Modeling layered intercalation compounds from first principles poses a problem, as many of their properties are determined by a subtle balance between van der Waals interactions and chemical or Madelung terms, and a good description of van der Waals interactions is often lacking. Using van der Waals density functionals we study the structures, phonons and energetics of the archetype layered intercalation compound Li-graphite. Intercalation of Li in graphite leads to stable systems with calculated intercalation energies of $-0.2$ to $-0.3$~eV/Li atom, (referred to bulk graphite and Li metal). The fully loaded stage 1 and stage 2 compounds LiC$_6$ and Li$_{1/2}$C$_6$ are stable, corresponding to two-dimensional $sqrt3timessqrt3$ lattices of Li atoms intercalated between two graphene planes. Stage $N>2$ structures are unstable compared to dilute stage 2 compounds with the same concentration. At elevated temperatures dilute stage 2 compounds easily become disordered, but the structure of Li$_{3/16}$C$_6$ is relatively stable, corresponding to a $sqrt7timessqrt7$ in-plane packing of Li atoms. First-principles calculations, along with a Bethe-Peierls model of finite temperature effects, allow for a microscopic description of the observed voltage profiles.
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