In several important approximate treatments in molecular collision theory, the S- and T-matrices are local in some subset of coordinates. Linear factorization relations and consistency conditions are derived for the matrix elements of such local operator and sometimes for the appropriately averaged square of the magnitude of these matrix elements. The coefficients in these relations and conditions are "spectroscopic" (i.e., dynamics independent). Using these relations, one can predict scattering information (S/T-matrices and degeneracy averaged cross sections) for any transition using corresponding information for transitions out of (or into) a fixed, but arbitrary, input state. The relevancy of these relations to various forms of the sudden approximation is explored;Factorization in the sudden approximation of atom-diatom phenomeno-logical cross sections, which are important in transport coefficient calculations, is also examined;Finally, the energy sudden (ES) factorization relations for the T- matrix elements in a dissociative collision are studied. The theory is applied to collinear collisions between an atom and a truncated square well, diatomic oscillator. Under certain conditions, vibrational enhancement/inhibition as a function of the final dissociative state;can be predicted without any dynamic calculation of the transition probabilities. Some results of calculations are discussed; *DOE Report IS-T-1066. This work was performed under Contract No. W-7405-Eng-82 with the U.S. Department of Energy.