2 GEORGE A. HAGEDORN tion that the electrons are in a discrete energy level that is isolated from the rest of the spectrum of the electronic Hamiltonian. The purpose of this paper is to study the phenom- ena that arise when this basic assumption is violated in the simplest possible way. The violations of this assumption that we allow are electronic level crossings. We assume there are two discrete electronic levels Ej^{X) and E$(X) that are isolated from the rest of the electronic spectrum for all nuclear configurations X in some region of interest. We also assume that E^(X) ^ Eg(X) except for X's that belong to some proper submanifold T of the nuclear configuration space. To keep the scope of the project within reason, we make two further assumptions. First, we assume that Ej^{X) and Eg(X) have the minimal degeneracy allowed by the symmetry group of the electronic Hamiltonian. Second, we assume that the electron Hamiltonian satisfies a genericity condition near the crossing manifold I\ This simply means that certain first order Taylor series coefficients are non-zero. In Section 2, we derive the classification and structure theory for generic, minimal multiplicity level crossings. We show that there are 11 distinct types of crossings, and we prove that the relevant part of the electron Hamiltonian can be put into a normal form near a generic crossing point. We show that the crossing manifold T generically has codimension 1, 2, 3, or 5 for minimal multiplicity crossings. The codimension depends on the type of the crossing. From conversations with chemists, we believe that all 11 types of crossings occur in familiar molecules in nature. However, to realize two of the crossing types, one must break the time reversal symmetry, e.g., by applying a suitable magnetic field. In the language of the chemical physics literature, the codimension is the number of nuclear configuration parameters that must be adjusted in order to arrive at the crossing submanifold. Near codimension 1 crossings, the process of finding the normal form for the electronic Hamiltonian involves a proper choice of an "adiabatic basis." Near a higher codi- mension crossings, it involves finding a special basis associated with the electron Hamiltonian that depends smoothly on the nuclear configuration X. By choosing the special basis cor- rectly, we show that the electronic Hamiltonian can be put into a special form. This special basis is neither an adiabatic basis nor a "diabatic basis." We prove existence of the special basis, so the non-existence of diabatic bases near higher codimension crossings [43,45,46] is not relevant to our analysis. Seven of the eleven types of crossings have the codimension of T equal to 1. These codi- mension 1 crossings arise when the levels Ej[(X) and E&(X) are associated with inequivalent representations or corepresentations of the symmetry group of the electronic Hamiltonian.
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