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We report a computational study of the anion states of the resveratrol (RV) and resorcinol (RS) molecules, also investigating dissociative electron attachment (DEA) pathways. RV has well known beneficial effects in human health, and its antioxidant activity was previously associated with DEA reactions producing H$_2$. Our calculations indicate a valence bound state ($\pi^*_1$) and four resonances ($\pi^*_2$ to $\pi^*_5$) for that system. While the computed thermodynamical thresholds are compatible with DEA reactions producing H$_2$ at 0~eV, the well known mechanism involving vibrational Feshbach resonances built on a dipole bound state should not be present in RV. Our results suggest that the shallow $\pi^*_1$ valence bound state is expected to account for H$_2$ elimination, probably involving $\pi_1^*$/$\sigma_{\text{OH}}^*$ couplings along the vibration dynamics. The RS molecule is also an oxidant and a subunit of RV. Since two close-lying hydroxyl groups are found in the RS moiety, the H$_2$-elimination reaction in RV should take place at the RS site. Our calculations point out a correspondence between the anion states of RV and RS, and even between the thresholds. Nevertheless, the absence of bound anion states in RS, indicated by our calculations, is expected to suppress the H$_2$-formation channel at 0~eV. One is lead to conclude that the ethene and phenol subunits in RV stabilize the $\pi^*_1$ state, thus switching on the DEA mechanism producing H$_2$., Comment: 29 pages, 13 figures