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

In the framework of time-dependent density functional theory (TDDFT), the exact exchange-correlation (xc) kernel ${f}_{\mathrm{xc}}(n,q,\ensuremath{\omega})$ 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 ${f}_{\mathrm{xc}}(n,q,\ensuremath{\omega})$ of 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 ${f}_{\mathrm{xc}}$ 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, ${lim}_{q\ensuremath{\rightarrow}0}{f}_{\mathrm{xc}}(n,q,\ensuremath{\omega})=\ensuremath{-}\ensuremath{\alpha}(\ensuremath{\omega}){e}^{2}/{q}^{2}$, of a weakly inhomogeneous electron gas, using MCP07 and other common xc kernels. The strong wave-vector and frequency dependence of the ``ultranonlocality'' coefficient $\ensuremath{\alpha}(\ensuremath{\omega})$ is demonstrated for a variety of simple metals and semiconductors. We examine how imposing exact constraints on an approximate kernel shapes $\ensuremath{\alpha}(\ensuremath{\omega})$. Comparisons to kernels derived from correlated-wave-function calculations are drawn.