MOLECULAR PROPAGATION THROUGH LEVEL CROSSINGS 25

By standard Taylor series results, MX(X) has the form M\(X) = N\(X) + 0(||X|| 2 ),

where

e-X + ifX\

c-X-id-X

0

-b-X

J

for some vectors 6, c, d, e, and / . Generically 6, c, d, e, and / are linearly independent. By

a rotation of the coordinate system we may assume that only the first five components of 6,

c, d, e, and / are non-zero.

If ?y(0) is a generic nuclear momentum vector, then we can rotate the first five coordinate

axes so that the projection of 77(0) into the five dimensional subspace spanned by 6, c, d, e,

and / lies along the positive X\ axis.

At this point, the Xj coordinates for j 5 no longer play a role in the structure of

Ni(X). Furthermore, without altering the basic structure obtained so far, we still have

the freedom to rotate the X2, -X3, X4, and X5 coordinate directions, and we can perform

those X-independent unitary transformations of the four dimensional space spanned by the

basic electronic wave functions ^i(X) , faiX), ip^(X), and tp^(X) that preserve the relations

ip2(X) = JCipi(X) and ip4(X) = Kip^(X). We claim that by doing such operations in

generic situations, we can arrange for the following five conditions to be satisfied:

1. The first component of b is non-zero.

2. c\ = C3 = C4 = C5 = 0, but C2 7^ 0.

3. d\ = ^2 = d\ = d$ = 0, but d% / 0.

4. e\ = t2 — e 3 = e5 — 0 but e4 7^ 0.

5. h = f2 = h = k = 0, but /

5

/ 0.

Thus, we may assume that N\(X) has the form

Ni(X) =

I b-X 0 c-X + id-X

0 fc-X -e-X + if-X

c-X-id-X -e-X-if-X -b-X

\e-X-if-X c-X + id-X 0