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
(4.39) |
where is the lattice displacement. The last two parts of the Hamiltonian are once again diagonalized via the unitary operator where
(4.40) |
Using the expansion of Eq. 4.29 we find for the transformed phonon and electron annihilation operators:
(4.41) |
(4.42) |
Substituting these transformed operators we find in the new basis
(4.43) |
and for the breathing-Hamiltonian
(4.45) |
where
(4.46) |
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
(4.49) |
This definition is identical to that found for the Holstein polaron besides the overall factor of .
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