Artificial Symmetries for Calculating Vibrational Energies of Linear Molecules

Abstract

Linear molecules usually represent a special case in rotational-vibrational calculations due to a singularity of the kinetic energy operator that arises from the rotation about the a (the principal axis of least moment of inertia, becoming the molecular axis at the linear equilibrium geometry) being undefined. Assuming the standard ro-vibrational basis functions, in the 3N−6 approach, of the form ∣ν1,ν2,ν3ℓ3;J,k,m⟩, tackling the unique difficulties of linear molecules involves constraining the vibrational and rotational functions with k=ℓ3, which are the projections, in units of ℏ, of the corresponding angular momenta onto the molecular axis. These basis functions are assigned to irreducible representations (irreps) of the C2v(M) molecular symmetry group. This, in turn, necessitates purpose-built codes that specifically deal with linear molecules. In the present work, we describe an alternative scheme and introduce an (artificial) group that ensures that the condition ℓ3=k is automatically applied solely through symmetry group algebra. The advantage of such an approach is that the application of symmetry group algebra in ro-vibrational calculations is ubiquitous, and so this method can be used to enable ro-vibrational calculations of linear molecules in polyatomic codes with fairly minimal modifications. To this end, we construct a—formally infinite—artificial molecular symmetry group D∞h(AEM), which consists of one-dimensional (non-degenerate) irreducible representations and use it to classify vibrational and rotational basis functions according to ℓ and k. This extension to non-rigorous, artificial symmetry groups is based on cyclic groups of prime-order. Opposite to the usual scenario, where the form of symmetry adapted basis sets is dictated by the symmetry group the molecule belongs to, here the symmetry group D∞h(AEM) is built to satisfy properties for the convenience of the basis set construction and matrix elements calculations. We believe that the idea of purpose-built artificial symmetry groups can be useful in other applications.

A highly accurate potential energy surface for carbonyl sulphide (OCS): how important are the ab initio calculations?

“…52 TROVE is a wellestablished program used extensively by the ExoMol database [53][54][55] to generate comprehensive molecular line lists (catalogues of transitions and their probabilities) for exoplanetary science. The methodology of TROVE is well described in the literature 52,[56][57][58][59][60][61] and it has been used to compute the spectra of several triatomic molecules including CO 2 , 62 SiO 2 , 63 CaOH, 64 KOH and NaOH 65 (amongst numerous other larger polyatomics). Full details of the TROVE approach for treating linear and quasi-linear molecules can be found in ref.…”

Ro-vibrational levels and their (e-f) splitting of acetylene molecule calculated from new potential energy surfaces

“…from the Hamiltonian (7), then all levels of a certain vibrational state in the block 1 Jp  would coincide exactly with the levels in the block 0 Jp  for K > 0 . In this case, it really makes sense to talk about the point group D∞h [37], [40], for which doubly degenerate representations K (K=1,… ∞) would occur. But this scenario is only true in an approximation where the Coriolis interaction corr H is neglected.…”

“…Note that in ref. [40] an artificial, formally infinite molecular symmetry group D∞h(AEM) has been introduced for linear molecules, but wave functions were finally classified according to one-dimensional representations.…”

Symmetry and Molecular Spectroscopy

“…An alternative scheme to classify the highly excited rovibrational states of linear molecules was proposed by Mellor et al in [9]. The authors note that for well-bent molecules such as H 2 S, a standard zero-order Hamiltonian system is expressed in terms of…”