  Starting from Holstein's work on small polaron hopping, the evolution
equations for localized and extended states in the presence of atomic
vibrations are systematically derived for an amorphous semiconductor. The
transition probabilities are obtained for transitions between all combinations
of localized and extended states. For any transition process involving a
localized state, the activation energy is not simply the energy difference
between the final and initial states; the reorganization energy of atomic
configuration is also included as an important part of the activation energy
(Marcus form). The activation energy for the transitions between localized
states decreases with rising temperature and leads to the Meyer-Neldel rule.
The predicted Meyer-Neldel temperatures are consistent with observations in
several materials. The computed field-dependence of conductivity agrees with
experimental data. The present work suggests that the upper temperature limit
of variable range hopping is proportional to the frequency of first peak of
phonon spectrum. We have also improved the description of the photocurrent
decay at low temperatures. Analysis of the transition probability from an
extended state to a localized state suggests that there exists a short-lifetime
belt of extended states inside the conduction band or valence band.
