  We propose an approximate method for evaluating the importance of
non-Born-Oppenheimer effects on the quantum dynamics of nuclei. The method uses
a generalization of the dephasing representation (DR) of quantum fidelity to
several diabatic potential energy surfaces and its computational cost is the
cost of dynamics of a classical phase space distribution. It can be implemented
easily into any molecular dynamics program and also can utilize on-the-fly ab
initio electronic structure information. We test the methodology on three model
problems introduced by Tully and on the photodissociation of NaI. The results
show that for dynamics close to the diabatic limit the decay of fidelity due to
nondiabatic effects is described accurately by the DR. In this regime, unlike
the mixed quantum-classical methods such as surface hopping or Ehrenfest
dynamics, the DR can capture more subtle quantum effects than the population
transfer between potential energy surfaces. Hence we propose using the DR to
estimate the dynamical importance of diabatic, spin-orbit, or other couplings
between potential energy surfaces. The acquired information can help reduce the
complexity of a studied system without affecting the accuracy of the quantum
simulation.
