Electron correlation energies from scaled exchange-correlation kernels: Importance of spatial versus temporal nonlocality

Within density functional theory, a coordinate-scaling relation for the coupling-constant dependence of the exchange-correlation kernel $f_{\mathrm{xc}}(\mathbf{r},{\mathbf{r}}^{\prime };\omega )$ is utilized to express the correlation energy of a many-electron system in terms of $f_{\mathrm{xc}}.$ As a test of several of the available approximations for the exchange-correlation kernel, or equivalently the local-field factor, we calculate the uniform-gas correlation energy. While the random phase approximation ${(f}_{\mathrm{xc}}$ $=$ 0) makes the correlation energy per electron too negative by about 0.5 eV, the adiabatic local-density approximation $[f_{\mathrm{xc}}$ $=$ $f_{\mathrm{xc}}(q$ $=$ $0,\omega$ $=$ 0)] makes a comparable error in the opposite direction. The adiabatic nonlocal approximation $[f_{\mathrm{xc}}$ $=$ $f_{\mathrm{xc}}(q,\omega$ $=$ 0)] reduces this error to about 0.1 eV, and inclusion of the full frequency dependence $[f_{\mathrm{xc}}$ $=$ $f_{\mathrm{xc}}(q,\omega )]$ in an approximate parametrization reduces it further to less than 0.02 eV. We also report the wave-vector analysis and the imaginary-frequency analysis of the correlation energy for each choice of kernel.