Fröhlich canonical transformation

An anti-Hermitian generator $S$ is chosen so $[S,H_0]=-H_{e\text{-}ph}$. Expanding $e^{S}H{e}^{-S}$ leaves $H_0+[S,H_{e\text{-}ph}]/2$ to second order. Eliminating virtual phonons produces an effective electron-electron interaction, plus energy shifts. Resonant denominators and higher orders limit this perturbative elimination.