Time-dependent Schrieffer-Wolff Transformation

Consider the transformation $$|psi'rangle=e^S|psirangle$$ We are looking for the transformed Hamiltonian $H'$ such that $$H'|psi'rangle=ihbar{dover dt}|psi'rangle Leftrightarrow H|psirangle=ihbar{dover dt}|psirangle$$ Replacing by the expression of $|psi'rangle$, one gets $$eqalign{ H'e^S|psirangle &=ihbar{dover dt}left(e^S|psirangleright)cr &=ihbar left({de^Sover dt}+e^S{dover dt}right)|psiranglecr &=left(ihbar{de^Sover dt}+e^SHright)|psiranglecr }$$ Multiplying both sides on the left by $e^{-S}$, $$e^{-S}H'e^S|psirangle =left(ihbar e^{-S}{dSover dt}+Hright)|psirangle$$ for any $|psirangle$, so $$e^{-S}H'e^S=ihbar e^{-S}{dSover dt}+H$$ and finally $$H'=ihbar {dSover dt}e^{-S}+e^SHe^{-S}$$