Spin-dependent processes involving nonadiabatic transitions between electronic states with different spin multiplicities play important roles in the chemistry of complex systems. The rates of these processes can be predicted based on the molecular properties at the minimum energy crossing point (MECP) between electronic states. We present the development of the MECP search technique within the fragment molecular orbital (FMO) method applicable to large complex systems. The accuracy and scalability of the new method is demonstrated on several models of the metal-sulfur protein rubredoxin. The effect of the model size on the MECP geometry and relative energy is discussed. The fragment energy decomposition and spin density delocalization analyses reveal how different protein residues and solvent molecules contribute to stabilization of the spin states. The developed FMO-MECP method can help to clarify the role of nonadiabatic spin-dependent processes in complex systems and can be used for designing mutations aimed at controlling these processes in metalloproteins, including spin-dependent catalysis and electron transfer.