  A general solution for vacancy-mediated diffusion in the
dilute-vacancy/dilute-solute limit for arbitrary crystal structures is derived
from the master equation. A general numerical approach to the vacancy lattice
Green function reduces to the sum of a few analytic functions and numerical
integration of a smooth function over the Brillouin zone for arbitrary
crystals. The Dyson equation solves for the Green function in the presence of a
solute with arbitrary but finite interaction range to compute the transport
coefficients accurately, efficiently and automatically, including cases with
very large differences in solute-vacancy exchange rates. The methodology takes
advantage of the space group symmetry of a crystal to reduce the complexity of
the matrix inversion in the Dyson equation. An open-source implementation of
the algorithm is available, and numerical results are presented for the
convergence of the integration error of the bare vacancy Green function, and
tracer correlation factors for a variety of crystals including wurtzite
(hexagonal diamond) and garnet.
