  Small-wavevector excitations in Coulomb-interacting systems can be decomposed
into the high-energy collective longitudinal plasmon and the low-energy
single-electron excitations. At the critical wavevector and corresponding
frequency where the plasmon branch merges with the single-electron excitation
region, the collective energy of the plasmon dissipates into single
electron-hole excitations. The jellium model provides a reasonable description
of the electron-energy-loss spectrum (EELS) of metals close to the
free-electron limit. The random phase approximation (RPA) is exact in the
high-density limit but can capture the plasmonic dispersion reasonably even for
densities with rs > 1. RPA and all beyond-RPA methods investigated here, result
in a wrong infinite plasmon lifetime for a wavevector smaller than the critical
one where the plasmon dispersion curve runs into particle-hole excitations.
Exchange-correlation kernel corrections to RPA modify the plasmon dispersion
curve. There is however a large difference in the construction and form of the
kernels investigated earlier. Our current work introduces recent model
exchange-only and exchange-correlation kernels and discusses the relevance of
some exact constraints in the construction of the kernel. We show that, because
the plasmon dispersion samples a range of wavevectors smaller than the range
sampled by the correlation energy, different kernels can make a strong
difference for the correlation energy and a weak difference for the plasmon
dispersion. This work completes our understanding about the plasmon dispersion
in realistic metals, such as Cs, where a negative plasmon dispersion has been
observed. We find only positive plasmon dispersion in jellium at the density
for Cs.
