Progress towards understanding ultranonlocality through the wavevector and frequency dependence of approximate exchange-correlation kernels

In the framework of time-dependent density functional theory (TDDFT), the exact exchange-correlation (xc) kernel $f_{xc}(n,q,\omega)$ determines the ground-state energy, excited-state energies, lifetimes, and the time-dependent linear density response of any many-electron system. The recently developed MCP07 xc kernel $f_{xc}(n,q,\omega)$ of A. Ruzsinszky et al. [Phys. Rev. B 101, 245135 (2020)] yields excellent uniform electron gas (UEG) ground-state energies and plausible plasmon lifetimes. As MCP07 is constructed to describe $f_{xc}$ of the UEG, it cannot capture optical properties of real materials. To verify this claim, we follow Nazarov et al. [Phys. Rev. Lett. 102, 113001 (2009)] to construct the long-range, dynamic xc kernel, $\lim_{q\to 0}f_{xc}(n,q,\omega) = -\alpha(\omega)e^2/q^2$, of a weakly inhomogeneous electron gas, using MCP07 and other common xc kernels. The strong wavevector and frequency dependence of the "ultranonlocality" coefficient $\alpha(\omega)$ is demonstrated for a variety of simple metals and semiconductors. We examine how imposing exact constraints on an approximate kernel shapes $\alpha(\omega)$. Comparisons to kernels derived from correlated-wavefunction calculations are drawn.