  Collisions with background gas can perturb the transition frequency of
trapped ions in an optical atomic clock. We develop a non-perturbative
framework based on a quantum channel description of the scattering process, and
use it to derive a master equation which leads to a simple analytic expression
for the collisional frequency shift. As a demonstration of our method, we
calculate the frequency shift of the Sr$^+$ optical atomic clock transition due
to elastic collisions with helium.
