MOLECULAR PROPAGATION THROUGH LEVEL CROSSINGS

19

can locally approximate h\(X) by the matrix

/P(X) j(X) 0 \

~hl(X)= l{X) -P(X) 0 I (2.4)

V o o h{{x)j

where 0(X) = biX1 + b2X2 + 0(X2) and j(X) = c2X2 + 0(X2) with &i 0, c2 0, and

6

2

^ 0 .

STRUCTURE O F T Y P E B CROSSINGS Suppose a

Cfc

electron Hamiltonian function

h(X) has a type B crossing of two simple eigenvalues E^(X) and E${X) at X = 0. In this

situation we mimic the construction of the vectors ipi(X) and V2p0 m ^ n e c a s e °f a Type I

crossing. Since there is no anitunitary operator /C £ G, we choose arbitrary orthogonal unit

vectors ^i(O) and ip2(0) from the range of P(X), and then proceed with the construction.

This yields an orthonormal basis {^i(X), ip2(X) } for the range of P(X).

In this basis, the restriction of

MX) = M*)- ^ ( x ) + £e(*))

to the range of P(X ) is represented by a self-adjoint traceless 2 x 2 matrix valued function

M\(X) whose entries are Ck functions that all vanish when X — 0. That is,

0(X) j(X) + i6(X)\

y(X) - i6(X) -f3(X) J '

where /?, 7, and 6 are C^ real valued functions. The difference between Ej^{X) and Eg(X)

is the same as the difference between the eigenvalues of Mi (A"). By direct computation, the

eigenvalues of M\(X) are

± y//?(X)2 +

7

(X) 2 + 8{Xf.

Thus, the eigenvalues Ej^{X) and Eg(X) cross precisely at those points X where (3(X) =

7(A) = 5(X) — 0. Generically this defines a codimension 3 submanifold T. Further-

more, it is clear that the eigenvalues Ej±(X) and Eg(X) are continuous, but generically not

differentiable near I\

By standard Taylor series results, Mi(X) has the form Mi(X ) = Ni(X) + 0(||X|| 2 ),

where

b • X c-X + id-X\

c-X-id-X -b-X ) '

Mi(X) =

Ni(X) =