Why does a time-dependent perturbed Hamiltonian commonly have diagonal elements equal to zero?
Because the perturbation is what is going to mix the eigenstates of the unperturbed Hamiltonian. But the diagonal elements could be non-zero: an actual physical perturbation could also shift the energy levels. Most often, it is easier to work with a perturbation that only mixes the states, so I would move the diagonal elements to the "unperturbed" Hamiltonian in such a case.
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