MOLECULAR PROPAGATION THROUGH LEVEL CROSSINGS
f(Y) = C lim min \ EBW)
In the notation of Sections 6, 7, and 8, the constant C has the value C = — . Intuitively
this quantity is inversely proportional to the length of the temporal boundary layer during
which the nuclei are passing through the region where the gap between the levels is small
enough for the standard time-dependent Born-Oppenheimer approximation to break down.
To leading order, the part of the nuclear probability density that ends up on the Ej±
surface just after the temoral boundary layer is the full nuclear probability density just prior
to the temporal boundary layer multiplied by 1 — e~'( . This is what one would expect
from a properly interpreted Landau-Zener formula [19,23-27].
There is very little mathematical literature concerning the validity of Born-Oppenheim-
er approximations. Interested readers should consult references [5-7,9,12-16,18,20,22,30-
32,34-37]. Reference  contains an announcement of some (but not all) of the results
of the present paper. It presents a fairly detailed analysis of some simple examples without
all the complication of the general situations. A reader may find it useful to see the simple
examples before reading Sections 5-8 below.
We conclude this introduction by presenting two numerical simulations of evolution
through codimension 2 crossings. These simulations are direct numerical integration of the
full Schrodinger equation for a molecular type system. The results are represented as contour
plots of the nuclear probability densities as functions of the variable Y just prior to, during,
and just after the temporal boundary layer. On the left we show the portion associated with
the lower electronic level. The portion associated with the upper electronic level is presented
on the right. The electronic Hamiltonian is the two level system given by the matrix
* * = ( & - % ) •
The value of e was taken to be very small, 0.0001, so that only the leading order behavior
would be discernible. With this small value of e, the total transition probability from the Ej±
surface to the Eg surface from our numerical simulation agrees to three significant figures
with the prediction from the zeroth order analysis of Section 6.
In all of the contour plots, the nuclei impinge upon the crossing submanifold X\ —
X2 — 0 along the negative Xi-axis. Since the independent variable in our plots is Y instead
of X, the crossing submanifold moves across the picture from right to left.