  We propose to measure nonadiabaticity of molecular quantum dynamics
rigorously with the quantum fidelity between the Born-Oppenheimer and fully
nonadiabatic dynamics. It is shown that this measure of nonadiabaticity applies
in situations where other criteria, such as the energy gap criterion or the
extent of population transfer, fail. We further propose to estimate this
quantum fidelity efficiently with a generalization of the dephasing
representation to multiple surfaces. Two variants of the multiple-surface
dephasing representation (MSDR) are introduced, in which the nuclei are
propagated either with the fewest-switches surface hopping (FSSH) or with the
locally mean field dynamics (LMFD). The LMFD can be interpreted as the
Ehrenfest dynamics of an ensemble of nuclear trajectories, and has been used
previously in the nonadiabatic semiclassical initial value representation. In
addition to propagating an ensemble of classical trajectories, the MSDR
requires evaluating nonadiabatic couplings and solving the Schr\"{o}dinger (or
more generally, the quantum Liouville-von Neumann) equation for a single
discrete degree of freedom. The MSDR can be also used to measure the importance
of other terms present in the molecular Hamiltonian, such as diabatic
couplings, spin-orbit couplings, or couplings to external fields, and to
evaluate the accuracy of quantum dynamics with an approximate nonadiabatic
Hamiltonian. The method is tested on three model problems introduced by Tully,
on a two-surface model of dissociation of NaI, and a three-surface model
including spin-orbit interactions. An example is presented that demonstrates
the importance of often-neglected second-order nonadiabatic couplings.
