Interestingly, Eq. ( ) also appears in the derivation of the EF in Ref. 11. Eqs. ( ) and ( ) still depend on |v 1 ⟩, but they can be reformulated using only |v 1 ⟩ and Û without invoking explicitly the R, t-dependent basis, so that dynamics is fully encoded in χ and Û . In summary, we have derived the equations of motion for the electronic and nuclear contributions of the EF by exploiting the close relationship between the BH and EF representations.