  We propose to measure the importance of spin-orbit couplings (SOCs) in the
nonadiabatic molecular quantum dynamics rigorously with quantum fidelity. To
make the criterion practical, quantum fidelity is estimated efficiently with
the multiple-surface dephasing representation (MSDR). The MSDR is a
semiclassical method that includes nuclear quantum effects through interference
of mixed quantum-classical trajectories without the need for the Hessian of
potential energy surfaces. Two variants of the MSDR are studied, in which the
nuclei are propagated either with the fewest-switches surface hopping or with
the locally mean field dynamics. The fidelity criterion and MSDR are first
tested on one-dimensional model systems amenable to numerically exact quantum
dynamics. Then, the MSDR is combined with "on-the-fly" computed electronic
structure to measure the importance of SOCs and nonadiabatic couplings (NACs)
in the photoisomerization dynamics of CH2NH2+ considering 20 electronic states
and in the collision of F + H2 considering six electronic states.
