MOLECULAR PROPAGATION THROUGH ELECTRON-ENERGY LEVEL-CROSSINGS

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(A)(X) and E(B)(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 GAMMA = {X : E(A)(X) = E(B)(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, minmal multiplicity electron level crossings. For crossings that have the codimension of GAMMA 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 GAMMA 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.