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) =
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