  We study the vibrational spectrum of the protonated water dimer, by means of
a divide-and-conquer semiclassical initial value representation of the quantum
propagator, as a first step in the study of larger protonated water clusters.
We use the potential energy surface from [Huang et al., J. Chem. Phys. 122,
044308 (2005)]. To tackle such an anharmonic and floppy molecule, we employ
fully Cartesian dynamics and carefully reduce the coupling to global rotations
in the definition of normal modes. We apply the time-averaging filter and
obtain clean power spectra relative to suitable reference states, that
highlight the spectral peaks corresponding to the fundamental excitations of
the system. Our trajectory-based approach allows us for physical interpretation
of the very challenging proton transfer modes. We find that it is important,
for such a floppy molecule, to selectively avoid to initially excite lower
energy modes, in order to obtain cleaner spectra. The estimated vibrational
energies display a mean absolute error (MAE) of $\sim29\text{cm}^{-1}$ with
respect to available Multi-Configuration time-dependent Hartree calculations
and $\text{MAE}\sim14\text{cm}^{-1}$ when compared to the optically active
experimental excitations of the Ne-tagged Zundel cation. The reasonable scaling
in the number of trajectories for Monte Carlo convergence is promising for
applications to higher dimensional protonated cluster systems.
