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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