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We unveil the microscopic origin of largely debated magnetism in theMo$_{3}$O$_{8}$ cluster systems. Upon considering an extended Hubbard model at1/6 filling on the anisotropic kagom\'e lattice formed by the Mo atoms, weargue that its ground state is determined by the competition between kineticenergy and intersite Coulomb interactions, which is controlled by thetrimerisation of the kagom\'e lattice into the Mo$_{3}$O$_{13}$ clusters. Basedon first-principles calculations, we show that the strong interaction limit isrealised in LiZn$_{2}$Mo$_{3}$O$_{8}$ revealing a plaquette charge order withunpaired spins at the resonating hexagons, whose origin is solely related tothe opposite signs of intracluster and intercluster hoppings, in contrast toall previous scenarios. On the other hand, both Li$_{2}$InMo$_{3}$O$_{8}$ andLi$_{2}$ScMo$_{3}$O$_{8}$ are demonstrated to fall into the weak interactionlimit where the electrons are well localised at the Mo$_{3}$O$_{13}$ clusters.While the former is found to exhibit long-range antiferromagnetic order, thelatter is more likely to reveal short-range order with quantum spin liquid-likeexcitations. Our results not only reproduce most of the experimentally observedfeatures of these unique materials, but will also help to describe variousproperties in other quantum cluster magnets.

Quantum spin liquid and cluster Mott insulator phases in the Mo$_{3}$O$_{8}$ magnets