## IMA Postdoc Seminar (October 21, 2008)

**Speaker:** Mark Herman (IMA)

**Title:** Born-Oppenheimer Corrections Near a Renner-Teller Crossing

**Abstract:** We perform a rigorous mathematical analysis of the bending modes of
a linear triatomic molecule that exhibits the Renner-Teller effect.
Assuming the potentials are smooth, we prove that the wave functions
and energy levels have asymptotic expansions in powers of epsilon,
where the fourth power of epsilon is the ratio of an electron mass to
the mass of a nucleus.To prove the validity of the expansion, we must prove
various properties of the leading order equations and their
solutions. The leading order eigenvalue problem is analyzed in
terms of a parameter b, which is equivalent to the parameter
originally used by Renner. For 0 < b < 1, we prove
self-adjointness of the leading order Hamiltonian, that it has
purely discrete spectrum, and that its eigenfunctions and their
derivatives decay exponentially. Perturbation theory and finite
difference calculations suggest that the ground bending vibrational
state is involved in a level crossing near b = 0.925. We also
discuss the degeneracy of the eigenvalues. Because of the crossing,
the ground state is degenerate for 0 < b < 0.925 and
non-degenerate for 0.925 < b < 1.

**Slides:** PDF