Lattice vibrations of point defects are essential for understanding
non-radiative electron and hole capture in semiconductors as they govern
properties including persistent photoconductivity and Shockley-Read-Hall
recombination rate. Although the harmonic approximation is sufficient to
describe a defect with small lattice relaxation, for cases of large lattice
relaxation it is likely to break down. We describe a first-principles procedure
to account for anharmonic carrier capture and apply it to the important case of
the \textit{DX} center in GaAs. This is a system where the harmonic
approximation grossly fails. Our treatment of the anharmonic Morse-like
potentials accurately describes the observed electron capture barrier,
predicting the absence of quantum tunnelling at low temperature, and a high
hole capture rate that is independent of temperature. The model also explains
the origin of the composition-invariant electron emission barrier. These
results highlight an important shortcoming of the standard approach for
describing point defect ionization that is accompanied by large lattice
relaxation, where charge transfer occurs far from the equilibrium
configuration.