# High-Temperature Thermodynamics Modeling of Graphite
## 4 Discussion  
When comparing our theory with other experiments, such as low-temperature phonon dispersions, Gibbs free energies (CALPHAD or EOS models representing experimental data), or specific heats, as shown in Figures 2-4, we found a rather satisfactory agreement in all cases. Nevertheless, van der Waals interactions, not included in the present density functional theory, play important roles in graphite. As we mentioned, freezing the axial c/a ratio in a hexagonal graphite structure mitigates some of the deficiencies of this theory and abates the need to include van der Waals contributions, which are computationally expensive and complex non-local electron exchange and correlations. Our theory relies on a far simpler LDA assumption for electron interactions. For the smaller 96-atom supercell, however, we studied the effect of the van der Waals interaction by including it in our VASP calculations within the Grimme DFT-D2 method [18]. As it turns out, it had almost no effect on the SCAILD phonon density of states, presumably because constraining both the volume and the c/a axial ratio suppressed the effect of the van der Waals interactions. The advantage of the LDA over the DFT-D2 method is computational expediency. Additionally, for this particular study, DFT-D2 has not yet been implemented in the FPLMTO all-electron code that we employ to calculate the electronic contributions.  
To better justify our simplification, we analyzed the calculated electronic structure and compared this with X-ray photoemission spectra.  
As shown in Figure 5, we contrast our calculated electronic density of states with X-ray photoemission experiments with 122 eV photons [41]. We notice that the density of states at the highest occupied energy level, the Fermi level, is very small. Consequently, the electronic contribution to the specific heat is also small, as mentioned in the previous section.  
Although not in agreement on a very detailed level with photoemission, the calculated density of states is consistent with the measurements. As discussed in [42], the appearance of the spectra between \(-20\) and \(-13\) eV, i.e., a stepwise increase from zero to a nearly constant value, indicates two-dimensional behavior. An analogous feature is observed also in the calculated density of states. Both measured and calculated spectra show a similarly significant peak located between \(-5\) and \(-10\) eV. The reasonable comparison between measured and modeled electronic structures, together with the acceptable accuracy of the  
Figure 5: Calculated (FPLMTO) electronic density of states (black line), broadened with a 20 mRy Gaussian, together with X-ray photoemission spectra with 122 eV photons (red line) [41]. The Fermi level is at zero energy, and it is indicated with a vertical dashed line.  
calculated thermodynamic properties discussed above, give us confidence in the quality of our graphite model.  
Lastly, we emphasize that constraining the behavior of the lattice vibrations to a quasi-harmonic model above about 1000 K leads to unacceptable inaccuracies in the thermodynamics of graphite.  
Conceptualization, P.S. and A.L.; methodology, P.S., A.L., R.Q.H., E.E.M. and A.P.; writing--original draft preparation, P.S.; writing--review and editing, P.S., A.L., R.Q.H., E.E.M., A.P., J.T.M. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
We acknowledge helpful discussions with [PERSON], [PERSON] and [PERSON]. This work was conducted under the auspices of the U.S. DOE by LLNL under Contract DE-AC52-07 NA27344.  
The authors declare no conflict of interest.