  On-the-fly quantum nonadiabatic dynamics for large systems greatly benefits
from the adiabatic representation readily available from the electronic
structure programs. However, frequently occurring in this representation
conical intersections introduce non-trivial geometric or Berry phases which
require a special treatment for adequate modelling of the nuclear dynamics. We
analyze two approaches for nonadiabatic dynamics using the time-dependent
variational principle and the adiabatic representation. The first approach
employs adiabatic electronic functions with global parametric dependence on the
nuclear coordinates. The second approach uses adiabatic electronic functions
obtained only at the centres of moving localized nuclear basis functions (e.g.
frozen-width Gaussians). Unless a gauge transformation is used to enforce
single-valued boundary conditions, the first approach fails to capture the
geometric phase. In contrast, the second approach accounts for the geometric
phase naturally because of the absence of the global nuclear coordinate
dependence in the electronic functions.
