In early 1980 we have initiated broken-symmetry (BS) MO theoretical calculations of transition-metal oxo species M=O (M=Ti,V,Cr,Mn,Fe,Ni,Cu) to elucidate the nature of dσ-pσ and dπ-pπ bonds. It has been concluded that high-valent M=O species such as [Mn(IV)=O]2+ and [Fe(IV)=O]2+ exhibit electrophilic property in a sharp contrast with nucleophilic character of low-valent M=O bonds: [M(II)O2-] 0, and closed-shell dπ-pπ bonds of high-valent M=O species often suffer the triplet-instability, giving rise to open-shell (BS) configurations with significant metal-diradical (MDR) character: •M-O•: note that these bonds are therefore regarded as typical examples of strongly correlated electron systems. Because of the MDR character, 1,4-metal diradical mechanism was indeed preferable to four-centered mechanism in the case of addition reaction of naked Mn(IV)=O to ethylene. Recently the manganese-oxo species have been receiving renewed interest in relation to catalytic cycle of oxygen evolution from water molecules in the photosynthesis II (PSII) system. Accumulated experimental results indicate that this process is catalyzed with four manganese oxide clusters coordinated with calcium ion (CaMn4O4). Past decade we have performed BS MO theoretical investigations of manganese oxide clusters related to CaMn4O 4. These calculations have elucidated that high-valent Mn(X)=O (X=IV,V) bonds exhibit intermediate MDR character (y=40-60%) in the case of total low-spin (LS) configuration but the MDR character decreases with coordination of Ca2+ and water molecules. While the MDR character of the Mn-oxo bonds becomes very high at the high-spin (HS) configuration. Our computational results enabled us to propose two possible mechanisms on the theoretical ground: (A) electrophilic (EP) mechanism and (B) radical coupling (RC) mechanism. The theoretical results indicate that the EP mechanism is preferable for the low-spin (LS) state in polar media like in the protein environments (native OEC), whereas the RC mechanism is feasible at the state without such environmental stabilization: local singlet and local triplet diradical mechanisms are proposed for the OO coupling process. Possibilities of EP and RC mechanisms are examined in comparison with a lot of experimental results accumulated and theoretical results with several groups.