Gas phase Elemental abundances in Molecular cloudS (GEMS) VIII. Unlocking the CS chemistry: the CH + S$\rightarrow$ CS + H and C$_2$ + S$\rightarrow$ CS + C reactions

We revise the rates of reactions CH + S -> CS + H and C_2 + S -> CS + C, important CS formation routes in dark and diffuse warm gas. We performed ab initio calculations to characterize the main features of all the electronic states correlating to the open shell reactants. For CH+S we have calculated the full potential energy surfaces for the lowest doublet states and the reaction rate constant with a quasi-classical method. For C_2+S, the reaction can only take place through the three lower triplet states, which all present deep insertion wells. A detailed study of the long-range interactions for these triplet states allowed to apply a statistic adiabatic method to determine the rate constants. This study of the CH + S reaction shows that its rate is nearly independent on the temperature in a range of 10-500 K with an almost constant value of 5.5 10^{-11} cm^3/s at temperatures above 100~K. This is a factor \sim 2-3 lower than the value obtained with the capture model. The rate of the reaction C_2 + S depends on the temperature taking values close to 2.0 10^{-10} cm^3/s at low temperatures and increasing to 5. 10^{-10} cm^3/s for temperatures higher than 200~K. Our modeling provides a rate higher than the one currently used by factor of \sim 2. These reactions were selected for involving open-shell species with many degenerate electronic states, and the results obtained in the present detailed calculations provide values which differ a factor of \sim 2-3 from the simpler classical capture method. We have updated the sulphur network with these new rates and compare our results in the prototypical case of TMC1 (CP). We find a reasonable agreement between model predictions and observations with a sulphur depletion factor of 20 relative to the sulphur cosmic abundance, but it is not possible to fit all sulphur-bearing molecules better than a factor of 10 at the same chemical time.