MOLECULAR PROPAGATION THROUGH LEVEL CROSSINGS

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meaning of this is defined later). This is the generic situation. We prove in subsequent

sections that the mean nuclear momentum rj(t) is continuous near the time of the crossing

so that there is a well defined non-zero vector 77(0) if the classical path associated with the

nuclear wave packet hits the crossing at time t = 0. Generically, 77(0) is not tangent to the

manifold on which the eigenvalues involved in the crossing coincide.

Many of the different types of crossings may be analyzed in similar ways. For the sake

of brevity we have organized the discussion below to exploit this fact. In addition, we have

described the structures of the various types in order of increasing complexity.

STRUCTURE O F CROSSINGS O F T Y P E S A AND C. Suppose two eigenvalues E^(X)

and Eg(X) of a C electron Hamiltonian function h{X) have a crossing of Type A or Type

C at X — 0. By properly labeling the eigenvalues, we may assume that E^(X) corresponds

to one irreducible representation or corepresentation U\ of G for all X, and that Eg(X)

corresponds to U2 for all X. Since h(X) commutes with the action of G, it follows that

h(X) commutes with the orthogonal projections P\ and P2 onto the mutually orthogonal

carrier subspaces associated with U\ and L^, respectively.

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

h(X) associated with both the eigenvalues Ej^{X) and Eg(X) as

p{x) = hj{z~h{x))~ldz'

where C is a contour that encloses E^(X) and Eg(X) but no other parts of the spectrum

of h(X). From this it follows that P{X) is a Ck, rank 2 operator valued function of X near

X = 0 that commutes with Pj and P^. Since U\ and U2 are inequivalent, it follows that

PA(X) = P\P(X) and P&(X) = P2P(X) are Ck, rank one orthogonal projections that

project onto mutually orthogonal subspaces.

For Type A crossings, we arbitrarily choose $.4(0) and $#(0) to be unit vectors in the

ranges of -PA(O) and -P^O), respectively. We then define

and

V(*B(O),PB(*)*B(O)'