Type of Document 
Dissertation 
Author 
Herman, Mark Steven

Author's Email Address 
mherman@vt.edu 
URN 
etd07032008115546 
Title 
BornOppenheimer Corrections Near a RennerTeller Crossing 
Degree 
PhD 
Department 
Mathematics 
Advisory Committee 
Advisor Name 
Title 
Hagedorn, George A. 
Committee Chair 
Ball, Joseph A. 
Committee Member 
Crawford, Daniel T. 
Committee Member 
Klaus, Martin 
Committee Member 

Keywords 
 Perturbation Theory
 BornOppenheimer Approximation
 RennerTeller Effect

Date of Defense 
20080703 
Availability 
unrestricted 
Abstract
We perform a rigorous mathematical analysis of the bending modes of a linear triatomic molecule that exhibits the RennerTeller effect. Assuming the potentials are smooth, we prove that the wave functions and energy levels have asymptotic expansions in powers of ε, where ε4 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 selfadjointness 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 nondegenerate for 0.925 < b˜ < 1.

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