  Spin tunneling in molecular magnets has attracted much attention, however
theoretical considerations of this phenomenon up to now have not taken into
account the many-spin nature of molecular magnets. We present, to our
knowledge, the first successful attempt of a realistic calculation of tunneling
splittings for Mn$_{12}$ molecules, thus achieving a quantitatively accurate
many-spin description of a real molecular magnet in the energy interval ranging
from about 100 K down to 10$^{-12}$ K. Comparison with the results of the
standard single-spin model shows that many-spin effects affect the tunneling
splittings considerably. The values of ground state splitting given by
single-spin and many-spin models differ from each other by a factor of five.
