The last two terms of the Holstein Hamiltonian (Eq. 4.1) can be diagonalized via the Lang-Firsov canonical transformation(39) defined by a unitary operator where

(4.28) |

Using the expansion

we find for the transformed phonon annihilation operator:

(4.30) |

and similarly for the electron operator

(4.31) |

The tilde mark is used to label the transformed operators. The Holstein Hamiltonian in the new basis can be written as

(4.32) |

(4.34) |

where

(4.35) |

For a single electron the density term in Eq. 4.33 simplifies, yielding

One can define a dimensionless coupling constant for the electron-phonon interaction as the ratio between the gained lattice deformation energy (see second term in )

and the bare electron kinetic energy, taken to be the half-bandwidth, ,