MOLECULAR PROPAGATION THROUGH LEVEL CROSSINGS 21
We need only show that by a proper choice of 0, the columns of I _ I can be forced
to be orthogonal. We choose 0 so that I . * 1 and I ) are an orthonormal basis
of eigenvectors for the real symmetric matrix AtA) where A = ( 1. A simple
\C3 dsJ
computation then shows that I ) and I _ I are orthogonal to one another.
W
\d3J
We can now rotate the Xi and X% coordinate directions so that c and d point along the
positive X2 and X3 directions, respectively. By adding TT to our choice of 0, we can change
the signs of both c and d. By interchanging ip\(X) and ip2(X) we can change the sign of d
without altering c.
Thus, we can arrange for c\ = 0, ci 0, C3 = 0, d\ 0, di 0, and d% 0. This
proves our claims.
STRUCTURE O F CROSSINGS OF T Y P E K Suppose a
Ck
electron Hamiltonian function
h(X) has a Type K crossing of two multiplicity 2 eigenvalues E^{X) and Eg(X) at X = 0.
As in the earlier constructions, we let P(X) be the spectral projection for h(X) corresponding
to both the eigenvalues E^(X) and EQ(X). This projection has rank 4, and its range is
the direct sum of a two dimensional subspace that lies in the carrier subspace for the D
representation of the subgroup H G G, and a two dimensional subspace that lies in the
carrier subspace for the C representation. We arbitrarily pick two orthonormal vectors ^i(0)
and ^2(0) t
n a
t lie in the range of P(0) and in the carrier subspace for the D representation.
We let
MX) = ,
P{X)M0)
.
y/(1l(0),P(X)M0))
We let P\(X) denote the orthogonal projection onto the span of ip\(X) and define
MX) =
(i-fiW)PW^(o)
V(V-2(o), (i-Pi(x))P(x)v2(o)
In a neighborhood of the origin, these two vectors form an orthonormal basis for the in-
tersection of the range of P(X) and the carrier subspace for the D representation. We let
ip$(X) = JCipi(X) and V4p0 = ^V2p0- Then ^(X) and ip^X) form an orthonormal
basis for the intersection of the range of P(X) and the carrier subspace for the C represen-
tation. The set of all four vectors is an orthonormal basis for the range of P(X).
In this basis, the restriction of
M X ) = h(X) - \(EA(X) + EB(X))
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