I am trying to understand the theory behind the optical calculation and encountered a problem. In the reference [Phys. Rev. B 67, 165332 (2003) - Linear and second-order optical response of III-V monolayer superlattices] they define the electron velocity in equation A5, but I don’t know what is the w_{mn}. I am wondering either it should be omega. Another problem is how to calculate the diagonal part. Since only the diagonal part appears in the final expression, I think this should be important.
\omega_{mn}(\boldsymbol{k}) = \omega_m(\boldsymbol{k}) - \omega_n(\boldsymbol{k}) is simply the energy difference between eigenstates. You can see its definition if you dive into the ref’d papers [61] (https://journals.aps.org/prb/pdf/10.1103/PhysRevB.53.10751). I think they used w instead of \omega by mistake.
Thank you very much for your kind reply and the article. They are really helpful. I understand how to calculate the off-diagonal part with Em not equal to En. However, I am still wondering how to get the diagonal part because the delta_{mn} in the expression is the difference between the diagonal parts.
Sorry for the late reply. I agree that the definition for electron velocity is OK. I change the resource code and let the ABINIT print the momentum matrix elements. However, I find that the diagonal part is 0 for GaAs and LiNbO3 primitive cells. I am wondering when the diagonal elements are nonvanishing.
The situation is a bit complex. The diagonal elements of the d/dk=r operator for Bloch wavefunctions can always be made zero thanks to a k-dependent phase factor (or a set of unitary transformations among degenerate wavefunctions), for every localized region around some k. However, this is not possible when one is interested in the computation of the static electric polarization, which must be done using the Berry phase approach, as the k-dependent phase factors cannot always be modified globally in the whole Brillouin Zone to have a vanishing diagonal component contribution, as one must respect the k-point periodicity. By contrast, for the linear optical reponse, it is clear that only the non-diagonal matrix elements contribute to the final value. For the non-linear optical response at finite frequency, this is not discussed in the Sharma et al papers. I have not followed the litterature recently, while I know some people have continued to work on this topics, for the MBPT treatment of non-linear optic. Anyhow, in the ABINIT implementation, one sticks to the formulation published by Sharma et al., where the diagonal matrix elements are neglected. Whether this is an approximation, or an exact result at that level of the formalism is not clear to me.
In Sharma’s papers, there are some terms including Delta_{mn} for the intraband and the modification contributions. I think these terms are related to the diagonal part of the transition dipole. When we integral over the whole BZ, these terms may not be 0 in some area of BZ, so we can not just ignore them. Is it the correct understanding?