The present communication is concerned with the calculation of rotation–vibration energies for triatomic molecules directly from the Born–Oppenheimer potential-energy function and the ‘inversion’ of this process: the refinement of the parameters in an analytical expression for the potential-energy function by least-squares fitting to experimental data. Three approaches to these problems are presented. First the non-rigid-bender model (see P. Jensen and P. R. Bunker, J. Mol. Spectrosc., 1986, 118, 18, and references therein) is briefly outlined, and secondly a Morse-oscillator-based model for equilateral triangular molecules is described. The primary molecules for which the latter model is applicable are the H+3 molecular ion and its isotopes (see P. Jensen and V. Špirko, J. Mol. Spectrosc., 1986, 118, 208, and references therein). With this method, a preliminary experimental potential-energy surface for H+3 has been obtained from experimental data. In the non-rigid bender model, the focus was on obtaining an accurate description of the bending and rotation of the triatomic molecule, whereas in the model for H+3, the primary aim was to improve the description of the stretches. The experience gathered in the work with these two models has been used as the basis for the Morse-oscillator–rigid-bender internal dynamics (MORBID) Hamiltonian (P. Jensen, J. Mol. Spectrosc., in press) which is presented as the third method for calculating the rotation–vibration energies of a triatomic molecule; MORBID combines the non-rigid-bender description of the bending and rotation motion with a Morse-oscillator description of the stretching. With MORBID, rotation–vibration energies for H2O are calculated from a published ab initio potential-energy surface.