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 fₗ₂ (r, r^';) is utilized to express the correlation energy of a many-electron system in terms of fₗ₂. 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 (fₗ₂ = 0) makes the correlation energy per electron too negative by about 0. 5 eV, the adiabatic local-density approximation fₗ₂ = fₗ₂ (q = 0, = 0) makes a comparable error in the opposite direction. The adiabatic nonlocal approximation fₗ₂ = fₗ₂ (q, = 0) reduces this error to about 0. 1 eV, and inclusion of the full frequency dependence fₗ₂ = fₗ₂ (q, ) 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.