Oscillators, 3-dimensional harmonic basis states

In variance with the hydrogen-like and Slater functions the potential employed to formally construct the gaussian basis states has nothing to do with the real potential acting upon an electron in an atom. On the other hand the solutions of this (actually three-dimensional harmonic oscillator problem) form a complete discrete basis in the space of orbitals in contrast to the hydrogen-like orbitals. 

The bending vibration of a linear ABC molecule may be treated as a two-dimensional isotropic harmonic oscillator. The v,l) basis set is particularly convenient, where v and l are respectively the vibrational and vibrational angular momentum quantum numbers. The allowed values of l are —v, —v + 2,. .. v. Since there are two quantum numbers, two pairs of creation, annihilation operators are needed to generate all basis states from the v = 0, 1=0) zero-point state. These are, following the notation of Cohen-Tannoudji, et al., (1977), a, ay and at,a9, where... 

Acetylene, H-C=C-H, has two identical HCC two-dimensional isotropic local-bender harmonic oscillators. In the normal mode basis set, the bending basis states are specified by four quantum numbers, v4l4V5l5), where mode 4 is the 7r9-symmetry trans-bend and mode 5 is the ttu-symmetry cis-bend. (The normal mode basis states do not have identical harmonic frequencies, u>5 — w4 = 121 cm-1.) There are four pairs of normal mode creation, annihilation operators, two for mode 4, a. d, a.dd, and a. g, a.ig, and two for mode 5, a, 5d,... 

This idea occurred to me during the Dimensional Scaling Workshop itself. I should like to state here that Martin Dunn independently and essentially simultaneously realised the benefits of computing the l/Z -expansion coefficients perturba-tively in a basis of harmonic oscillator eigenfunctions. 

Calculations done on this reaction with a two-dimensional bound state basis set of harmonic oscillator functions showed... 

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