In this work the analysis of the equivalent rotations from the permutation inversion group formalism is revisited. We emphasize that explicit knowledge of changes in the Euler angles are not required in order to determine the transformation that a given symmetry operation causes to the rotational functions when dealing with the permutation inversion group formalism. Indeed, matrix elements of the equivalent rotations are provided by a single Wigner's D ^(j)(R) function. Taking advantage of this, we propose a symmetry projection approach to build the rovibrational functions of methane. This approach focuses on the relevance of the isomorphism between permutations and equivalent rotations. In our method, symmetry adapted functions are obtained by simultaneous diagonalization of a set of commuting operators, whose representation is given in terms of direct products of Wigner's D functions and vibrational matrix representations provided by a local scheme. The proposed approach is general and permits us to obtain in a systematic fashion an orthonormal set of symmetry-projected functions, with good total angular momentum, and carrying the irreducible representations of the molecular symmetry group.

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 transition moments, 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.

The potential energy surface (PES) for the ground state of the ¹²C ¹⁶O₂ molecule is estimated through an algebraic approach based on unitary groups. It is shown that a purely algebraic approach may lead to a unphysical surface even when the fit turns out to be of a remarkable high quality. The vibrational description is obtained in the framework of the U(2) × U(3) × U(2) model, where the U(2) algebras are associated with the CO bond stretching and the U(3) algebra with the OCO bending. The algebraic representation of the Hamiltonian is obtained through the connection between the coordinates and momenta and the generators of the dynamical algebra. It is shown that through this connection it is possible to choose the appropriate set of spectroscopic parameters leading to the right PES. The fit from which the PES is estimated involves 101 energy levels with an standard deviation of RMS = 0.53 cm⁻¹.

An approach to connect the dynamical group – used to describe the bending modes of linear molecules – with configuration space is discussed. The group may be seen as a consequence of adding a scalar boson to the space of two degenerate harmonic oscillators. The resulting group becomes the dynamical group for 2D systems The connection of the model with configuration space however is not obvious. Our approach is based on establishing a mapping between the algebraic and configuration states. The identification in the algebraic space of coordinates and momenta leads to a new identification of the group chains: they are associated with energy, coordinate and momentum representations. In addition an analysis of local-to-normal mode transition is presented. This provides a criterion to decide between a local or normal mode descriptions in an spectroscopic description. As an example we consider the situation of the bending modes of acetylene taking a Hamiltonian up to quartic order. The different results for the force constants obtained from the local and normal mode treatments leads to the conclusions that previous applications of the bending modes of acetylene dealing with a local mode treatment is on the verge of applicability.

The eigenfunction approach used for discrete symmetries is deduced from the concept of quantum numbers. We show that the irreducible representations (irreps) associated with the eigenfunctions are indeed a shorthand notation for the set of eigenvalues of the class operators (character table). The need of a canonical chain of groups to establish a complete set of commuting operators is emphasized. This analysis allows us to establish in natural form the connection between the quantum numbers and the eigenfunction method proposed by J.Q. Chen to obtain symmetry adapted functions. We then proceed to present a friendly version of the eigenfunction method to project functions.