Geometric amplitude factors in adiabatic quantum transitions

The exponentially small probability of transition between two quantum states, induced by the slow change over infinite time of an analytic hamiltonian $\hat{H}$ = H($\delta $t)$\cdot \hat{\boldsymbol{S}}$ (where $\delta $ is a small adiabatic parameter and $\hat{\boldsymbol{S}}$ is the vector spin-$\frac{1}{2}$ operator), contains an additional factor exp {$\Gamma _{\text{g}}$} of purely geometric origin (that is, independent of $\delta $ and $\hslash $). For $\Gamma _{\text{g}}$ to be non-zero, $\hat{H}$ must be complex hermitian rather than real symmetric, and the hamiltonian curve H($\tau $) must not lie in a plane through the origin nor be a helix identical (up to rigid motion) with its time reverse. An expression is given for $\Gamma _{\text{g}}$, as an integral from the real t axis to the complex time of degeneracy of the two states. Explicit examples are given. The geometric effect could be observed in experiments with spinning particles.