16

GEORGE A. HAGEDORN

identifying H ^ (C 0 (T2 © Ran (1 - P(X)), we can locally represent /i(X) by the matrix

(2.2)

'EA{X) 0 0 0 \

^

V

x , 0 EB(X) 0 0

"W

=

I 0 0 EB(X) 0

0 0 0

hL{X)J

Throughout our discussion no restrictions have been imposed on the Ck functions EA(X)

and Efi(X), except that they take the same value at the origin. Thus, they could be any

two Ck functions whose values coincide at the origin. Generically the values of two such

functions coincide on a submanifold T of codimension 1.

In our applications in the subsequent sections, we have a nuclear momentum vector rj(0)

that is not tangent to the manifold I\ We rotate the nuclear coordinate system so that the

gradient of EA{X) — EB(X) is parallel to the Xi-axis and 77(0) has a positive component

in the X\ direction.

STRUCTURE O F CROSSINGS O F T Y P E S D, E, AND H. Suppose two eigenvalues

EA{X) and EB{X) of an electron Hamiltonian function h(X) have a crossing of type D,

E, or H at X = 0. By mimicking the constructions used for Type F and G crossings, we see

that we can choose four smooth, mutually orthogonal unit-vector valued functions £^ i{X),

*MX) = K * A l ( * ) ' *B,l(X), and *Bt2(X) = C ^ i ( X ) , such that $

A l

( X ) , and

$.4 2P O a r e eigenvectors of h(X) with eigenvalue EA{X), and $ 3 1 (X), and $32{X) are

eigenvectors of h(X) with eigenvalue EB(X). Furthermore, whenever a type III corepre-

sentation is involved, the eigenvector with second subscript 1 belongs to one representation

of the subgroup H and the eigenvector with the second subscript 2 belongs to the other

representation of the subgroup.

As in the earlier constructions, by using these vectors as part of a basis, and identifying

H = (D2 © (C2 0 Ran (1 - P(X)), we can locally represent h{X) by the matrix

h(X) =

(EA{X)

0

0

0

^ 0

0

EA(X)

0

0

0

0

0

EB{X)

0

0

0

0

0

EB(X)

0

0

\

0 1

0

0

h\X)J

(2.3)

Throughout our discussion no restrictions have been imposed on the functions EA(X) and

EB(X), except that they take the same value at the origin. Thus, they could be any two Ck

functions whose values coincide at the origin. Generically the values of two such functions

coincide on a submanifold T of codimension 1.