves the way to establish a polyad breaking approach as a next step to improve the description.

Fidelity, entropy, and Poincaré sections as tools to study the polyad breaking phenomenon

In search of a region where a local mode model stops being adequate to estimate the local force constants, the correlation diagram of the vibrational energy spectra associated with the stretching modes of triatomic molecules such as CO2 and H2O is analyzed by means of two interacting Morse oscillators. By considering a linear dependence of the structure and force constants, it is shown that the fidelity, entropy and Poincaré sections detect the polyad breaking process manifested in the transition from local to normal mode behaviors. Additionally Poincaré sections show a transition to chaos where the local polyad cannot be defined.

Vibrational Resonance in the Classical Morse Oscillator Driven by Narrow-Band and Wide-Band Frequency Modulated Signals

The phenomenon of vibrational resonance (VR) in the classical Morse oscillator influenced by narrow band and wide band frequency modulated signals is numerically studied. Vibrational resonance is found to occur when the amplitudes f and g and frequencies ω and Ω of the signals are varied. The dynamics of the system is studied in the presence of both signals separately. Vibrational resonance and dynamics of the system are characterized using the response amplitude and bifurcation diagram.

Vibrational energies and full analytic potential energy functions of PbI and InI from pure microwave data

Pure rotational spectra of PbI and InI are interpreted to yield a full analytic potential energy function for each molecule. Rotational spectra for PbI have been retrieved from literature sources to perform the analysis. Rotational transition frequencies for excited vibrational states of InI (0<v<11) are measured during this work. Ignoring hyperfine splittings, Bv and Dv values are used to generate a set of “synthetic” pure R(0) transitions for each vibrational level. These are then fitted to an “Expanded Morse Oscillator” (EMO) potential using the direct-potential-fit program, dPOTFIT. The well-depth parameter, De, is fixed at a literature value, while values of the equilibrium distance re and EMO exponent-coefficient expansion (potential-shape) parameters are determined from the fits. Comparison with potential functions determined after including older mid-IR and visible electronic transition data shows that our analysis of the pure microwave data alone yields potential energy functions that accurately predict (to better than 1%) the overtone vibrational energies far beyond the range spanned by the levels for which the microwave data is available.

Accurate non-adiabatic quantum dynamics from pseudospectral sampling of time-dependent Gaussian basis sets

Quantum molecular dynamics requires an accurate representation of the molecular potential energy surface from a minimal number of electronic structure calculations, particularly for nonadiabatic dynamics where excited states are required. In this paper, we employ pseudospectral sampling of time-dependent Gaussian basis functions for the simulation of non-adiabatic dynamics. Unlike other methods, the pseudospectral Gaussian molecular dynamics tests the Schrödinger equation with N Dirac delta functions located at the centers of the Gaussian functions reducing the scaling of potential energy evaluations from O(N2) to O(N). By projecting the Gaussian basis onto discrete points in space, the method is capable of efficiently and quantitatively describing the nonadiabatic population transfer and intra-surface quantum coherence. We investigate three model systems: the photodissociation of three coupled Morse oscillators, the bound state dynamics of two coupled Morse oscillators, and a two-dimensional model for collinear triatomic vibrational dynamics. In all cases, the pseudospectral Gaussian method is in quantitative agreement with numerically exact calculations. The results are promising for nonadiabatic molecular dynamics in molecular systems where strongly correlated ground or excited states require expensive electronic structure calculations.