Wigner function of the rotating Morse oscillator

We present an analytical expression of the Wigner distribution function (WDF) for the bound eigenstates of the rotating Morse oscillator (RMO). The effect of rotational excitation on the WDF on the quantum phase space has been demonstrated. This effect has been visualized by a series of contour diagrams for given rovibrational quantum states. Rotations of the molecule have been proved to qualitatively and quantitatively change the Wigner function. As a result, the most probable distance between atoms in a rotating molecule changes, and depends on the parity of the vibrational quantum number. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

The Wigner function of the rotating Kratzer oscillator mapped onto the Morse oscillator

In this article, the rotating Kratzer oscillator in quantum phase space is studied. The Langer transformation is used to map the Kratzer oscillator with centrifugal term onto a one‐dimensional Morse oscillator. As a result, the Wigner distribution functions for the Morse oscillator are obtained. The quantum states of the system are visualized in the phase space for a few vibrational and rotational quantum numbers. The results obtained in the phase space correspond to those derived in the standard quantum theory. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Ladder operators for the Kratzer oscillator and the Morse potential

Taking a close look at the Infeld–Hull ladder operators for the Kratzer oscillator system, V(x) = [x² + β(β ? 1)x^?2]/2, we deduce and explicitly construct energy‐raising and ‐lowering operators for the generalized Morse potential system V(z) = (Ae^?4αz ? Be^?2αz)/2, through a canonical transformation that exists between the two systems. For the Morse potential system, we obtain a system of raising and lowering operators P_±(n) (n = 0, 1, 2, 3, … , n_max) with the specific property that P_±(n)Φ_n = c_±(n)Φ_n±1, where Φ_n denotes the nth energy eigenfunction. While P_?(0) annihilates the ground‐state Φ₀, the operator P₊(n_max), instead of annihilating the highest bound‐state Φ, actually knocks it out of the L₂ space spanned by the discrete bound states and becomes inadmissible. Yet, raising and lowering operators _± with proper end‐of‐spectrum behavior (i.e., _?|0〉 = 0 and ₊|n_max〉 = 0) can be constructed in a straightforward way in the energy representation. We show that the operators ₊, _?, and ₀ (where ₀ ≡ (1/2)[ ₊, _?]) form a su(2) algebra only if we restrict them to the (N ? 1)‐dimensional subspace spanned by the lowest (N ? 1) basis vectors, but not in the full (N + 1)‐dimensional space spanned by the discrete bound states [N ≡ n_max ≡ integral part of (1/2)(B/(2α ) ? 1)]. Realization of this su(2) algebra in the position representation (when restricted to the (N ? 1)‐dimensional subspace) is also given. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

On the exact revival of Morse oscillator wave packets

Journal of Mathematical Chemistry - The exact analytic expressions of the autocorrelation function and Husimi distribution function for a Morse oscillator wave packet have been derived and we use...     

The Morse oscillator under time-dependent external fields

A method to solve the equations for the Morse oscillator under intense time-dependent external fields is presented. Exact analytical formulas for the dipole matrix elements are calculated by the use of the hypergeometric algebra. The continuum is described by an expansion using Laguerre functions. The full algorithm for the calculation of wave functions can be controlled by the convergence of series and by the errors of a first order integration method. We apply our technique to the selective preparation of high overtones by femtosecond laser pulses. The population of the target state is optimized as a function of the intensity and frequency. Introducing a second simultaneous laser, we study the effects of relative frequency and phase over the target state population and dissociation channels. The calculations exhibit a rich interference pattern showing the enhancement and the suppression of the target population by varying the laser parameters.     

Algebraic approximation to the Franck–Condon factors for the Morse oscillator

An algebraic approach is proposed to calculate the Franck–Condon factors for the Morse potential of diatomic molecules. The Morse oscillator is approximated by means of a fourth-order anharmonic oscillator. In the second-quantized formalism, this anharmonic Hamiltonian is diagonalized by way of the Bogoliubov–Tyablikov transformation. The Franck–Condon factors are estimated using the harmonic frequency equivalent and the recurrence relations for the Franck–Condon factors of the harmonic oscillator. Overlap integrals are shown for three band systems and compared with values calculated with an RKR potential. Excellent agreement is achieved.     

The supersymmetric quantum mechanics theory and Darboux transformation for the Morse oscillator with an approximate rotational term

Anharmonic potentials with a rotational terms are widely used in quantum chemistry of diatomic systems, since they include the influence of centrifugal force on motions of atomic nuclei. For the first time the Taylor-expanded renormalized Morse oscillator is studied within the framework of supersymmetric quantum mechanics theory. The mathematical formalism of supersymmetric quantum mechanics and the Darboux transformation are used to determine the bound states for the Morse anharmonic oscillator with an approximate rotational term. The factorization method has been applied in order to obtain analytical forms of creation and annihilation operators as well as Witten superpotential and isospectral potentials. Moreover, the radial Schrödinger equation with the Darboux potential has been converted into an exactly solvable form of second-order Sturm–Liouville differential equation. To this aim the Darboux transformation has been used. The efficient algebraic approach proposed can be used to solve the Schrödinger equation for other anharmonic exponential potentials with rotational terms.     

Energy shifts in the relative coordinates treatment of the collinear collision between an atom and an oscillator. Morse interaction potential

The use of energy shifted functions in the realtive coordinates treatment of the collinear collision between an atom and a harmonic osciilator described previously is tested for the Morse interaction potential. Good results are obtained for the transition probabilities when the interaction potential is not too deep. A simple analysis is presented of the compound state resonances with may produce dips in the transition probabilities.     

Perturbational analysis of the regioselectivity in the acrolein dimerization

Using intermolecular SCF-perturbation theory, the controlling factors determining the regioselectivity in the acrolein dimerization are derived by a rigorous decomposition of the interaction energy. This partitioning of the interaction energy is performed up to a level where a comparison with frontier orbital models is possible. The regioselectivity is found to be determined by orbital interactions occuring via the terminal atoms and atoms involved in secondary Woodward-Hoffmann interactions.     

Perturbational calculations of parity-violating effects in nuclear-magnetic-resonance parameters

We investigate the effects of the parity-violating electroweak interaction in the spectral parameters of nuclear magnetic resonance. Perturbational theory of parity-violating effects in the nuclear magnetic shielding is presented to the order of G(F)alpha, and in the indirect spin-spin coupling, to the order of G(F)alpha3. These leading-order parity-violating corrections are evaluated using analytical linear-response theory methods based on Hartree-Fock and density-functional theory reference states. Parity-violating contributions to spin-spin couplings are evaluated for the first time at the first-principles level. Calculations are carried out for two chiral halomethanes, bromochlorofluoromethane and bromofluoroiodomethane.     

Perturbational relativistic theory of electron spin resonance g-tensor

We carry out a complete treatment of the leading-order relativistic one-electron contributions, arising from the Breit-Pauli Hamiltonian, to the g-tensor of electron spin resonance spectroscopy. We classify the different terms and discuss their interpretation as well as give numerical ab initio estimates for the F2(-), Cl2(-), Br2(-), and I2(-) series, using analytical response theory calculations with a multiconfigurational self-consistent field reference state. The results are compared to available experimental data. (c) 2004 American Institute of Physics     

Ladder operators for the Morse potential

A realization of the raising and lowering operators for the Morse potential is presented. It is shown that these operators satisfy the commutation relations for the SU(2) group. Closed analytical expressions are obtained for the matrix elements of different operators such as 1/y and d/dy. The harmonic limit of the SU(2) operators is also studied and an approach previously proposed to calculate the Franck–Condon factors is discussed. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Perturbational calculation of the exchange forces in the two lowest states of the hydrogen molecule

The interaction energies in the ground (X¹Σ) and the first excited (b³Σ)states of the hydrogen molecule were calculated using the modified Jansen and Byers Brown perturbation method. Calculations were performed for nine values of the internuclear distance R in the range 5 a.u. ≦ R ≦ 9 a.u. The present results were compared with the interaction energies calculated by four other perturbation methods as the expectation values of the H – E⁰ operator with the function accurate up to the first order. The results show that the method can give satisfactory values of the interaction energies.     

Quasi-bound states of coupled Morse oscillators

Quasi-bound (resonance) states are present in the continuous spectrum of the Hamiltonian of two coupled Morse oscillators. Two different methods for approximating these as localized states are compared. The algebraic approach is shown to be in very good accord with the other method which is formulated in coordinate space and hence is differential in character. For these highly excited states an intermultiplet mixing term is included in the algebraic Hamiltonian.     

Modified Morse potential for unification of the pair interactions

We designed a novel model potential that unifies the pair interactions including the well known Morse and Lennard-Jones potentials. Using two parameters, the interactions at the minimum, short range, and long range of the new model potential can be controlled separately, so the potential is very flexible to fit various systems. It is found that for potentials with similar range with the Lennard-Jones potential at the minimum, due to the difference at the short and long ranges, the favorite structures can be very different, and some previously unknown magic numbers are located.     

A new approach to the exact and approximate anharmonic vibrational partition function of diatomic and polyatomic molecules utilizing Morse and Rosen–Morse oscillators

Exact closed forms of the equilibrium partition functions in terms Jacobi elliptic functions are derived for a particle in a box and Rosen–Morse (Poschl–Teller) oscillator (perfect for modeling bending vibrational modes). An exact form of the equilibrium partition function of Morse oscillator is reported. Three other approximate forms of Morse partition function are presented. Having an exact closed‐form for the vibrational partition function can be very helpful in evaluating thermodynamic state functions, e.g., entropy, internal energy, enthalpy, and heat capacity. Moreover, the herein presented closed forms of the vibrational partition function can be used for obtaining spectroscopic and dynamical information through evaluating the two‐ and four‐point dipole moment time correlation functions in anharmonic media. Finally, a closed exact form of the rotational partition function of a particle on a ring in terms of the first kind of complete elliptic integral is derived. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Resonances and antibound states in a Morse potential

We calculate the resonant and antibound state energies for a Morse potential with a centrifugal barrier using Siegert boundary conditions. Starting with a complex wave number k (purely imaginary for bound and antibound states), we integrate numerically from the origin up to a matching point using Numerov's method. The inward integration is performed using the corresponding (first-order) Riccati equation. The complex eigenvalues are found by matching the two logarithmic derivatives. We find narrow shape resonances within the well, above the dissociation limit, and broad resonances above the centrifugal barrier. Antibound states are found even with J = 0. © 1997 John Wiley & Sons, Inc.