Download Homework 4.

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.