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Aims. We aim to improve a previous model for the prediction of columndensities and deuterium fractions of non- and singly deuterated methanol.Thereby, we try to identify crucial chemical and physical parameters, for whichthe study of deuteration could provide valuable additional constraints.  Methods. We employed a gas-grain chemical code to devise a model that is inagreement with the observed column density and deuterium fraction profiles ofthe innermost region of the pre-stellar core L1544. For that purpose, wedeveloped a new treatment of reactive desorption, deriving an individualreactive desorption efficiency for every product species in a chemicalreaction, that depends on the reaction enthalpy and type of underlying surface.Furthermore, we explored several options to promote the diffusion of hydrogenand deuterium atoms over the surface of interstellar dust grains, in order toincrease methanol formation.  Results. Our fiducial model employs diffusion by quantum tunneling ofhydrogen and deuterium atoms, resulting in CH$_3$OH and CH$_2$DOH columndensities that are approximately an order of magnitude lower than the observedvalues, which improves the results compared to the previous model by a factor10. The $N$(CH$_2$DOH)/$N$(CH$_3$OH) ratio is reproduced within a factor of 1.2for the centre and 1.8 for the position of the methanol peak. Given the largeuncertainties that chemical models typically have, we consider our predictionsto be in agreement with the observations. In general, we conclude that adiffusion process with a high diffusion rate needs to be employed to obtainmethanol column densities that are in accordance with the observed values.Also, we find that the introduction of abstraction reactions into the methanolformation scheme suppresses deuteration, when used in combination with a highdiffusion rate.

Modelling Deuterated Isotopologues of Methanol toward the Pre-Stellar Core L1544