Algebraic approach to the vibrational excitations in methane

We present a description of the vibrational excitations of methane by means of an algebraic analysis of a model of coupled anharmonic oscillators

UNITARY GROUP APPROACH FOR EFFECTIVE POTENTIALS IN 3D SYSTEMS

An algebraic approach based on the unitary $U(4)$ algebra is proposed to describe 3D systems for effective potentials. Our approach is based on the 3D vibron model where the addition of a scalar boson is introduced into the space of a 3D harmonic oscillator keeping constant the total number of bosons $N$. However instead of dealing directly with the dynamical symmetries we proceed to identify the coordinates and momenta in the algebraic space. Our approach is based on the mapping between the $U(4) \supset U(3) \supset O(3)$ dynamical symmetry and the harmonic oscillator states. A minimization procedure is used in order to determine the coefficients involved in the algebraic expansion of the coordinates and momenta. This allows the kets associated with the different subgroup chains to be linked to energy, coordinate and momentum representations. This identification provides useful tools to obtain the matrix representation of 3D Hamiltonians in a simple form through the use of the transformation brackets connecting the different bases. The exact energy and wave functions are obtained in the $N$ large limit. As an application of this approach the eigensystem of the 3D-Morse potential is analyzed, whose wave functions are contrasted with the approximate analytical solutions for null angular momentum. The analyses of inertia moments as well as the dipole moment strengths are also included. This approach provides results which contrasts to the 3D-vibron model where the Morse functions are identified with a dynamical symmetry

CONNECTION BETWEEN THE SU(3) ALGEBRAIC MODEL AND CONFIGURATION SPACE FOR BENDING MODES OF LINEAR MOLECULES: APPLICATION TO ACETYLENE

An approach to connect the $su(3)$ dynamical group- used to describe the bending modes of linear molecules- with configuration space is discussed. _x000d_ The $SU(3)$ group may be seen as a consequence of adding a scalar boson to the $SU(2)$ space of two degenerate harmonic oscillators. The resulting $SU(3)$ group becomes the dynamical group for the bending degrees of freedom of linear molecules, but the connection to configuration space is not obvious. This work aims at providing this connection. Our approach is based on the basis of establishing a mapping between the algebraic and configuration states. An arbitrary operator in configuration space is then expanded in terms of generators of the dynamical algebra. The coefficients are determined through a minimization procedure and given in terms of matrix elements defined in configuration space. As an application we consider the vibrational description of the bending modes of the acetylene molecule, where the force constants are estimated in the framework of the $U(3) times U(3)$ model

ALGEBRAIC APPROACHES AND THEIR CONNECTION WITH PHASE SPACE METHODS: APPLICATIONS TO SPECTROSCOPY

First the salient features of the $U(\nu+1)$ algebraic approach associated to $\nu$ equivalent oscillators are presented. Then we introduce the 1D case through the connection of the $U(2)$ algebra with the Morse/P\"oshl-Teller potentials with the goal of describing the vibrational degrees of freedom of non linear polyatomic molecules. The coordinates and momenta are then identified and generalized to any potential, providing the possibility to solve the 1D Schr\"odinger equation for general potentials by purely algebraic means using the concept of transformation brackets. A new procedure to calculate of Franck-Condon factors is presented. Because of their importance in linear molecules the $U(3)$ model is introduced, emphasizing its connection with configuration space. It is shown the application of the $U(2) \times U(3) \times U(2)$ algebraic approach to describe the Raman spectroscopy of the CO$_2$ molecule. The $U(3)$ model is applied to consider general potentials to describe linear-to-bend transition in triatomic molecules. Finally the $U(4)$ model is introduced to describe 3D systems for general potentials. The Hydrogen atom as well as the 3D Morse systems are analyzed by purely algebraic means as a benchmark to show how to apply the algebraic method for potentials with spectroscopic interest

Algebraic description of atom-molecule interactions

We describe a new method based on algebraic techniques, which leads to a model of atom-diatom collisions

An algebraic alternative for the accurate simulation of CO2 Raman spectra

We present an accurate simulation of the Raman spectrum of the carbon dioxide molecule in the 1150–1500 cm−1 spectral range, comparing the results obtained using the three polyad schemes found in the literature of this molecule. The description of the molecule with the algebraic U1(2) × U(3) × U2(2) local model encompasses both stretching and bending degrees of freedom. A detailed analysis of the Hamiltonian interactions for the three polyad schemes provides fittings with root mean square deviations in the range 0.14–0.20 cm−1, involving 19 parameters taking into account the 178 experimental term energies found in the literature. Using a limited subset of 9 experimental transitionmoments, we optimize 5 partial derivatives of the mean polarizability and simulate the Raman spectrum of CO2 for the three polyad schemes. Comparing the calculated results with the experimental spectrum, we obtain an overall good agreement for the three polyads. However, an inspection in detail of the spectrum seems to show a slight preference for polyad P212 albeit not due to the interaction characterizing the polyad but due to anharmonic effects and energy distribution. Finally, we assess the effect of the Fermi resonance over CO2 Raman line intensities.Centro de Estudios Avanzados de Física, Matemática y Computación. Universidad de Huelva, Grant/Award Number: FEDER/MINECO UNHU-15CE-28; CMST COST action, Grant/Award Number: CM1405 MOLIM; Consejería de Conocimiento. Investigación y Universidad, Junta de Andalucía and European Regional Development Fund, Grant/Award Number: SOMM17/6105/UGR; Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México, Grant/Award Number: IN-22701

Accidental degeneracy in a simple quantum system: A new symmetry group for a particle in an impenetrable square-well potential

The two-dimensional square-well potential is one of the simplest quantum-mechanical systems that exhibits accidental degeneracy. We show that the double degeneracy present is a consequence of a dynamical symmetry and derive a new symmetry group associated with the syste

Accidental degeneracy and hidden symmetry: Rectangular wells with commensurate sides

Two-dimensional quantum square wells and rectangular billiards with commensurate sides are simple systems which exhibit accidental degeneracies. We show that a recent analysis for square wells can be similarly applied to rectangular wells with commensurate side