A novel connection between algebraic spectroscopic parameters and force constants in the description of vibrational excitations of linear triatomic molecules M. Sánchez-Castellanos a , R. Lemus a, * , M. Carvajal b , F. Pérez-Bernal b a Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Ciudad Universitaria, Apartado Postal 70-543, CP 04510 México, Distrito Federal, Mexico b Departamento de Física Aplicada, Facultad de Ciencias Experimentales, Universidad de Huelva, 21071 Huelva, Spain article info Article history: Received 22 August 2008 In revised form 10 October 2008 Available online 18 October 2008 Keywords: Algebraic description U(3) Dynamical algebra Bending modes Linear molecules Carbon dioxide abstract A connection between an algebraic approach to the dynamics of triatomic molecules based on the U(2)  U(3)  U(2) Lie algebra and the traditional description in configuration space is presented. The connection is established in four steps. First, the molecular Hamiltonian is expanded in symmetrized local coordinates. Second, the Hamiltonian is transformed into an algebraic representation by introducing the realization of coordinates and momenta in terms of bosonic creation and annihilation operators of normal character. The third step is to perform a canonical transformation applied to the bosons associ- ated with the stretching degrees of freedom in order to obtain a unified representation in a local scheme. Finally, an anharmonization procedure is applied to identify the U(2)  U(3)  U(2) dynamical algebra. The main advantage of the proposed approach is that it provides relations between the spectroscopic parameters and the molecular structure and force constants. As an application, the analysis of the vibra- tional excitations of CO 2 in its ground electronic state is considered. In this scheme, each stretching degree of freedom is identified as an interacting Morse oscillator, with an associated U(2) dynamical alge- bra, and the doubly degenerate bending degree of freedom is modelled with a U(3) dynamical algebra, obtaining as a final result a reasonable set of force constants. Ó 2008 Elsevier Inc. All rights reserved. 1. Introduction Vibrational spectroscopy constitutes a powerful tool to probe the structure and dynamics of molecules in gases, liquids, and interfaces [1–4]. To this end, the use of effective Hamiltonians is crucial in the assignment of spectral lines, making possible to calculate force constants by means of perturbation methods [1,2]. Phenomenological approaches that expand the Hamiltonian in second quantization operators are a valuable alternative to tradi- tional methods when trying to characterize molecular vibrational spectra [2,5]. They are particularly useful when the generation of the spectrum using integro-differential techniques to calculate the potential energy surface and solve the Schrödinger equation implies a formidable computing task. Although full variational methods, with exact kinetic energy operators, have been inten- sively developed, they are computationally very demanding and cannot be used for large, and even medium-size, molecules [6]. In contrast, because of its phenomenologic character and the pos- sibility to take advantage of group theoretical techniques, algebraic methods are capable of providing a deep physical insight into the problem keeping the necessary calculations relatively simple. Its simplicity is maximal when a normal mode basis is used. In such case, the diagonal contribution of the Hamiltonian presents har- monic contributions as well as anharmonic terms proportional to higher powers of the normal quantum numbers, with energies gi- ven as a Dunham expansion as a first approximation [7]. In general, this simple expansion turns out to be inadequate, and couplings between zeroth-order states must be introduced. The Darling– Dennison [8] and Fermi [9] resonances are among the most striking examples of such couplings. It is important to emphasize that these methods have the remarkable feature that they provide the inter- action operators in such a way that its action on the basis states can be precisely stated, without the interference of additional con- tributions that are usual in the traditional configuration space ap- proach. However, there is a downside behind this simplicity: the connection with the configuration space is blurred and the infor- mation about force constants and potentials is not trivially ex- tracted from the spectroscopic parameters. Methods based on normal harmonic oscillator basis are useful as long as we are interested in the low lying region of the spectrum [10,11]. For highly excited vibrational states, a local perspective represents a better alternative [12–14]. Local models appear in natural form when using internal coordinates for the molecular vibrational modeling. Usually Morse oscillators [15] are associated with stretching coordinates, since they reflect more accurately than harmonic oscillators the main physical properties of a pure 0022-2852/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jms.2008.10.001 * Corresponding author. Fax: +52 55 56 16 22 33. E-mail address: renato@nucleares.unam.mx (R. Lemus). Journal of Molecular Spectroscopy 253 (2009) 1–15 Contents lists available at ScienceDirect Journal of Molecular Spectroscopy journal homepage: www.elsevier.com/locate/jms