  We investigate the role of the geometric phase (GP) in an internal conversion
process when the system changes its electronic state by passing through a
conical intersection (CI). Local analysis of a two-dimensional linear vibronic
coupling (LVC) model Hamiltonian near the CI shows that the role of the GP is
twofold. First, it compensates for a repulsion created by the so-called
diagonal Born-Oppenheimer correction (DBOC). Second, the GP enhances the
non-adiabatic transition probability for a wave-packet part that experiences a
central collision with the CI. To assess the significance of both GP
contributions we propose two indicators that can be computed from parameters of
electronic surfaces and initial conditions. To generalize our analysis to
N-dimensional systems we introduce a reduction of a general N-dimensional LVC
model to an effective 2D LVC model using a mode transformation that preserves
short-time dynamics of the original N-dimensional model. Using examples of the
bis(methylene) adamantyl and butatriene cations, and the pyrazine molecule we
have demonstrated that their effective 2D models reproduce the short-time
dynamics of the corresponding full dimensional models, and the introduced
indicators are very reliable in assessing GP effects.
