  Lithium-rich complex transition-metal oxides Li$_2$MoO$_3$, Li$_2$RuO$_3$,
Li$_3$RuO$_4$, Li$_3$NbO$_4$, Li$_5$FeO$_4$, Li$_5$MnO$_4$ and their
derivatives are of interest for high-capacity battery electrodes. Here, we
report a first-principles density-functional theory study of the atomic and
electronic structure of these materials using the Heyd-Scuseria-Ernzerhof (HSE)
screened hybrid functional which treats all orbitals in the materials on equal
footing. Dimerization of the transition-metal ions is found to occur in layered
Li$_2$MoO$_3$, in both fully lithiated and partially delithiated compounds. The
Ru--Ru dimerization does not occur in fully lithiated Li$_2$RuO$_3$, in
contrast to what is commonly believed; Ru--Ru dimers are, however, found to
occur in the presence of lithium vacancies caused by lithium loss during
synthesis and/or lithium removal during use. We also analyze the electronic
structure of the complex oxides and discuss the delithiation mechanism in these
battery electrode materials.
