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.
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