Perturbed Morse Oscillator Research Articles

Dynamics of the Morse Oscillator: Analytical Expressions for Trajectories, Action-Angle Variables, and Chaotic Dynamics

We consider the one degree-of-freedom Hamiltonian system defined by the Morse potential energy function (the “Morse oscillator”). We use the geometry of the level sets to construct explicit expressions for the trajectories as a function of time, their period for the bounded trajectories, and action-angle variables. We use these trajectories to prove sufficient conditions for chaotic dynamics, in the sense of Smale horseshoes, for the time-periodically perturbed Morse oscillator using a Melnikov type approach.

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.

Calculation of spectroscopic constants of diatomic molecules with the help of the potential functions of the perturbed Morse oscillator

The spectroscopic molecular constants were calculated for the ground electronic states of copper, silver, and gold dimers. The calculation was performed based on the potential functions of the perturbed Morse oscillator whose parameters were determined in this work. The calculated values of the vibrational energy, the rotational constant, and the centrifugal-distortion constant were compared with the experimental data.

INTENSITY STUDY OF LIF SPECTRUM FOR THE B0 +u→X0 +g SYSTEM OF A TELLURIUM NATURAL SAMPLE

The fluorescence spectrum of a tellurium sample in its natural isotopic abundance, induced by the 4764.8 Å line of an argon ion laser is studied. The study of natural tellurium samples allows us to bring out and analyze the intensity of three fluorescence series not previously reported. Photographic plates and densitometric techniques were used when recording experimental intensities. The radiative transition probabilities for the observed fluorescence series were calculated by using hybrid Perturbed-Morse-Oscillator (PMO)-Rydberg–Klein–Rees (RKR)- Long Range potential energy curves for the B0 + u and X0 + g states and a function for the transition dipole moment in the framework of the r-centroid approximation. Radiative lifetimes of the rovibrational levels for the upper state pumped by the laser line have also been calculated. The transition probabilities and lifetimes are in good agreement with the corresponding observed measurements usually within the experimental uncertainty.

A quantal entropy signature for the dynamics of pure states: Studies on some model problems

Von Neuman-Shannon entropy is zero for a pure state and remains so at every stage of its unitary evolution. However, a pseudo or quantal entropy ( S Q) of a pure state reveals significant dynamical features during a unitary evolution. S Q can be identified, in a way with the entropy of ‘basis mixing’. The quantum evolution of a pure state is generally shown to be accompanied by an early loss of information followed by an information gain at a later stage. This happens irrespective of the adiabaticity or the lack of it in the evolution process. Several model problems studied tend to suggest generality of this behaviour. These include, the photo-dissociation dynamics of a diatomic molecule modeled by a Morse oscillator, photo-dissociation of a randomly perturbed Morse oscillator, destabilization of a random linear oscillator perturbed by quartic anharmonicity, generalized Rabi oscillations in a three-level metastable system, etc. Adiabatically switching from a classically regular to a classically chaotic region of a suitable Hamiltonian which is known to generate quantum chaos is shown to leave a recognizable signature on the quantum dynamics of S Q. This leads us to suggest that S Q can perhaps be used as a dynamic descriptor for the onset of quantum chaos.

Analysis and transition probabilities of the B1Πu → X1Σg + system of Na 2 excited by the 5017 Å line of the argon ion laser

The fluorescence spectrum of Na 2 induced by the 5017.16 Å line of an argon ion laser has been analyzed. This work extends previous observations on Na 2 by using this excitation line. We have observed seven fluorescence series for the B 1 Π u → X 1 Σ g + band system corresponding to the excitation transitions: ν′ = 5, J′ = 37 ← ν″ = 6, J″ = 38; ν′ = 5, J′ = 88 ← ν″ = 5, J″ = 87; ν′ = 5, J′ = 124 ← ν″ = 3, J″ = 125; ν′ = 0, J′ = 42 ← ν″ = 2, J″ = 43; ν′ = 10, J′ = 78 ← ν″ = 9, J″ = 77; ν′ = 14, J′ = 68 ← ν″ = 12, J″ = 68 and ν′ = 15, J′ = 58 ← ν″ = 13, J″ = 58. The latter four series are reported for the first time. The radiative transition probabilities for the observed fluorescence series are calculated by using hybrid Perturbed-Morse-Oscillator (PMO)-Rydberg-Klein-Rees (RKR)-long range potential energy curves for the B 1 Π u and X 1 Σ g + states and ab initio transition dipole moment function. Radiative lifetimes of the rovibrational levels of the upper state pumped by the laser line have also been calculated. The transition probabilities and lifetimes are in good agreement with the corresponding observed measurements usually within the experimental uncertainty. From the rotational satellite structure of some bands, collision-induced transition rates and average cross sections have been determined.

Perturbation theory using series expansions and the Riccati equation

An algebraic procedure is proposed for the analytical solution of Schrödinger equations that can be viewed as a factorizable equation with an adequately chosen perturbation. This procedure relies on the solution of the Riccati equation associated with the given eigenequation and the use of power series of suitable functions which are specific to each factorization type. As illustrative examples, analytical solution of the symmetric anharmonic oscillator, perturbed Morse oscillator and singular anharmonic oscillator equations are carried out. Further applications are pointed out.

On nonintegral E corrections in perturbation theory: application to the perturbed Morse oscillator

A new formulation of the Rayleigh–Schrödinger perturbation theory is applied to the derivation of the vibrational eigenvalues of the perturbed Morse oscillator (PMO). This formulation avoids the conventional projection of the Ψ corrections on the basis of unperturbed eigenfunctions [Formula: see text], or the projection of the nonhomogeneous Schrödinger equations on [Formula: see text], it gives simple expressions for each E correction [Formula: see text] free of summations and integrals. When the PMO is characterized by the potential U = UM + UP (where UM is the unperturbed Morse potential), the eigenvalue of a vibrational level ν is given by: [Formula: see text]. According to the new formulation the correction £, [Formula: see text] is given by [Formula: see text], where σp(r) is a particular solution of the nonhomogeneous differential equation y″ + f y = sp; here [Formula: see text], sp is well known for each p: for p = 0, [Formula: see text]; for [Formula: see text]. For the numerical application one single routine is used, that of integrating y″ + f y = s, where the coefficients are known as well as the initial values. An example is presented for the Huffaker PMO of the (carbon monoxide) CO-X1Σ+ state. The vibrational eigenvalues Eν are obtained to a good accuracy (with p = 4) even for high levels. This result confirms the validity of this new formulation and gives a semianalytic expression for the PMO eigenvalues.

Energy levels of a perturbed Morse oscillator

Analytic formulae for perturbation corrections (up to fourth order) to the eigenvalues of a perturbed Morse oscillator are derived. They are applied to calculate accurate rovibrational energy levels of diatomic molecules.

Accurate potentials for X1Σ+, A3Π1and B3Π+0 states of l35Cl from experimental data

A detailed investigation of the potential-energy curves up to dissociation for the low-lying X 1Σ+, A 3Π1 and B 3Π+0 states of l 35Cl has been performed on the basis of perturbed Morse oscillator–Rydberg Klein Rees–long-range potential calculations. New molecular constants necessary for construction of the potential functions were obtained from the available spectroscopic information. As a critical test on the accuracy of the potentials, eigenvalues, rotational constants Bv and centrifugal distortion terms Dv, Hv and Lv were calculated by numerical Integration of the radial wave equation. The computed eigenvalues agree well with experimental results. For the adiabatic B 3Π+0 state with a potential barrier, probability density distributions are also shown.

Potential energy curves of diatomic molecules from the Hill determinant method

The Hill determinant method is shown to be suitable for constructing potential energy curves of diatomic molecules. Both the Dunham and the perturbed Morse oscillator potentials are used to fit spectroscopic data. Results are shown for ionic and covalent molecules.

The potential energy function for the ground state of CCl + calculated from spectroscopic data

The dissociation energy ( D e= 57754±872 cm −1 has been estimated for the ground state of CCl + by fitting a Hulburt-Hirschfelder potential to the RKR turning points. This value of D e has been used together with molecular constants B e, ω e, a i ( i= 1–6) and R e obtained by Gruebele and co-workers to construct a potential energy function for CCl + in the form of a perturbed Morse oscillator.

On the calculation of model potential parameters from molecular spectra

In this paper we propound a procedure for inversion of an energy spectrum in order to obtain the original Hamiltonian function, applicable to nonresonant multidimensional systems. We use the Lie transform method to find the appropriate EBK-quantizable Hamiltonian, explicitly dependent on the unknown potential parameters. Comparison with the observed spectra allows the determination of explicit expressions for those parameters in terms of the spectral parameters. We first illustrate these concepts with the construction of power series and perturbed Morse oscillator model potentials for diatomics, and its extension to a simple model of triatomics including bending motion. We finally apply the procedure to the construction of a Morse potential model for linear polyatomic molecules.

A unified approach to the perturbed morse oscillator and Simons-Parr-Finlan diatomic potential functions

The Schrödinger equations corresponding to the Morse and Kratzer potential functions may each be transformed into harmonic form by a suitable combination of nonlinear coordinate transformation and similarity transformation, such that the resulting equations differ only in terms which couple the transformed position and momentum coordinates. Generalizations, such as the perturbed Morse oscillator (PMO) and the Simons-Parr-Finlan (SPF) potentials, are represented as Dunham-style expansions in the product of the transformed coordinate times the square root of the anharmonicity constant. Inclusion of rotational motion leads to expansions weighted by the anharmonicity constant itself. The method is well suited to a variational analysis, since all necessary matrix elements are analytic, and is illustrated by an application of various basis sets to the Morse oscillator in the transformed variables.

Comment on ‘‘Accurate analytic model potentials for D2 and H2 based on the perturbed-Morse-oscillator model’’

Huffaker and Cohen (ref.1) claim that the perturbed−Morse−oscillator (PMO) model, for the potential energy function for hydrogen, gives very high accuracy results; surpassing that of the RKR potential. A more efficient approach to formulating analytical functions based on the PMO model is given, and some defects of the PMO model are discussed.(AIP)

Perturbative calculation of vibrational energy levels for local modes

A perturbative technique for obtaining the vibrational energy levels in the local modes description of molecular vibrations is presented. The method allows to treat the anharmonicity of the polyatomic molecules, using a perturbed Morse oscillator as potential function for the local coordinates. The vibrational spectrum of the H2O molecule is obtained and compared with experimental values and with the results derived using other procedures.

Highly accurate energy levels and expectation values for perturbed morse oscillators

A perturbative method for obtaining the energy levels of perturbed Morse oscillators is formulated. The method uses commutation relations to derive recursion expressions which allow us to calculate high order energy corrections with little computational effort. Moreover, these expressions can be used to obtain expectation values of exponential operators. In order to test the method we make a numerical application for the CO molecule in its electronic ground state comparing our results with those derived by direct numerical integration.

PMO-RKR-van der Waals potential energy curves for the X 1Σ + and A 1Σ + states of the isotopic lithium hydrides

The potential energy curves for the X 1Σ + and A 1Σ + states of lithium hydrides and deuterides have been calculated by a new procedure which uses: (a) a perturbed Morse oscillator for the minimum zone; (b) a Rydberg-Klein-Rees potential for the zone of energy levels which can be represented from the experimental data; (c) a van-der Waals-type potential for the asymptotic part of the potential. All the calculated potentials have been checked by solving numerically the radial Schrödinger equation. From the wavefunctions generated we have obtained the vibrational energy eigenvalues and B v values, observing a very good agreement with the experimental data.

Hypervirial calculation of perturbed—Morse-oscillator potentials for diatomic molecules

The rotational-vibrational behavior of diatomic molecules has been studied using a hypervirial treatment for the perturbed Morse oscillator (PMO). Relations needed to calculate the PMO parameters from spectral data are derived. Rotational effects are included and the Dunham coefficients are obtained in terms of model parameters. The procedure is applied to CO in the ground-electronic state and the results are compared with others available data.

Perturbed morse oscillator in diatomic molecules: an hypervirial treatment

The rovibrational behavior problem of diatomic molecules is solved by an hypervirial perturbed treatment, by using as potential a perturbed Morse oscillator (PMO). Formulae have been derived from spectral data in order to calculate the PMO parameters, and numerical calculations have been made with the RbH molecule in its electronic ground state, comparing our results with those obtained by others authors.