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

• Received 24 May 2021
• Revised 30 July 2021
• Accepted 23 August 2021
• Corrected 16 December 2021

DOI:https://doi.org/10.1103/PhysRevB.104.125112