Resurrection of the original approach to the Renner-Teller effect for a linear tri-atomic molecule

What is it about?

30. March 2018: Co-author John M. Brown very sadly died 2009. The following is therefore written by me, Flemming Jørgensen, alone. If the electronic state in a molecule is non-degenerate, the movement of the nuclei is governed by the Born-Oppenheimer (BO-) potential. In a linear tri-atomic molecule with a non-zero angular momentum around the axis, we have two electronic states, one for each sign of the parity under reflection in the plane formed by the three nuclei when the molecule bends and thus breaks the exact rotational symmetry around the axis. This gives two BO-potentials which touch each other in the strict linear configuration. In Renner's original approach from 1933 for the resulting energy levels he considered a molecular model with no stretching vibrations and no end-over-end rotation. If we consider his results in the language of an effective Hamiltonian this was that of a vibration in the mean potential plus an expression formulated in terms of certain 2x2 matrices and the difference between the two potentials. Assuming these to be harmonic and fairly close to each other he determined the energy levels by second order perturbation theory. Doing the same with a modern approach to an effective Hamiltonian, the original calculations are so substantially simplified that we with a modest effort reach the order four - with inclusion of quartic anharmonicities in the two bending potentials. The 2x2 matrices are expressed in terms of Pauli matrices, considered as operators in their own right and with a simple physical interpretation.

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Why is it important?

Electronic angular momentum about the axis usually goes with a single electron in an open shell - that is with free radicals for which it for many years was impossible to obtain a gas pressure high enough for spectroscopy. Therefore about 25 years passed after Renner's paper before experimental observation. When thereafter a large number of high quality data were obtained, researchers constructed their own more ad hoc in the effective Hamiltonians to fit the results. One reason for this is probably that Renner had ignored electron spin - and the spin-orbit interaction of the single last electron is, actually, as strong as the Renner-Teller effect itself. Another reason seems to be not recognizing the strict systematic in Renner's old formulation with its mixture of differential operators and 2x2 matrices. This strict systematic does, however, lead to expressions for the various parameters in the effective Hamiltonian which, in principle at least, are ready for ab initio calculation.

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