MOLECULAR PROPAGATION THROUGH LEVEL CROSSINGS

5

f(Y) = C lim min \ EBW)

+

eY)-EA«t)

+

eY)\\

K e—0

t

e2

7T

In the notation of Sections 6, 7, and 8, the constant C has the value C = — . Intuitively

46if)0

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 [18] 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.