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Observations of the second solar spectrum(SSS)revealed the existence of prominent linear polarization signals due to lines of the C2 molecule.Interpretation of the SSS is the only tool to obtain the weak and turbulent magnetic field which is widespread in the Quiet Sun.However,this interpretation is conditioned by the determination of accurate collisional data.In this context,we present a formulation of the problem of the calculation of the polarization transfer rates by collisions of polarized C2 states with electrons.The obtained formulae are applied to determine,for the first time,the polarization transfer rates between the C2 states of the Swan band electronic system(a 3Πu-d3Π a)and electrons for temperatures going up from 1000 to 10000K.However,due to the closeness of the electronic states of the C2 molecule,the two electronic d 3rIg and a 3Πu cannot be disconnected from the other electronic levels and,thus,a model based on only two states is not sufficient to describe the formation of the lines in the Swan band.Consequently,we also calculated the collisional polarization transfer rates in the case where the first eight electronic states of C2 are taken into account.All rates are given as functions of the temperature by power laws.Our results should be useful for future solar applications.