Despite the rarity, large negative linear compressibility (NLC) was observed in metal-organic framework material Zn(HO3PC4H8PO3H)∙2H2O (ZAG-4) in experiment. We find a unique NLC mechanism in ZAG-4 based on first-principle calculations. The key component to realize its large NLC is the deformation of H3O(+) tetrahedron. With pressure increase, the oxygen apex approaches and then is inserted into the tetrahedron base (hydrogen triangle). The tetrahedron base subsequently expands, which results in the b axis expansion. After that, the oxygen apex penetrates the tetrahedron base and the b axis contracts. The negative and positive linear compressibility is well reproduced by the hexagonal model and ZAG-4 is the first MOFs evolving from non re-entrant to re-entrant hexagon framework with pressure increase. This gives a new approach to explore and design NLC materials.