# [SOLVED] The degenerate states in the nonlinear optical calculation

Based on the reference of the optic utility, the transition dipole moments are calculated from the momentum matrix element. However, the momentum matrix elements are always zero between two degenerate states. Since the optic utility calculates the optical properties from the momentum matrix, I am wondering how the degeneracy is handled in the code.

Dear Yunfan,
In the case of gapped materials, the optical properties are always computed from transitions starting from occupied states to unoccupied states. Two states that are both occupied or two states that are both unoccupied do not contribute. So, the momentum matrix elements between two degenerate states are irrelevant to compute the optical spectrum. Things are different for metals (the Drude contribution), but this is another story.
Best,
X.

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Thank you very much for your kind reply. I agree that the linear optical response is only related to the transition between valence and conduction states, but I still have a question about second-order nonlinear optical calculation.

According to the equation A9-A11 in the literature [Phys. Rev. B 67, 165332 (2003) - Linear and second-order optical response of III-V monolayer superlattices]. The nonlinear optical response may be related to the transition between degenerate states because l can be both valence and conduction states. I don’t know how to deal with this situation. Is it still irrelevant to the calculation?

You are right that such transitions between degenerate states should be considered in non-linear response. Unfortunately, they are not considered in the Sharma2003 paper (Appendix A) that you cite… In this paper, the transition between one level and itself is set to zero. I am not sure the case of degenerate states has been considered in the litterature … Usually, when there are degenerate states, there is also the freedom to make a unitary transform, so that the transition matrix becomes diagonal, and then, in a second step,

Thank you very much. I think that makes sense. The diagonal part of the transition dipole doesn’t have a contribution and we can always find a suitable superposition of the degenerate state to diagonal the matrix. Therefore the transition dipole between degenerate states should not contribute.