4
GEORGE A. HAGEDORN
tron energy levels to be Ck away from crossings or absorption into the continuous spectrum.
Some of these restrictions are not satisfied by realistic electron Hamiltonians with Coulomb
potentials. However, one should be able to accommodate Coulomb potentials by using the
regularization techniques of [ 14,15,31 ].
The time-dependent Schrodinger equation that we study is
ie2^ = H{*)1, (1.3)
for t in a fixed interval. The factor of e2 on the left hand side of this equation indicates
a particular choice of time scaling. Other choices could be made, but this choice is the
"distinguished limit" [4] that produces the most interesting leading order solutions. With
this scaling, all terms in the equation play significant roles at leading order, and the nuclear
motion has a non-trivial classical limit. This is also the scaling for which the mean initial
nuclear kinetic energy is held constant as e tends to zero.
The nuclear wave packets we use are discussed in Section 2. They are concentrated near
a classical configuration a(t), and have position and momentum uncertainties proportional
to e. Because of this localization and an assumption that a(t) goes non-tangentially through
the crossing manifold T with non-zero velocity, the nuclei only feel the effect of the crossing
when the time is in an interval whose length is on the order of e. Away from this temporal
boundary layer, the standard time-dependent Born-Oppenheimer approximation applies.
In the cases of codimension 2, 3, or 5 crossings, we find that the Landau-Zener formula
[19,23-27] can be used to determine the leading order nuclear probability densities for the
system to be found in the Ej± or Eg state immediately after the wave packet has passed
through the crossing. Although we have not stated this result explicitly in Sections 6, 7, or 8,
it follows from the techniques presented there. Suppose the initial wave packet is associated
with the Ej± level prior to the temporal boundary layer. While in the boundary layer, the
nuclear wave packet does not have time to significantly change shape, i.e., when viewed as a
function of t and Y = , it is approximately independent of t for t in the boundary
e
layer.
To leading order, the nuclear probability density depends only on Y as the nuclei prop-
agate through the crossing manifold during the temporal boundary layer. As a function
of y , the portion of the nuclear probability density that is associated with the Eg surface
immediately after the temporal boundary layer is the product of the full nuclear probability
density immediately before the temporal boundary layer times e~^ \ where
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