By standard perturbation theory, these unit-vector valued functions are C in a neighbor-
hood of the origin. The vectors $A(X) and $g(X) belong to the ranges of PA(X) and
PQ(X), respectively, for each X. One can easily show that $A(X) and $#(X ) are eigen-
vectors of h(X) that correspond to eigenvalues EA(X) and £g(X) , respectively. Standard
arguments also show that EA{-) and Eg(-) are Ck functions in a neighborhood of the origin.
For Type C crossings, we perform the same construction, but impose an additional
constraint. By decomposing G = H U KH, we have selected a special antiunitary element K
of G. A simple calculation shows that we may choose the phases of the vectors $^(0) and
^^(O) so that /C$^(0) $4(0) and /C$g(0) = $5(0). By making such choices we obtain
$A(X) a n d $B(X) t h a t
satisfy K$A(X) =
$A(X) a n d 1C®B(X)
®B(X)- I n a
subsequent section of the paper, we note that this choice trivializes the adiabatic connections
on the vector bundles of eigenvectors for these two eigenvalues.
Let h-L(X) denote the restriction of h(X) to the subspace orthogonal to the range
of P(X). By using $A(X) a n d ®B(X) a s a basis for the range of P(X) and identifying
H ^ ( D 0 ( D 0 Ran (1 - P(X)), we can locally represent h(X) by the matrix
(EA{X) 0 0 \
h(X) = 0 EB(X) 0 . (2.1)
V 0 0 h±(X)J
Throughout our discussion, no restrictions have been imposed on the functions EA(X) and
Eg(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 F of codimension 1.
In our applications in the subsequent sections, a non-zero nuclear momentum vector
77(0) occurs in our analysis. We assume rj(Q) is not tangent to I\ So, we can rotate the
nuclear coordinate system so that the gradient of EA(X) Eg(X) is parallel to the X\
coordinate axis, and 77(0) has a positive component in the X\ direction.
G. Suppose two eigenvalues EA(X)
and Eg(X) of a Ck electron Hamiltonian function h{X) have a crossing of Type F or Type G
at X 0. We may assume the eigenvalues are labeled so that the corepresentation associated
with EA(X) if of type /an d the corepresentation assiciated with Eg(X) is of type IIor III.
Let U\ and U2 denote the irreducible corepresentations of G associated with EA(X) and
Eg(X), respectively, and note that the dimension of the U2 is 2. As in the case of Type A or
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