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 (Formula presented) is utilized to express the correlation energy of a many-electron system in terms of (Formula presented) 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 (Formula presented) 0) makes the correlation energy per electron too negative by about 0.5 eV, the adiabatic local-density approximation (Formula presented) 0)] makes a comparable error in the opposite direction. The adiabatic nonlocal approximation (Formula presented) 0)] reduces this error to about 0.1 eV, and inclusion of the full frequency dependence (Formula presented) 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.