We have carried out collinear and noncollinear electronic structure calculations to investigate the structural, electronic, and magnetic properties of isolated Cr atoms, dimers, and compact trimers. We find that the Cr monomer prefers to adsorb on the fcc hollow site with a binding energy of 3.13 eV and a magnetic moment of 3.93 B . The calculated Kondo temperature of 0.7 K for the monomer is consistent with the lack of a Kondo peak in scanning tunneling microscopy ͑STM͒ experiments at 7 K. The compact Cr dimer orders antiferromagnetically and its bond length contracts to 1.72 Å close to the value for the free-standing Cr dimer. The very low magnetic moment of 0.005 B for the Cr atoms in the dimer is due to the strong d-d hybridization between the Cr adatoms. Thus, these calculations reveal that the absence of the Kondo effect observed in STM experiments is due to the small local moments rather than the Kondo quenching of the local moments suggested experimentally. The Cr compact trimer exhibits noncollinear coplanar magnetism with vanishing net magnetic moment in agreement with experiment.