The expressions (13) of the classical force and ( ) of the evolution of the electronic coefficients define the CTMQC algorithm. The main approximations introduced to derive them are: (i) in the definition of the electron-nuclear coupling operator, Eq. ( ), the first term is neglected , as well as its contribution in the expression of the TDPES to maintain the gauge invariance; (ii) the spatial variation of the coefficients is expressed in the form ∇ ν C α j (t) = i f α ν,j C α j (t), with the force for state j accumulated along the trajectory f α ν,j = -t 0 ∇ ν E α j dτ ; (iii) the TDVP along a trajectory, using the expansion of the electronic amplitude in the adiabatic basis in its definition (6), is approximated as