Extensive　density　functional　theoretical　(DFT)　and　ab　initio　[CCSD(T)]　calculations　were　combined　to　investigate　the　geometric　and　electronic　structure　of　a　series　of　dinuclear　hafnium　oxide　clusters,　Hf2On-/0　(n=1　similar　to　6).　DFT　calculations　were　performed　to　search　for　the　lowest　energy　structures　for　both　the　anionic　clusters　and　the　neutral　counterparts.　The　search　for　the　global　minima　was　performed　using　analytical　gradients　with　the　Stuttgart　relativistic　small　core　potential　and　the　valence　basis　sets　augmented　with　two　f-type　and　one　g-type　polarization　functions　for　Hf　and　the　aug-cc-pVTZ　basis　set　for　oxygen.　The　relative　energies　of　the　low-lying　structures　(within　ca.　0.35　eV)　were　further　evaluated　via　single-point　calculations　at　the　coupled　cluster　[CCSD(T)]　level　with　the　Hf/Stuttgart+2flg/O/aug-cc-pVTZ　basis　sets　at　the　B3LYP　geometries.　Generalized　Koopmans＇　theorem　(GKT)　was　applied　to　predict　VDEs　and　simulate　the　anion　photoelectron　spectra　(PES).　The　trends　of　structural　evolution　and　the　behavior　of　sequential　oxidation　of　the　Hf2On　(n=　1　similar　to　6)　clusters　were　observed.　For　the　anionic　species,　starting　from　the　C-2v　triangular　structure　Hf2O-　the　next　O　atom　in　the　Hf2O2-　cluster　was　again　bridge-bonded,　forming　a　rhombus　structure.　The　third　O　atom　occupied　the　terminal　site.　The　fourth　O　atom　favored　the　bridging　site　and　the　fifth　O　atom　occupied　the　terminal　site.　In　Hf2O6-,　the　additional　O　atom　was　bonded　to　a　terminal　site.　Molecular　orbital　analyses　were　performed　to　elucidate　the　chemical　bonding　and　the　structural　evolution　in　Hf(2)On(-)　(n=　1　similar　to　4)　clusters.　Spin　density　analyses　revealed　oxygen　radical,　diradical　and　superoxide　characters　in　the　oxygen-rich　clusters,　except　for　the　singlet　Hf2O5　cluster.　We　showed　that　Hf2O3　contains　a　localized　Hf2+　site,　which　can　readily　react　with　O-2　to　form　the　Hf2O5　cluster.　The　Hf2O6　-and　Hf2O6　clusters,　which　can　be　viewed　to　be　formed　by　the　interaction　of　Hf2O4-/0　and　O-2,　may　be　utilized　as　molecular　models　to　understand　dioxygen　activation　on　Hf2O4-　and　Hf2O4　clusters.