We found from DFT calculations that Ag-Ag orbital interactions as well as Ag-O electrostatic interactions determine the structures of three silver cations inside a nanometer-sized cavity of ZSM-5 (Ag-3-ZSM-5) in lower and higher spin states. Both interactions strongly depend on the number of Al atoms substituted for Si atoms on the ZSM-5 framework (ZSM-5(Al-n), where n ranges from 1 to 3. In smaller n, stronger Ag-Ag orbital interactions and weaker Ag-O electrostatic interactions operate. Accordingly; there are significant dependencies of the structures of three silver Cations on the number of Al atoms In lower spin states of Ag-3-ZSM-5(Al-1) and Ag-3-ZSM-5(Al-2), D-3h-like triangle clusters are contained inside ZSM-5 whereas their higher spin states have triangle clusters distorted significantly from the D-3h structure. In lower spin states, the totally symmetric orbital consisting of 5s(Ag) orbitals is responsible for cluster formation, whereas in higher spin states occupation of a 5s(Ag)-based orbital with one node results in significant distortion of the triangle clusters. The: distortion can be partially understood by analogies to Jahn-Teller distortion of the bare D-3h Ag3+ cluster in the triplet spin state. When n is 3, we found that three silver cations are isolated in a lower spin state and that a linear cluster consisting of two silver cations is formed in a higher spin state. Thus, we demonstrate from DFT calculations that the number of Al atoms can control the properties of three silver cations inside a ZSM-5 cavity. Since the structural and electronic features of the enclosed silver clusters can link to their catalytic properties, the DFT findings can help us to understand the catalytic activity of Ag-ZSM-5.