Based on the equivalence of the gauge transformations for the nuclear and electronic wave functions, the physical meaning of the geometric phase effect has been disclosed for a two coupled-state system. It is found that the geometric phase A(R) is defined by the argument of the complex electronic vector state in the complex plane spanned by the two real-valued electronic components. Such an angle is identical (up to a constant) to the mixing angle gamma(R). Novel generalized Born-Oppenheimer equations for the two coupled-state problem in the vicinity of the crossing seam have been derived, and numerical calculations of vibrational spectra done for H-3. The results demonstrate significant differences in relation to those obtained from the assumption that A(R)=phi/2.