# Thermodynamics Modeling for Actinide Monocarbides and Mononitrides from First Principles
## 2 Computational Methods
### Electronic-Structure Methods
Our anti-ferromagnetic calculations for PuC and PuN give atomic volumes closer to experiments, whereas the volumes for UC and UN are less sensitive to the magnetic ordering, see Table 1.  
For our thermodynamics calculations we assume ferromagnetism, regardless of what the actual (room temperature) configuration may be. This is not an important concern, however, because at the very high temperatures of interest here, magnetic order is entirely lost. Our all-electron FPLMTO-computed (low temperature) magnetic moments are listed in Table 2. Because the 5f shell is less than half occupied for U and Pu, the spin and orbital moments align anti-parallel so that the total moment is relatively small for the uranium and plutonium compounds. The orbital moment is significantly enhanced, resulting from the orbital polarization of the 5f electrons [39], and the OP interaction is therefore essential for good agreement between theory and experiment for the net magnetic moment in UN. The orbital-polarization mechanism is not only important for the magnetic properties, but it is critical also for the thermodynamical properties of PuC [15]. For the atomic-force calculations, associated with lattice dynamics (see next subsection), we took advantage of an approach computationally cheaper than the more accurate FPLMTO. Namely, the pseudopotential plane-wave Vienna ab initio simulation package (VASP) and the projector-augmented-wave (PAW) method [40; 41; 42]. The VASP-PAW computational parameters consist of an energy cut-off of 450 eV, energy convergence of 100 eV, and a Monkhorst-Pack scheme with a total of 8 k points for the 54-atom supercell we employ for the lattice-dynamics method. Ferromagnetic order is assumed for the VASP-PAW magnetic calculations, in analogy with the FPLMTO calculations. The electron exchange and correlation functional is that of the GGA and includes, for the plutonium compounds, spin-orbit coupling, and orbital polarization with a fixed 5f-electron Racah parameter (0.05 eV) as implemented and presented [28]. The value is decided from the Racah parameters that are self-consistently determined within the FPLMTO method. For comparison with our FPLMTO all-electron results we show VASP-PAW-calculated atomic volumes and spin moments for ThC, ThN, UC, UN, PuC, and PuN in Table 3. Here, the thorium compounds are non-magnetic while the others are treated as ferromagnetic. For these calculations we increased the number of k points to about 400 to better resolve energies and magnetic moments. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **ThC** & **ThN** & **UC** & **UN** & **PuC** & **PuN** \\ \hline Theory-NM & 19.20 & 17.27 & 15.24 & 14.53 & 14.71 & 14.70 \\ Theory-AF & – & – & 15.27 & 14.85 & 16.00 & 15.00 \\ Theory-FM & – & – & 15.33 & 14.85 & 16.19 & 15.18 \\ Experiment & 18.84 & 17.41 & 15.26 & 14.62 & 15.44 & 14.89 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The atomic volumes in Å\({}^{3}\)/atom from present theory and experimental data [16; 31; 32; 33; 34; 35]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Magnetic Moment** & **UC** & **UN** & **PuC** & **PuN** \\ \hline Spin & \(-0.04\) & \(-1.31\) & \(-3.81\) & \(4.10\) \\ Orbital & 0.08 & 2.09 & 3.92 & \(-3.60\) \\ Total & 0.04 & 0.78 & 0.11 & 0.50 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The calculated (low temperature) magnetic moments in units of \(\upmu_{\text{B}}\). The only available experimental value is that of UN, 0.75 \(\upmu_{\text{B}}\)[38]. Overall, the two computational approaches give similar results with some differences that are expected from two relatively different electronic-structure methods. We consider the all-electron results to be most accurate and the FPLMTO method is used to calculate all properties in this paper, except the forces that are utilized in the SCAILD lattice-dynamics scheme.