  We have studied deuterium fractionation on interstellar grains with the use
of an exact method known as the direct master equation approach. We consider
conditions pertinent to dense clouds at late times when the hydrogen is mostly
in molecular form and a large portion of the gas-phase carbon has already been
converted to carbon monoxide. Hydrogen, oxygen and deuterium atoms, as well as
CO molecules, are allowed to accrete on to dust particles and react there to
produce various stable molecules. The surface abundances, as well as the
abundance ratios between deuterated and normal isotopomers, are compared with
those calculated with the Monte Carlo approach. We find that the agreement
between the Monte Carlo and the direct master equation methods can be made as
close as desired. Compared with previous examples of the use of the direct
master equation approach, our present method is much more efficient. It should
now be possible to run large-scale gas-grain models in which the diffusive dust
chemistry is handled `exactly'.
