  By using a second quantized formulation of level crossing, which does not
assume adiabatic approximation, a convenient formula for geometric terms
including off-diagonal terms is derived. The analysis of geometric phases is
reduced to a simple diagonalization of the Hamiltonian in the present
formulation. If one diagonalizes the geometric terms in the infinitesimal
neighborhood of level crossing, the geometric phases become trivial for any
finite time interval $T$. The topological interpretation of Berry's phase such
as the topological proof of phase-change rule thus fails in the practical
Born-Oppenheimer approximation, where a large but finite ratio of two time
scales is involved.
