# Thermodynamics of Plutonium Monocarbide from Anharmonic and Relativistic Theory
## 3 Results
### Ground-State Properties and Thermodynamics  
Here, we discuss our predicted ground-state properties, including the elastic constants, and our lattice-dynamics results. In Table 1 we show atomic volumes and elastic constants for PuC from the full theoretical treatment (DFT+SO+OP) and for the simpler theory that excludes spin-orbit coupling and orbital polarization (DFT). First, we recognize that the two theoretical approaches overestimate the atomic volume relative to both [PERSON]'s and [PERSON]'s experimental values (14.8 and 15.4 A\({}^{3}\), respectively) [54; 55]. The same trend, with a predicted too-large volume for PuC, was observed in a hybrid-DFT study [11] where they argued that the discrepancy was due to an overestimation of 5f localization. We believe the reason is that the approximation for the electron exchange and correlation, GGA, is not ideal for PuC. It is known that GGA is accurate for the light actinide metals and Pu [56; 57], but for carbon (graphite), the local-density approximation (LDA) is better for the structural properties [58]. Consequently, neither LDA nor GGA is perfectly suitable for PuC and, therefore, the GGA atomic volume is imperfect. Next, we compare our elastic constants in Table 1 with results obtained from Born-Mayer and Coulomb model potentials that [PERSON] et al. reported for PuC [59]. Their zero-temperature moduli are considerably smaller (\(C_{11}=63.9\), \(C_{12}=27.2\), \(C_{44}=27.2\) GPa) than ours. We expect that our elastic constants for PuC in Table 1 are reasonable because they are rather accurately calculated, using the same theoretical framework, for the known phases of plutonium metal [60]. The Poisson's ratio, \(\
u\), in Table 1 is obtained from the standard formula for cubic crystals:  
\[\
u=\frac{3B-2G}{2(3B+G)} \tag{3}\]  
Unfortunately, there are no published experimental data on the single-crystal elastic constants (\(C_{ij}\)) that could help distinguish between the present and previous [59] modeling for PuC. For the isothermal bulk modulus (\(B\)), however, an experimental value has been reported (118 GPa) [61] that is far closer to our best theory (125 GPa) than that of the model potential (39.4 GPa) [59]. In regard to our first-principles results in Table 1, we find some differences between the full theory (DFT+SO+OP) and the simpler scalar-relativistic approach (DFT). The former produces a somewhat larger atomic volume and a correspondingly softer bulk modulus, while the shear modulus is about the same. Most notably is the fact that the tetragonal shear constant, \(C^{\prime}=(C_{11}-C_{12})/2\), is very small for the scalar-relativistic approximation (27 vs. 70 GPa). The small \(C^{\prime}\) indicates that the crystal is close to a mechanical instability and this is linked to softer phonons for the related phonon branches. Softer phonon modes imply more entropy and a greater phonon contribution to the free energy, and that is also observed in our calculated free energies shown in Figure 1. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Method** & \(\mathbf{V}\) & \(\mathbf{B}\) & \(\mathbf{B}^{\prime}\) & \(\mathbf{C_{11}}\) & \(\mathbf{C_{12}}\) & \(\mathbf{C_{44}}\) & \(\mathbf{G}\) & \(\mathbf{\
u}\) \\ \hline DFT+SO+OP & 16.2 & 125 & 2.7 & 218 & 78.0 & 75.0 & 73.0 & 0.256 \\ DFT & 15.6 & 141 & 0.65 & 177 & 123 & 141 & 73.9 & 0.277 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Ground-state properties obtained from the full theoretical treatment (DFT+SO+OP) and for a treatment that ignores spin-orbit coupling and orbital polarization (DFT). The atomic volume (\(V\)) is given in units of Å\({}^{3}\), while the bulk (\(B\)), shear (\(G\)), and elastic (\(C_{ij}\)) moduli are given in units of GPa. The shear modulus is a Voigt–Reuss–Hill average and the Poisson’s ratio, \(\
u\), is obtained from \(B\) and \(G\) (see main text).