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.