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Problem 8

Consider a diatomic molecule with one orbital per atom. Assume the orbitals are orthogonal. Use linear combinations of these orbitals to construct the one-electron eigenstates for the molecule.

(1) Assuming only the on-site Coulomb matrix element U is important (i.e. neglecting all others):

(a) Find the energy of the singlet and triplet two-electron states within the Hartree-Fock approximation. For what values of U/t is ferromagnetism favored (i.e. triplet energy lower than singlet energy)?

(b) Same as (a) using the Heitler-London approximation for the singlet state. When do you expect the Heitler-London state to be a better approximation to the exact singlet ground state than the Hartree-Fock one?

(c) Same as (a) with the exact ground states of the system.

(d) What is the singlet-triplet energy splitting for large U/t?

(2) Including now all 2-center matrix elements of the Coulomb interaction, find conditions on the parameters for ferromagnetism to be favored

(a) Within the Hartree-Fock approximation

(b) Exactly

(c) What is now the singlet-triplet energy splitting and the condition for ferromagnetism for large U/t?

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