  The essential properties of graphite-based 3D systems are thoroughly
investigated by the first-principles method. Such materials cover a simple
hexagonal graphite, a Bernal graphite, and the stage-1 to stage-4 Li/Li$^+$
graphite intercalation compounds. The delicate calculations and the detailed
analyses are done for their optimal stacking configurations, bong lengths,
interlayer distances, free electron $\&$ hole densities, Fermi levels,
transferred charges in chemical bondings, atom- or ion-dominated energy bands,
spatial charge distributions and the significant variations after
intercalation, Li-/Li$^+$- $\&$ C-orbital-decomposed DOSs. The above-mentioned
physical quantities are sufficient in determining the critical orbital
hybridizations responsible for the unusual fundamental properties. How to
dramatically alter the low-lying electronic structures by modulating the
quest-atom/quest-ion concentration is one of focuses, e.g., the drastic changes
on the Fermi level, band widths, and number of energy bands. The theoretical
predictions on the stage-n-dependent band structures could be examined by the
high-resolution angle-resolved photoemission spectroscopy (ARPES). Most
important, the low-energy DOSs near the Fermi might provide the reliable data
for estimating the free carrier density due to the interlayer atomic
interactions or the quest-atom/quest-ion intercalation. The van Hove
singularities, which mainly arise from the critical points in
energy-wave-vector space, could be directly examined by the experimental
measurements of scanning tunneling spectroscopy (STS). Their features should be
very useful in distinguishing the important differences among the stage-$n$
graphite intercalation compounds, and the distinct effects due to the atom or
ion decoration.
