Direct methods for non-adiabatic dynamics: connecting the single-set variational multi-configuration Gaussian (vMCG) and Ehrenfest perspectives

In this article, we outline the current state-of-the-art on-the-fly methods for non-adiabatic dynamics, highlighting the similarities and differences between them. We derive the equations of motion for both the Ehrenfest and variational multi-configuration Gaussian (vMCG) methods from the Dirac-Frenkel variational principle. We explore the connections between these two methods by presenting an alternative derivation of the vMCG method, which gives the Ehrenfest equations of motion when taking the appropriate limits.