Fröhlich Hamiltonian

A second-quantized electron-phonon Hamiltonian with free electrons, harmonic phonons and terms linear in phonon displacement, $M_q{c}_{k+q,\sigma }^{†}{c}_{k,\sigma }(b_q+b_{-q}^{†})$. The name is used for both general linear electron-phonon models and the more specific long-range coupling to polar optical modes; the intended dispersion and matrix element must be stated.

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.