Two-state approximation in the adiabatic and sudden-perturbation limits

The properties of the two-state approximation are considered from the point of view of atomic collision theory in the limit of large and small values of a characteristic collision time $T$. For large $T$ (the adiabatic limit) asymptotically exact expressions are obtained for the elastic-scattering phase shifts and for the nonadiabatic transition probability due to the pseudocrossing of terms. This approximation is carried out under fairly general assumptions about the Hamiltonian, enabling us to consider such processes as transitions between $\Sigma -\Pi$ terms caused by rotation of an internuclear axis. Such general problems of the adiabatic approximation as the applicability of adiabatic perturbation theory, the introduction of a dynamical basis, and the properties of the electronic wave functions in the pseudocrossing region are discussed. For small $T$ (the sudden-peturbation limit) the evolution operator to zeroth and first order in $T$ is calculated. We introduce a general and unambiguous definition of an adiabatic basis as a basis of eigenvectors of the evolution matrix to zeroth order in $T$.