## 3 Results
At lower to moderate temperatures one expects a quasi-harmonic approximation of the lattice vibrations to be sufficient [38] while at temperatures well above 1000 K anharmonic effects become increasingly relevant. With the self-consistent phonon approach, it is possible to address strong phonon-phonon coupling that leads to anharmonic lattice vibrations that go beyond the quasi-harmonic behavior. In Figure 2 we show the free energy for UN obtained from SCAILD. Results from only the lattice-vibration contribution, at a fixed 14.85 A\({}^{3}\) atomic volume, are displayed with open squares. When adding the electron contribution, including the electronic entropy, the free energy drops somewhat (solid circles). Finally, evaluating the free-energy contributions at zero pressure (minimizing the free energy with respect to volume), results in the solid squares in Figure 2. For comparison we also show (solid line) the results from a CALPHAD assessment of available experimental data. Clearly, the first-principles treatment is consistent with the CALPHAD result, suggesting a rather accurate theoretical model. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **C\({}_{\bf{11}}\) (GPa)** & **C\({}_{\bf{12}}\) (GPa)** & **C\({}_{\bf{44}}\) (GPa)** \\ \hline DFT-FM (present) & 390 & 77.9 & 79.6 \\ DFT-GGA [39] & 428 & 136.6 & 29.7 \\ DFT-GGA [71] & 404.6 & 124.2 & 45.0 \\ Expt & 391 & 90 & 80 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Calculated, present, and published [39; 71], and experimental [18] elastic constants for UN. In addition to the free energy, it is interesting to directly contrast our calculated heat capacity with experimental data. In Figure 3 we show two sets of calculations, one assuming quasi-harmonic lattice vibrations, acquired from the GIBBS2 package [72; 73; 74; 75], and the other, allowing for anharmonic phonons, determined by the SCAILD method. For both these sets we add the electronic contribution obtained from the electronic structure. The C\({}_{\rm v}\) data from SCAILD were converted to C\({}_{\rm p}\) by applying the difference in these quantities computed by GIBBS2. There are also numerous sets of experimental data [4; 7; 8; 12; 13; 15; 19] included in the figure. The anharmonic phonon model compares very well with available experimental data except those from [7] that appears to deviate from all other experimental data above 1700 K. The quasi-harmonic model does reasonably well at lower temperatures up to 1000 K but beyond that it is insufficient for reproducing the trend of the experimental data. Figure 3: Theoretical models for the heat capacity at ambient pressure, assuming quasi-harmonic and anharmonic phonons. The experimental data are collected from References [4; 7; 8; 12; 13; 15; 19]. Note, the heat capacity is calculated per gram-atom (g-at.) of N\({}_{2}\) and 1 mol of N\({}_{2}\), which is equivalent to 1 gram-molecule of N\({}_{2}\), contains 2 gram-atoms of N\({}_{2}\). Figure 2: DFT and CALPHAD (from 300 K) free energies for UN. Open squares refer to SCAILD lattice energies at constant atomic volume (14.85 Å\({}^{3}\)). The DFT electron free energy that includes electronic entropy was added for the solid circles. The solids squares refer to results from volume relaxation (pressure is zero) that includes both electronic and lattice contributions. The lines connecting the symbols are guides to the eye only. The solid line without symbols refers to our CALPHAD results. The Debye temperature is another quantity for which the models can be tested against experiments. The phonon spectrum of UN is known from neutron scattering [76] and from this spectrum a Debye temperature \(\Theta_{\rm D}\) = 181 K can be deduced [40]. Our anharmonic treatment of UN predicts \(\Theta_{\rm D}\) = 190 K with only a very weak temperature dependence, while our quasi-harmonic treatment (GIBBS2) results in a substantially larger \(\Theta_{\rm D}\) = 269 K in worse agreement with neutron scattering.