  Ru and Rh are interesting cases for comparing equations of state (EOS),
because most general purpose EOS are semi-empirical, relying heavily on shock
data, and none has been reported for Ru. EOS were calculated for both elements
using all-electron atom-in-jellium theory, and cold compression curves were
calculated for the common crystal types using the multi-ion pseudopotential
approach. Previous EOS constructed for these elements used Thomas-Fermi (TF)
theory for the electronic behavior at high temperatures, which neglects
electronic shell structure; the atom-in-jellium EOS exhibited pronounced
features from the excitation of successive electron shells. Otherwise, the EOS
matched surprisingly well, especially considering the lack of experimental data
for Ru. The TF-based EOS for Ru may however be inaccurate in the
multi-terapascal range needed for some high energy density experiments. The
multi-ion calculations predicted that the hexagonal close-packed phase of Ru
remains stable to at least 2.5 TPa and possibly 10 TPa, and that its c/a should
gradually increase to the ideal value. A method was devised to estimate the
variation in Debye temperature from the cold curve, and thus estimate the
ion-thermal EOS without requiring relatively expensive dynamical force
calculations, in a form convenient for adjusting EOS or phase boundaries. The
Debye temperature estimated in this way was similar to the result from
atom-in-jellium calculations. We also predict the high-pressure melt loci of
both elements.
