12

GEORGE A. HAGEDORN

Typ e K Crossings The two irreducible corepresentations of G that correspond to

Ej\[X) and E$(X) are both of Type ///an d are unitarily equivalent to one another. Both

eigenvalues are multiplicity 2 away from the crossing.

REMARKS 1. If one ignores electron spin, then the all standard realistic electron Hamil-

tonians without external magnetic fields are time-reversal invariant, and the square of the

time-reversal operator is +1 . If one includes spin then hamiltonians without external mag-

netic fields are again time-reversal invariant. If the number of electrons is even, then the

square of the time-reversal operator is +1 . In the case of an odd number of electrons it is

- 1 .

2. Crossings of types A, C, D, E, F, G, and H all involve eigenvalues associated with two

different symmetry subspaces. To the extent that they are discussed in the chemical physics

literature, these types are all grouped together as one case. As we shall prove below, these

crossings all occur generically on codimension 1 submanifolds of the nuclear configuration

space. We shall also prove in the later sections of this paper that they give rise to similar

phenomena in the time-dependent Born-Oppenheimer approximation.

3. We prove below that crossings of Types B and K occur generically on codimension 3

submanifolds. For type B crossings, this is well known in the chemical physics literature.

Crossings of Types B and K give rise to similar phenomena in the time-dependent Born-

Oppenheimer approximation.

4. Crossings of Type I are familiar to the chemical physicists. They occur generically on

codimension 2 submanifolds in time-reversal invariant systems with an even number of elec-

trons [38].

5. Crossings of Type J are discussed in the chemical physics literature in the context of

crossings of Kramers doublets [38,40]. These crossings occur generically on codimension 5

submanifolds in time-reversal invariant systems with odd numbers of electrons.

Having described the classification of quantum eigenvalue crossings, we now turn to the

detailed structure of the electron Hamiltonian function h(X) near a generic crossing in each

of the cases. We derive normal forms for each of the crossing situations that are particularly

suited to our purposes in the subsequent sections of this paper. In our analysis of the time-

dependent Born-Oppenheimer approximation, we restrict attention to cases in which the

nuclear wave packets actually pass through the crossing with non-vanishing speed (a precise