Energy spectrum of the bound polaron

An eigenvalue problem for an electron interacting with a Coulomb center and a field of LO phonons is solved by a method of optimized canonical transformation. This method can be applied to arbitrary values of the electron-phonon coupling constant α. The energy eigenvalues for the 1s through 4f states have been calculated as function of α and of the ratio R of the donor rydberg $m_e$ $e^4$/2${ħ}^2$ ${\epsilon }_0^2$ to the LO-phonon energy ħω. These values are the upper bounds to the energy $E_{1s}$ of the ground state as well to all the energy levels of the excited states lying below $E_{1s}$+ħω. In a broad range of α and R, the present upper bounds are lower than previous variational results for the states 1s, 2s, and 2p. The energy levels for the 3s–4f states have been calculated for the first time by variational means. The calculated energy eigenvalues $E_{\mathrm{nl}}$ lie always below the corresponding hydrogenlike levels, i.e., $E_{\mathrm{nl}}$/ħω≤-α-R/$n^2$, where n and l are the principal and angular momentum quantum numbers, respectively. For all values of α and R, the following sequence of the energy levels for a given n has been obtained: $E_{\mathrm{nl}}$≤$E_{\mathrm{nl}\mathcal{’}}$ if l>l’. In particular, it leads to the positive Lamb shift $E_{2s}$-$E_{2p}$. The model of the bound polaron has been applied to the description of shallow donor spectra. The calculated values agree rather well with the measured 1s-2p transition energies for CdTe and ZnSe, and 1s-2s transition energies for CdS. For AgBr, AgCl, and ${\mathrm{CdF}}_2$ the upper bounds for the 1s level are too low, but the 2p-3p energy differences agree well with the experimental data. It means that the short-range donor potential neglected in the polaron model is repulsive for the considered impurities in the ionic crystals.