Na-ion batteries offer an attractive low-cost alternative to Li-ion batteries. Although graphite is used as the negative electrode in conventional Li-ion batteries, attempts to use it in Na-ion batteries have been hampered by the inability of Na to form graphite intercalation compounds (GICs) under moderate conditions. It is generally considered that this is due to the size mismatch between Na and the graphite interlayer spacings, but here we show with detailed first-principles calculations of Li, Na, K, Rb, and Cs GICs that the major reason is the change in chemical bonding between alkali metal (AM) ion and C atoms. van der Waals correction terms are introduced to better reproduce the layered graphite structure, and calculated formation energies of GICs AMC₆, where AM = Li, Na, K, Rb, and Cs, are found to become less negative (less stable) as ion size decreases from Cs to Na as a result of weakening ionic bonding, until the formation energy of NaC₆ becomes positive. The much smaller Li ion represents an exception to this trend, as its bonds with C atoms contain a covalent component, resulting in a negative formation energy. These subtle differences in staging for the different alkali metals explain why graphite is a good intercalation material for Li and K but not for Na.

A theoretical investigation has been made on the phonon spectrum and heat capacity of polymorphs of carbon and boron nitride with special interests on the variation of Debye temperature and stiffness with temperature. A part of optical phonon branches of graphite exhibits higher frequencies than those of diamond. As a consequence, graphite shows smaller heat capacity and higher Debye temperature than diamond above a crossover temperature of $1000\mathrm{K}$. This supports experimental reports of heat capacity although available experimental data are widely scattered. The higher Debye stiffness of graphite at above $1000\mathrm{K}$ is not contradictory to the fact that conventional stiffness of diamond is much larger than that of graphite, since the Debye stiffness is determined by both acoustic and optical phonons, whereas only acoustic phonons contribute to the conventional stiffness. The same trend was found between hexagonal and cubic boron nitrides with a crossover temperature of $600\mathrm{K}$.

We evaluate mean frequencies for atomic jumps in a crystal from first principles based on transition state theory, taking lithium diffusion by the interstitial and vacancy mechanisms in ${\text{LiC}}_6$ as a model case. The mean jump frequencies are quantitatively evaluated from the potential barriers and the phonon frequencies for both initial and saddle-point states of the jumps under the harmonic approximation. The lattice vibrations are treated within quantum statistics, not using the conventional treatment by Vineyard corresponding to the classical limit, and the discrepancy between the two treatments is quantitatively discussed. The apparent activation energies and the vibrational prefactors of the mean jump frequencies essentially depend on temperature, unlike in the case of the classical approximation. The discrepancies of the activation energies correspond to the changes in zero-point vibrational energy at 0 K, and there remains the effect even at 1000 K. With regard to the vibrational prefactors, the classical approximation extremely overestimates the prefactors at low temperatures while the discrepancies rapidly decrease with increasing temperature, e.g., by 30% at room temperature and by 5% at 1000 K. The calculated chemical diffusion coefficients of lithium atoms by the interstitial and vacancy mechanisms are $1\times {10}^{-11}$ and $1\times {10}^{-10}{\text{cm}}^2/\text{s}$, respectively.