General requirements for obtaining Dy(III) single-molecule magnets (SMM) were studied by CASSCF+RASSI calculations on both real and model systems. A set of 20 Dy(III) complexes was considered using their X-ray crystal structure for our calculations. Theoretical results were compared with their experimental slow relaxation data, and general conclusions about the calculated key parameters related with SMM behavior are presented. The effect of the coordination geometry and nature of ligands is discussed based on calculations on real and model systems. We found two different patterns to exhibit SMM behavior: the first one leads to the largest axial anisotropy in complexes showing heterolepticity of the ligand environment (more important than symmetric requirements), while the second one corresponds to sandwich-shaped complexes with a smaller anisotropy. Thus, most existing mononuclear zero-field SMMs adopting a heteroleptic coordination mode mixing neutral and anionic ligands present the same pattern in the electrostatic potential induced by their ligands, with a lower potential island related to the presence of neutral ligands inside a high potential background related with anionic groups. The existence of different electrostatic regions caused by the ligands induces a preferential orientation to reduce the electron repulsion for the electron density of the Dy(III) cations, resulting in the magnetic anisotropy.