High-order geometric integrators for representation-free Ehrenfest dynamics

Ehrenfest dynamics is a useful approximation for ab initio mixed quantum-classical molecular that can treat electronically nonadiabatic effects. Although severe to the exact solution of time-dependent Schr\"odinger equation, symplectic, time-reversible, and conserves exactly total energy as well norm electronic wavefunction. Here, we surpass apparent complications due coupling classical nuclear quantum motions present efficient geometric integrators "representation-free" dynamics, which do not rely on diabatic or adiabatic representation states are arbitrary even orders accuracy in time step. These numerical integrators, obtained by symmetrically composing second-order splitting method solving kinetic potential propagation steps, norm-conserving, time-reversible regardless step used. Using simulation region conical intersection an example, demonstrate these preserve properties and, if highly accurate solutions desired, be more than most popular non-geometric integrators.