MOLECULAR PROPAGATION THROUGH LEVEL CROSSINGS

17

In our applications, we have a nuclear momentum vector 7/(0) that is not tangent to the

manifold V We rotate the nuclear coordinate system so that the gradient of EA(X) — Eg(X)

is parallel to the Xi-axis and rj(0) has a positive component in the X\ direction.

STRUCTURE O F T Y P E I CROSSINGS Suppose a

Ck

electron Hamiltonian function

h(X) has a type I crossing of two simple eigenvalues EA(X) and Eg{X) at X = 0. For X in

a neighborhood of the origin, one can write the spectral projection P(X) for h(X) associated

with the eigenvalues EA(X) and Eg(X) as an integral of the resolvent of h(X). From this it

follows that P(X) is a Ck, rank 2 operator valued function of X near X = 0. Furthermore,

it follows that EA(X) + EB(X) = trace (h(X)P(X)) is a Ck function of X. Thus,

fcl(X) = h{X)-$(EA{X) + EB{X))

is a C operator-valued function whose restriction to the range of P(X) is traceless.

Let { Vi, ^2 } D e a basis for the range of P(0). By altering the phases of these two

vectors, we may assume that JCipi = ip\ and JCip2 = "02 where K is the antiunitary

operator chosen for the decomposition G = H U /C#. Define ipi(X) for X by

V ( ^ i , P ( A - ) ^ i

Since P(X ) is Cfc and commutes with the action of G, ip\(X) is well defined and Ck in

some neighborhood of the origin and satisfies JCipi(X) = ij)\{X). Let P\(X) denote the

projection onto the subspace spanned by ipi(X). It is a C operator-valued function in a

neighborhood of X that commutes with P(X) and the action of G. We define

MX) = Pixni-wm

^feP(X)(l-F,(IM)

This vector valued function is also Ck in a neighborhood of the origin; JCip2(X) = ^ P O i

and { ipi(X), ip2(X) } is an orthonormal basis for the range of P(X), for X in a neighborhood

of the origin.

In the basis {ipi(X), ^ P O } t n e restriction of h\(X) to the range of P(X) is repre-

sented by a real symmetric, traceless 2 x 2 matrix valued function M\(X) whose entries are

Ck functions that all vanish when X — 0. That is,

Ml{X)

' U w -P(X)J