Ground state energy of the large polaron in a crystal with many LO phonon branches

The large polaron is studied in polyatomic crystals where more than one LO phonon branch is present. An upper bound to the ground state energy is obtained which is valid for all values of the electron-phonon coupling constants. For small coupling, we obtain the usual result that the ground state energy is the sum of the contributions of the different phonon branches. For large coupling, there is an extra contribution due to crossed terms between the different branches. Results are also given for intermediate coupling.