Quantum simulations are reported of the dynamics following the photoexcitation
Ba(1S)->Ba(1P) in Ba(Ar)10 and Ba(Ar)20 clusters. The
evolution in time is studied in a framework that treats quantum-mechanically
all the coupled degrees of freedom. The focus is on the role of 
non-adiabatic transitions between  the three adiabatic surfaces corresponding
to the P-states of the Ba atom. The timescales of electronic relaxation
and of electronic depolarization (orbital reorientation) are computed,
and the competition between adiabatic and non-adiabatic effects is assesed.
The calculations are carried out by a new scheme that extends the recent
Classically-Based Separable Potentials (CSP) method. Semiclassical
surface-hopping simulations are used to produce effective single-mode
potentials on which nuclear "orbitals" are then generated. The full
wavepacket is constructed from the electronic states involved, and from
these nuclear wavefunctions.
Among the main results we find that non-adiabatic transitions become
appreciable around 1 ps after photoexcitation, and they 
are stronger in the smaller cluster. Comparing
Tully's semiclasical method with the quantum simulations, good qualitative
agreement is found. Quantitatively, the semiclassical predictions for the electronic
states branching rations deviate from the quantum results roughly by a factor of
two after 1 ps. In the smaller cluster direct dissociation of the Ba
atom dominates over energy redistribution within the cluster, 
the opposite being true for the large system. This example demonstrates the
feasibility of quantum simulations of non-adiabatic processes in large systems with the
new method.