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Constraint-based Wavevector- and Frequency-dependent Exchange-Correlation Kernel of the Uniform Electron Gas

According to time-dependent density functional theory, the exact exchange-correlation kernel f$_{xc}$(n, q, $ω$) determines not only the ground-state energy but also the excited-state energies/lifetimes and time-dependent linear density response of an electron gas of uniform density n $=$ 3/(4$π$r$^3_s$). Here we propose a parametrization of this function based upon the satisfaction of exact constraints. For the static ($ω$ = 0) limit, we modify the model of Constantin and Pitarke at small wavevector q to recover the known second-order gradient expansion, plus other changes. For all frequencies $ω$ at q $=$ 0, we use the model of Gross, Kohn, and Iwamoto. A Cauchy integral extends this model to complex $ω$ and implies the standard Kramers-Kronig relations. A scaling relation permits closed forms for not only the imaginary but also the real part of f$_{xc}$ for real $ω$. We then combine these ingredients by damping out the $ω$ dependence at large q in the same way that the q dependence is damped. Away from q $=$ 0 and $ω$ $=$ 0, the correlation contribution to the kernel becomes dominant over exchange, even at r$_s$ $=$ 4, the valence electron density of metallic sodium. The resulting correlation energy from integration over imaginary $ω$ is essentially exact. The plasmon pole of the density response function is found by analytic continuation of f$_{xc}$ to $ω$ just below the real axis, and the resulting plasmon lifetime first decreases from infinity and then increases as q grows from 0 toward the electron-hole continuum. A static charge-density wave is found for r$_s$ $>$ 69, and shown to be associated with softening of the plasmon mode. The exchange-only version of our static kernel confirms Overhauser's 1968 prediction that correlation enhances the charge-density wave.