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The effective on-site Coulomb interaction (Hubbard U) is an important parameter for studying strongly correlated systems.While U is determined empirically by fitting to bulk values, its value for a cluster with finite number of atoms remains uncertain.Here, we choose Gd13 as a prototypical strongly correlated cluster whose magnetic moment is smaller than its bulk value, which is contrary to the well-known results in transition metal clusters.Using density functional theory and the linear response approach, we determine U self-consistently for the cluster and apply it to explain its anomalous magnetic properties.We demonstrate that the interaction between core and shell atoms of Gd13 cluster strongly depends on the the Hubbard U.For U=0 eV magnetism is governed by direct f-f electron interaction between core and shell atoms, while for U=5.5 eV it is the indirect RKKY interaction that prevails.We also demonstrate that the non-collinear spin arrangement of each atom in the cluster strongly depends on the Hubbard U.Monte Carlo calculations further confirm that magnetic moments decrease with temperature, thus addressing a long standing disagreement in experimental results.