  Dimethyl ether is one of the most abundant interstellar complex organic
molecules. Yet its formation route remains elusive. In this work, we have
performed electronic structure and kinetics calculations to derive the rate
coefficients for two ion-molecule reactions recently proposed as a gas-phase
formation route of dimethyl ether in interstellar objects, namely CH$_3$OH +
CH$_3$OH$_2^+$ $\rightarrow$ (CH$_3$)$_2$OH$^+$ + H$_2$O followed by
(CH$_3$)$_2$OH$^+$ + NH$_3$ $\rightarrow$ CH$_3$OCH$_3$ + NH$_4^+$. A
comparison with previous experimental rate coefficients for the reaction
CH$_3$OH + CH$_3$OH$_2^+$ sustains the accuracy of the present calculations and
allow a more reliable extrapolation at the low temperatures of interest in
interstellar objects (10-100 K). The rate coefficient for the reaction
(CH$_3$)$_2$OH$^+$ + NH$_3$ is, instead, provided for the first time ever. The
rate coefficients derived in this work essentially confirm the prediction by
Taquet et al. (2016) concerning dimethyl ether formation in hot cores/corinos.
Nevertheless, this formation route cannot be efficient in cold objects (like
prestellar cores) where dimethyl ether is also detected, because ammonia has a
very low abundance in those environments.
