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))