The energy structure of the V$_{15}$ is studied using exact diagonalization methods, and the adiabatic changes of the magnetization in the system with the sweeping field are investigated. We confirm that the Dzyaloshinskii-Moriya interaction leads to an energy gap which allows the adiabatic change of the magnetization which has been found experimentally by Chiorescu et al., and also predict novel dynamics of the magnetization due to this energy level structure.
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