MOLECULAR PROPAGATION THROUGH LEVEL CROSSINGS

23

If 77(0) is a generic nuclear momentum vector, then we can rotate the first three coor-

dinate axes so that the projection of 7/(0) into the three dimensional subspace spanned by 6,

c, and d lies along the positive X\ axis.

At this point, the Xj coordinates for j 3 no longer play a role in the structure of

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

freedom to rotate the X2 and X3 coordinate directions, and we can perform X-independent

unitary transformations of the two dimensional space spanned by the basic electronic wave

functions ip\(X) and ip2(X). If we do such unitary transformations, we also redefine ips(X)

and ^ ( X ) to preserve the relations ips(X) = )Cipi(X) and ^ ( X ) = JCfoiX). We do

these operations, mimicking the procedure used in our discussion of Type B crossings, to see

that the following three conditions can be satisfied:

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

2. The first and third components of c are zero, but its second component is positive.

3. The first and second components of d are zero, but its third component is positive.

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

Ni(X)

=

I 3

;

J

bjXj C2X2 + 2^3X3

i = i

3

c2X2 - ^ 3 X 3 - ^

bjXj

3=1

\

0 0

V

y ^ ^Xj C2X2 - ^3X 3

3

c

2

X

2

+ id3X3 - ^2 bjXj

3=1 )

(2.6)

STRUCTURE O F T Y P E J CROSSINGS Suppose a Ck electron Hamiltonian function

h(X) has a Type J crossing of two multiplicity 2 eigenvalues Ej^{X) and Eg(X) at X = 0.

As in the earlier constructions, we let P{X) be the rank 4 spectral projection for h(X)

corresponding to both the eigenvalues Ej[(X) and Eg(X). We arbitrarily pick a unit vector

^l(O) that lies in the range of P(0), and we define ^2(0) = ^ ^ i ( 0 ) - We then choose

another unit vector ^3(0) that is in the range of P(0), but is orthogonal to both ^1(0) and

^2(0). We then let ^4(0) = K ^3(0). For Type II corepresentations of minimal multiplicity,

D(K?) = — 1, and it follows that the four vectors form an orthonormal basis for the range