Local vertex corrections from exchange-correlation kernels with a discontinuity

The fundamental gap of an interacting many-electron system is given by the sum of the single-particle Kohn-Sham gap and the derivative discontinuity. The latter can be generated by advanced approximations to the exchange-correlation (XC) energy and is the key quantity to capture strong correlation with density functional theory (DFT). In this work we derive an expression for the derivative discontinuity in terms of the XC kernel of time-dependent density functional theory and demonstrate the crucial role of a discontinuity in the XC kernel itself. By relating approximate XC kernels to approximate local vertex corrections we then generate beyond-GW self-energies that include a discontinuity in the local vertex function. The quantitative importance of this result is illustrated with a numerical study of the local exchange vertex on model systems.