First-principles wave-vector- and frequency-dependent exchange-correlation kernel for jellium at all densities

We propose a spatially and temporally nonlocal exchange correlation (XC) kernel for the spin-unpolarized fluid phase of ground-state jellium for use in time-dependent density functional and linear response calculations. The kernel is constructed to satisfy known properties of the exact XC kernel to accurately describe the correlation energies of bulk jellium and to satisfy frequency-moment sum rules at a wide range of bulk jellium densities, including those low densities that display strong correlation and symmetry breaking. These effects are easier to understand in the simple jellium model than in real systems. All exact constraints satisfied by the recent MCP07 kernel [A. Ruzsinszky et al., Phys. Rev. B 101, 245135 (2020)] are maintained in the revised MCP07 (rMCP07) kernel, while others are added. The revision ${f}_{\mathrm{XC}}^{\text{rMCP07}}(q,\ensuremath{\omega})$ differs from MCP07 only for nonzero frequencies $\ensuremath{\omega}$. Only at densities much lower than those of real bulk metals is the frequency dependence of the kernel important for the correlation energy of jellium. As the wave vector $q$ tends to zero, the kernel has a $\ensuremath{-}4\ensuremath{\pi}\ensuremath{\alpha}(\ensuremath{\omega})/{q}^{2}$ divergence whose frequency-dependent ultranonlocality coefficient $\ensuremath{\alpha}(\ensuremath{\omega})$ vanishes in jellium, and is predicted by rMCP07 to be extremely small for the real metals Al and Na.