Exact solutions of the Holstein model with different site energies

The two-site Holstein model with different site energies, the simplest model to mimic disorder, is analytically studied by a diagonalization method of coherent states. The solutions obtained in our work are exact, and agree well with those derived by a modified Lang-Firsov (MLF) method in the weak- and strong-coupling regimes. The deviation from the MLF solution in the intermediate-coupling regime implies that our solution, including the higher order correlation terms overlooked in the MLF treatment, gives a more accurate description. To check the validity of our approach, we have also calculated the case with the same site energies, which shows that our exact solution could be consistent with a previous treatment of the Holstein model using the coherent expansion method [Phys. Rev. B 65, 174303 (2002)]. The method and the results in the present work would be useful for testing various approximate methods for the Holstein model, and are applicable to more complicated situations.