# Thermodynamics of Plutonium Monocarbide from Anharmonic and Relativistic Theory
## 2 Computational Methods
### Lattice-Dynamics Methods  
Our main tool for lattice dynamics is the SCAILD methodology [23] that we often refer to as the self-consistent phonon method. The idea behind the scheme is to employ the small-displacement method to calculate the phonons in a first step and then apply thermally induced (fixed finite temperature) \"frozen\" phonons on the atoms and calculate corresponding DFT atomic forces. The next step is to recalculate new phonons informed by these DFT forces and repeat until convergence [23]. This self-consistent phonon approach is appealing because it correlates displacements of all atoms with each other (unlike the frozen-phonon method) and, therefore, it can account for strong phonon-phonon coupling and anharmonic behavior. We have used SCAILD for uranium mononitride and the cubic \(\gamma\) phases of uranium and plutonium in the past [5; 14; 30; 31] with satisfying results. For SCAILD, one needs to define a supercell of the crystal structure and for PuC we specify a 3 by 3 by 3 supercell for a total of 54 atoms. The DFT forces are gathered from VASP calculations and the self-consistentscheme requires about 200 iterations to converge the lattice-vibration contribution to the free energy to the meV level. We perform several simulations: For each temperature (T = 600, 800, 1000, 1200, 1500, and 2000 K), we choose three to five (depending on temperature) lattice constants in order to determine the equilibrium volume on the basis of the Gibbs-energy minimum. To assess the importance of 5f-electron correlations beyond conventional DFT-GGA, we employ two approaches; one full treatment that includes spin-orbit coupling and orbital polarization \"DFT+SO+OP\" and one bare treatment that excludes these interactions \"DFT\". In Figure 1 we show results from SCAILD for PuC at a temperature of 800 K, for the two levels of electronic-structure theory. The lattice constants are held fixed, corresponding to the zero-temperature equilibrium volumes (16.2 and 15.6 A\({}^{3}\), see Table 1 in Section 3.2). We notice in Figure 1 that the SCAILD vibrational free energy is well converged and that the energy from the lesser theory (DFT) lies below that of the full treatment (DFT+SO+OP). This result is related to softness of some phonon branches in the DFT approach that we will discuss further in the Results section. One can judge the importance of anharmonic phonons by comparing calculated thermal properties from anharmonic and lower-level quasi-harmonic theory. For this reason, we conduct a limited set of calculations employing two implementations of the Debye-Gruneisen quasi-harmonic approach [32]. The Debye-Gruneisen scheme is very efficient and only requires the DFT cold curve as input in addition to the atomic mass of the atoms in the material. There are, however, some assumptions within this model that must be made. First, [PERSON] et al. [32] suggested that the Debye temperature can be derived from this simple relationship:  
\[\theta_{D}=Const\sqrt{\frac{rB}{M}} \tag{1}\]  
Figure 1: Vibrational free energies (eV/cell) as functions of self-consistent ab initio lattice dynamics (SCAILD) iterations for Vienna ab initio simulation package (VASP)-force calculations that include spin-orbit coupling and orbital polarization (DFT+SO+OP) and not (DFT). The atomic volumes are kept constant and equal to 16.2 Å\({}^{3}\) and 15.6 Å\({}^{3}\), respectively (see main text). where \(r\) is the atomic radius and \(M\) the atomic mass in atomic units, \(B\) is the bulk modulus in kbar, and _Const_ a scaling factor. The _Const_ coefficient can be chosen to optimize agreement with experimental data or calculated assuming an isotropic material. Second, there are two philosophies regarding the formulation of the Gruneisen parameter. One that is supposedly [32] better at higher temperatures, \(\gamma_{HT}\), and one that is more suitable when comparing to experimental specific-heat data at lower temperatures, \(\gamma_{LT}\).