Treating linear molecules in calculations of rotation-vibration spectra

Abstract

In this article, a numerical implementation of the exact kinetic energy operator (KEO) for triatomic molecules (symmetric of XY 2type and asymmetric of YXZ-type) is presented. The implementation is based on the valence coordinates with the bisecting (XY 2type molecules) and bond-vector (YXZ) embeddings and includes the treatment of the singularity at linear geometry. The KEO is represented in a sum-of-product form. The singularity caused by the undetermined angle at the linear configuration is resolved with the help of the associated Legendre and Laguerre polynomials used as parameterized bending basis functions in the finite basis set representation. The exact KEO implementation is combined with the variational solver theoretical rovibrational energies, equipped with a general automatic symmetry-adaptation procedure and efficient basis step contraction schemes, providing a powerful computational solver of triatomic molecules for accurate computations of highly excited ro-vibrational spectra. The advantages of different basis set choices are discussed. Examples of specific applications for computing hot spectra of linear molecules are given.

Artificial Symmetries for Calculating Vibrational Energies of Linear Molecules

“…where ρ measures the bending displacement from the linear equilibrium configuration and the stretching vibrations are described by the bond lengths r 1 and r 2 . The bending function ϕ v 3 , (ρ) shown in Equation ( 3) plays the central role in the 3N − 6 formalism via two main properties: (i) it is constructed to possess the correct dependence at ρ → 0 to resolve the singularity of the KEO [13,19]:…”

“…Let us consider a ro-vibrational basis set of a centrosymmetric linear triatomic (CO 2 ) in a symmetry adapted form (compare to Equation (3)) and a bisecting embedding [19]:…”

“…The corresponding ro-vibrational basis set becomes dependent on the projections of the vibrational and rotational angular momenta (in units ofh) onto the molecular axis a (k and , respectively), subject to the constraint k = [4,[16][17][18]. Among the popular choices of the bending basis functions with correct behaviour are the associated Legendre polynomials [12,19] L v , with v and as the corresponding vibrational (bending) quantum numbers.…”

“…Updates of TROVE have been recently reported in Refs. [19,43,44], with the most recent extension [19] being the implementation in TROVE of the 3N − 6 methodology with a singularity resolution based on the inclusion of interaction between the bending motions and the rotation about the a axis. Additionally, the adaptation of linear-molecule symmetry that was described in Ref.…”

“…The theory and programming work reported in Ref. [19] and in the present work have already been employed for generating hot molecular line lists for SiO 2 [45] and CO 2 [46] as well as to produce a quadrupole spectrum of H 2 O [47]. TROVE uses an automatic approach for constructing a symmetry-adapted basis set to be used in setting up a matrix representation of the molecular rotation-vibration Hamiltonian [48] In order to compute the vibrational energies for a linear molecule, we solve the Schrödinger equation for the HamiltonianĤ (0) vib,lin , which involves the bending and stretching motions and, as explained in Introduction, the rotational motion about the a axis.…”