# Thermodynamics Modeling for Actinide Monocarbides and Mononitrides from First Principles
## 2 Computational Methods  
### Electronic-Structure Methods  
The electronic structures are obtained from density-functional theory where the necessary electron exchange and correlation is the generalized gradient approximation (GGA). Generally, with some exceptions noted, DFT is extended to include an orbital-polarization mechanism that accounts for electron orbital-orbital coupling that is absent in conventional DFT. For the thorium compounds, the strength of this interaction is limited, but for the magnetic compounds and plutonium, it cannot be ignored [15; 27; 28]. Fortunately, the implementation is efficient, and the computational penalty for including OP is not severe enough to dissuade us from applying the interaction for all presently studied compounds. We employed two codes for the electronic-structure calculations in this paper. First, for the most accurate electronic structure, we rely on an all-electron full-potential linear muffin-tin orbital (FPLMTO) method [30]. FPLMTO is used to calculate all electronic structures and energies except the atomistic forces that are needed for the lattice-dynamics method. For computing these forces, we utilized the more expedient pseudopotential plane-wave approach (see below). The FPLMTO method is characterized by its atomic-like basis functions that for actinides include 6s, 6p, 7s, 7p, 6d, and 5f orbitals that are allowed to hybridize in one energy window. Some quantities in the method are expanded in series (basis functions, electron densities, and potentials) of spherical harmonics in the muffin-tin sphere position at each atomic site. The radial part of the basis functions inside these spheres are calculated from a wave equation that addresses all relativistic corrections including the spin-orbit coupling for d and f states but not for the p states, as is appropriate and accurate for actinides [28]. Guided by the self-consistent Racah parameters obtained from the FPLMTO method we define one Racah parameter, associated with the 5f electrons, that is kept fixed (0.05 eV) for use in the pseudopotential method. The theoretical treatment is spin polarized (magnetic) and we found that the ferromagnetic configuration is lower in energy than the most trivial anti-ferromagnetic configuration (a doubling of the unit cell along the z axis), except for the thorium compounds where the spin moment collapses to zero. For this reason, and the fact that the magnetic moment disorders at very high temperatures, we only consider ferromagnetism in our thermodynamical modeling. In Table 1, we show our all-electron FPLMTO-calculated atomic volumes for the non-magnetic (NM), anti-ferromagnetic (AF), and ferromagnetic (FM) configurations together with experimental data [16; 31; 32; 33; 34; 35]. Overall, the theory reproduces the experiment fairly well, suggesting that GGA is a reasonable choice for these compounds. GGA is not as perfect as it is for the actinide metals [28], but that is not expected for these compounds because at least for carbon, GGA overestimates the atomic volumes significantly [36]. The volumes for all present monocarbides are consequently also overestimated, see Table 1. The atomic volumes in Table 1 are calculated at zero temperature and not corrected for thermal expansion (measurements are often taken at room temperature). Such a correction is small but will not uniformly improve the agreement between theory and experiment. For the uranium and plutonium compounds, there are few direct measurements of magnetism but UN and PuC are found to be anti-ferromagnetic [37; 38]. The only reported experimental magnetic moment is for UN, and it is 0.75 \(\upmu_{\text{B}}\)[38]. Apparently, the measurements were not of high enough fidelity to reveal the magnetic moment for PuC [37].