The discussion centers on the relationship between energy bands in k-space and their representation in real space. It establishes that energy bands exist solely in k-space due to the diagonal nature of the effective mean field one-particle Hamiltonian, specifically the Fock operator/Kohn-Sham operator. Transforming the eigenstates of this operator into real space yields Wannier orbitals, which resemble atomic orbitals but do not diagonalize the Hamiltonian, indicating a lack of a direct e(r) relationship. The conversation emphasizes the complexity of electron motion in a periodic structure under an external electric field, as described by the E vs k relationship.
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cgk said:The bands only exist in k-space, since the effective mean field one-particle Hamiltonian (Fock operator/Kohn-Sham operator), of which the e(k) are the eigenvalues, is diagonal in k-space, but not in real space.
If you transform the eigenstates of this operator (the crystal orbitals) into real space, you get the Wannier orbitals, which look closely like normal atomic orbitals (in particular, they are identical in each unit cell). But these Wannier orbitals do not diagonalize the effective one-particle Hamiltonian anymore, so there is no e(r) relation in this sense.