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We present new models of the deuterium chemistry in protoplanetary disks,including, for the first time, multiply deuterated species. We use these modelsto explore whether observations in combination with models can give us clues asto which desorption processes occur in disks. We find, in common with otherauthors, that photodesorption can allow strongly bound molecules such as HDO toexist in the gas phase in a layer above the midplane. Models including thisprocess give the best agreement with the observations. In the midplane, cosmicray heating can desorb weakly bound molecules such as CO and N$_2$. We find theobservations suggest that N$_2$ is gaseous in this region, but that CO must beretained on the grains to account for the observed DCO$^+$/HCO$^+$. This couldbe achieved by CO having a higher binding energy than N$_2$ (as may be the casewhen these molecules are accreted onto water ice) or by a smaller cosmic raydesorption rate for CO than assumed here, as suggested by recent theoreticalwork.  For gaseous molecules the calculated deuteration can be greatly changed bychemical processing in the disk from the input molecular cloud values. On thegrains singly deuterated species tend to retain the D/H ratio set in themolecular cloud, whereas multiply deuterated species are more affected by thedisk chemistry. Consequently the D/H ratios observed in comets may be partlyset in the parent cloud and partly in the disk, depending on the molecule.Comment: Accepted for publication in ApJ. 48 pages, 8 figure

The Chemistry of Multiply Deuterated Molecules in Protoplanetary Disks. I. The Outer Disk