  Density-functional-theory (DFT) calculations have been performed on the Li-Si
and Li-Ge systems. Lithiated Si and Ge, including their metastable phases, play
an important technological r\^ole as Li-ion battery (LIB) anodes. The
calculations comprise structural optimisations on crystal structures obtained
by swapping atomic species to Li-Si and Li-Ge from the X-Y structures in the
International Crystal Structure Database, where X={Li,Na,K,Rb,Cs} and
Y={Si,Ge,Sn,Pb}. To complement this at various Li-Si and Li-Ge stoichiometries,
ab initio random structure searching (AIRSS) was also performed. Between the
ground-state stoichiometries, including the recently found Li$_{17}$Si$_{4}$
phase, the average voltages were calculated, indicating that germanium may be a
safer alternative to silicon anodes in LIB, due to its higher lithium insertion
voltage. Calculations predict high-density Li$_1$Si$_1$ and Li$_1$Ge$_1$
$P4/mmm$ layered phases which become the ground state above 2.5 and 5 GPa
respectively and reveal silicon and germanium's propensity to form dumbbells in
the Li$_x$Si, $x=2.33-3.25$ stoichiometry range. DFT predicts the stability of
the Li$_{11}$Ge$_6$ $Cmmm$, Li$_{12}$Ge$_7$ $Pnma$ and Li$_7$Ge$_3$ $P32_12$
phases and several new Li-Ge compounds, with stoichiometries Li$_5$Ge$_2$,
Li$_{13}$Ge$_5$, Li$_8$Ge$_3$ and Li$_{13}$Ge$_4$.
