  We analyze two approaches to the quantum-classical Liouville (QCL) formalism
that differ in the order of two operations: Wigner transformation and
projection onto adiabatic electronic states. The analysis is carried out on a
two-dimensional linear vibronic model where geometric phase (GP) effects
arising from a conical intersection profoundly affect nuclear dynamics. We find
that the Wigner-then-Adiabatic (WA) QCL approach captures GP effects, whereas
the Adiabatic-then-Wigner (AW) QCL approach does not. Moreover, the Wigner
transform in AW-QCL leads to an ill-defined Fourier transform of double-valued
functions. The double-valued character of these functions stems from the
nontrivial GP of adiabatic electronic states in the presence of a conical
intersection. In contrast, WA-QCL avoids this issue by starting with the Wigner
transform of single-valued quantities of the full problem. Since the WA-QCL
approach uses solely the adiabatic potentials and non-adiabatic derivative
couplings as an input, our results indicate that WA-QCL can capture GP effects
in general two-state crossing problems using first-principles electronic
structure calculations without prior diabatization or introduction of explicit
phase factors.
