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Electronic structure of correlated electron systems home work for lectures 5 and 6 due March 2 2011 1. Show that for a model in which we have assume a uniformly polarizable medium with a very large dielectric constant, outside a radius R from an atom, the reduction of U cannot lead to a net attractive interaction U between two valence electrons. 2. Show that the net calculation of U in the models used for the polarizability can be done by either calculating the energy of two electron or holes on the same atom relative to the energy of two electron or holes on well separated atoms or by calculating the ionization potential and electron affinity in the polarizable medium. 3. Calculate the polarizability of an atom assuming a point charge nucleus in a spherical uniform distribution of electron charge with a radius R. 4. Calculate the increase in the singlet triplet splitting for a Hydrogen molecule in the Heitler London limit (U>>t)in free space relative to one lying flat at one Angstrom above the surface of a metal. 5. Describe how, in terms of mixing of atomic wave functions like s and p, a dipole moment is formed if an external electric field is applied. Determine the polarizability in terms of matrix elements involving the electric field and the energy separations of atomic states. Will the response of such a system be linear in electric field? 6. Show that the point charge contribution to the crystal field in first order perturbation theory for d orbitals in cubic symmetry is equal for the two eg orbitals and also for the three t2g orbitals but different for eg and t2g.