The electron-phonon interaction term in the B Hamiltonian (see last term in Eq. 4.2) can, by changing the order of summation, be re-write it in the form
where is the lattice displacement. The last two parts of the Hamiltonian are once again diagonalized via the unitary operator where
Using the expansion of Eq. 4.29 we find for the transformed phonon and electron annihilation operators:
Substituting these transformed operators we find in the new basis
and for the breathing-Hamiltonian
In the case of a system only containing a single electron the density terms contained in Eq. 4.44 simplify, yielding
Hence, the lattice deformation energy is found to be
In contrast to the H model (Eq. 4.37) we incur an additional factor of due to a coupling with two neighboring oscillators. For higher dimensionality the number of neighboring oscillators is given by the coordination number , thus replacing the factor in Eq. 4.48 with .
We thus define the dimensionless coupling constant for the electron-breathing-phonon interaction
This definition is identical to that found for the Holstein polaron besides the overall factor of .