Constraint-based wave vector and frequency dependent exchange-correlation kernel of the uniform electron gas

According to time-dependent density functional theory, the exact exchange-correlation kernel ${f}_{xc}(n,q,\ensuremath{\omega})$ for wave vector q and frequency $\ensuremath{\omega}$ 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 $n=3/(4\ensuremath{\pi}{r}_{s}^{3})$. Here we propose a parametrization of this function based upon the satisfaction of exact constraints. For the static $(\ensuremath{\omega}=0)$ limit, we modify the model of Constantin and Pitarke to recover at small q the known second-order gradient expansion, and to correct its approach to the large q limit. For all $\ensuremath{\omega}$ at $q=0$, we use the model of Gross, Kohn, and Iwamoto. A Cauchy integral extends this model to complex $\ensuremath{\omega}$. Scaling relations are identified. We then combine these ingredients, damping out the $\ensuremath{\omega}$ dependence at large q. Away from $q=0$ and $\ensuremath{\omega}=0$, the correlation contribution to the kernel becomes dominant over exchange, even at ${r}_{s}=4$. The resulting correlation energies for $1\ensuremath{\le}{r}_{s}\ensuremath{\le}10$ from integration over imaginary $\ensuremath{\omega}$ are essentially exact. The plasmon pole of the density response function is found by analytic continuation of ${f}_{xc}$ to $\ensuremath{\omega}$ just below the real axis, and the resulting plasmon lifetime at ${r}_{s}=$ 4 is found for $q<{k}_{F}$. A static charge-density wave is found for ${r}_{s}>69$, and shown to be associated with softening of the plasmon mode.