Problem: Hybridization of Energy Bands

    In certain materials a narrow energy band, due to the d orbitals of the atoms, lies within a broad energy band, originating from s-type atomic orbitals. (For example, in copper the Fermi energy is in the middle of the broad band and the narrow band lies somewhat below ; in nickel cuts into the narrow band.) A simple one-dimensional model can be constructed from a tight-binding Hamiltonian for the broad band, , and from , to describe nonoverlapping, localized electrons. Here and create and annihilate an s-wave-type electron in unit cell i; and create and annihilate a localized electron. The electron operators obey the usual anticommutation relations. The total Hamiltonian includes a term to describe the possible hopping of the electron from the localized orbit to the delocalized state and back:  

The spin indices are not used, since none of the processes involve spin flips.

Calculate the energy spectrum of the of the electrons for (a) and (b) for .