The principal results of this paper involve the extension of the time-dependent Born-
Oppenheimer approximation to accommodate the propagation of nuclei through generic,
minimal multiplicity electron energy level crossings. In preparation for these results, we
present a general discussion of quantum mechanical energy level crossings.
We begin by deriving a classification theory for level crossings of quantum Hamiltonians
h(X) that depend on (multi-dimensional) external parameters X, under the assumption that
the two levels E^(X) and Eg(X) that cross have the minimal multiplicity allowed by the
symmetry group of h(X). We prove that there are 11 distinct types of crossings, and for
each type, we show that h{X) can be put into a normal form near the crossing submanifold
r = {X : Ej{(X) = Eg(X)}. Depending on the type of crossing, this submanifold
generically has codimension 1, 2, 3, or 5.
Our main results involve the evolution of molecular systems, in which h(X) is the
electron Hamiltonian and X is the nuclear configuration variable. We analyze the asymptotic
behavior of the full molecular wave function in the Born-Oppenheimer limit as the nuclei
propagate through generic, minimal multiplicity electronic level crossings. For crossings that
have the codimension of T equal to 1, the leading order propagation is not affected by the
presence of the second level, but as the nuclei propagate through the crossing, a first order
correction term associated with the second level is generated. For crossings that have the
codimension of T equal to 2, 3, or 5, a Born-Oppenheimer wave packet initially associated
with one level splits to leading order into non-trivial components associated with both levels
as the nuclei move through the crossing.
K E Y WORDS AND PHRASES: Molecular propagation, Born-Oppenheimer approximation,
Semiclassical quantum mechanics, Adiabatic approximation, Electron energy levels, Level
crossings, Landau-Zener formula.
Previous Page Next Page