In this section we classify and derive normal forms for the various types of generic cross-
ings of eigenvalues of minimal degeneracy that can occur in quantum mechanical systems.
Except for a slight change of notation, the analysis we present is the same as that in the con-
ference proceedings article [21]. The structure of these crossings depends on the symmetry
properties of the Hamiltonian functions, and as we shall see, eleven distinct situations can
occur. Fortunately, several of these situations give rise to essentially the same behavior in
the time-dependent Born-Oppenheimer approximation.
In the chemical physics literature, only four eigenvalue crossing situations are usually
mentioned [ 38 ] because the chemical physicists regard several distinct situations as being of
the same type.
Throughout this section we assume we are given an electron Hamiltonian h(X) that
depends parametrically on the nuclear configuration X G M n . In the various different
situations, we assume the dimension n is large enough so that certain types of crossings can
occur generically. We show below that in each generic crossing situation, two eigenvalues
coincide on a submanifold T of some specific codimension. If n is less than this codimension,
then that type of crossing generically does not occur. We do not discuss non-generic crossings.
We assume that the X dependence of h(X) is Ck for some k 2 in the sense that the
resolvent (z /i(X)) - 1 is Ck in X for z £ IR. We let G denote the symmetry group associated
with this electron Hamiltonian. As has been known since the early days of quantum mechan-
ics [ 48 ], G is the group of all unitary and antiunitary operators that are X-independent in
some representation of the electronic Hilbert space, and that commute with the operators
h(X), for all X in some open set of interest. We let H denote the subgroup of unitary ele-
ments of G. The elements of G that do not belong to H are sometimes called time reversing
symmetry operators.
Since the product of two antiunitary operators is unitary, there are clearly two cases:
Either G H or H is a subgroup of G of index 2.
When G = H, standard group representation theory applies, and each distinct eigen-
value of h(X) is associated with a unique representation of G. Minimal multiplicity eigen-
values correspond to 1-dimensional representations, and if two simple eigenvalues Ej\[X)
and EB(X) cross, then there are two possibilities:
Previous Page Next Page