18

GEORGE A. HAGEDORN

where j3 and 7 are real valued Ck functions. The eigenvalues Ej^{X) and Eg(X) cross

precisely at those points X where f3(X) = j(X) = 0. Generically this defines a codimension

2 submanifold I\ Furthermore, the difference between Ej[(X) and E^{X) is the same as

the difference between the eigenvalues of M\(X). By direct computation, the eigenvalues of

MX(X) are

± y V ) 2 + 7(X)2.

Generically this function is continuous, but not differentiable near T. One can easily show

that the eigenvectors are not even continuous near I\

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

where

M / v

, fbX c-X \

N™

= [c-X -b-X ) '

for some vectors b and c. Generically b and c are linearly independent. By a rotation of

the coordinate system we may assume that only the first two components of b and c are

non-zero.

If 77(0) is a generic nuclear momentum vector, then we can rotate the first two coordinate

axes so that the projection of 77(0) into the two dimensional subspace spanned by b and c

lies along the positive X\ axis.

At this point, the Xj coordinates for j 2 no longer play a role in the structure of

N(X). Furthermore, the form of N(X) is not altered if we do X-independent orthogonal

transformations of the two dimensional space spanned by the basic electronic wave func-

tions ipi(X) and ^2{X). We replace X/JI(X) by cos(6)ipi(X) + sin(6)^2(X) and tp2(X) by

— sin(^)'0i(X) + cos(9)ipi(X). A simple calculation shows that we can choose 0 so that

the Xi-component of c is zero. Finally, by possibly interchanging the order of ipi(X) and

il2(X) or multiplying one of them by —1, we can assume that the Xi-component of b and

the ^-component of c are both positive (recall that b and c are linearly independent, so b

must have a non-zero Xi-component).

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

N

(X) =

(blXi

+ b2X2 c2X2 \

\ c2X2 -b\Xi - b2X2 J '

where 61 and c2 have the same sign. So, by identifying ?i = (C©(C© Ran (1 — P(X))i we