# Thermodynamics of Plutonium Monocarbide from Anharmonic and Relativistic Theory
## 2 Computational Methods  
### Electronic-Structure Methods  
We apply three methods, each with their own strengths, for calculating the electronic structure of PuC. Two of them are \"all-electron\" approaches (i.e., no core-electron approximation), that in one case is implemented with a \"full potential\" (i.e., no geometrical or structural approximations). The other all-electron method is constructed with a Green's function technique that allows for realistic alloy and disorder treatment. The third method is not all-electron but a plane-wave pseudopotential method that is efficient and fast for calculating forces on atoms in large cells. All three approaches rely on DFT and the generalized gradient approximation (GGA) [19] for the electron exchange and correlation energy functional. First, we spotlight the all-electron full-potential linear muffin-tin orbital (FPLMTO) method for which implementation-details can be found the literature [20]. It adopts no approximations for the core states that exist at deeper energy levels than the valence states. This is a more accurate treatment than that of the plane-wave methods where the core electrons are replaced by a pseudopotential. Some quantities are expanded in the series (basis functions, electron densities, and potentials) of spherical harmonics inside non-overlapping spheres centered at each atomic position. The radial part of the basis functions inside these spheres are calculated from a wave equation that addresses all relativistic corrections including spin-orbit coupling for d and f states but not for the p states, as is appropriate for actinides [6]. The orbital-polarization mechanism, considered here for the d and f states [6; 21], is not included in conventional density-functional theory but is known to be important for some f-electron systems, particularly plutonium [5; 15]. Generally, the set-up parameters of the present calculations for PuC are close to those applied for plutonium metal [5]. The FPLMTO method is employed to calculate the PuC formation enthalpy, the elastic constants, and the so-called \"cold curve\", i.e., the total-energy variation with atomic volume. The latter provides the fundamental input to the Debye-Gruneisen quasi-harmonic simulations (see below). The elastic constants are calculated, adopting conventional strains for cubic crystals while the shear modulus is the Voigt-Hill-Reuss average [22] of the single-crystal moduli. Furthermore, the presented electronic density of states and the related simulated photoelectron spectra are obtained from FPLMTO. Lastly, the Racah 5f parameters, that enter in the orbital-polarization formalism [21], are self-consistently calculated with FPLMTO for use as fixed parameters (-0.05 eV) in our plane-wave calculations [6]. For the computationally more demanding supercell calculations that couple to self-consistent ab initio lattice dynamics (SCAILD) [23], we utilize an efficient electronic-structure approach. Namely, the pseudopotential plane-wave Vienna ab initio simulation package (VASP) with the projector-augmented-wave method with the plane-wave basis set, as implemented in VASP [24; 25; 26]. The computational set-up is defined by an energy cut-off of 400 eV and an energy convergence of 100 eV. The VASP approach furthermore includes non-collinear magnetism with spin-orbit coupling and orbital polarization, as implemented recently [6]. PuC is sub-stoichiometric, as mentioned, and the sensitivity to the deviation of the (number of atoms) Pu/C ratio from unity is explored by VASP as well as utilizing a technique that incorporates efficient alloy theory within the coherent-potential approximation (CPA) [27]. The exact muffin-tin orbital (EMTO) method relies on [PERSON]'s function formalism where the one-electron potential is represented by optimized overlapping muffin-tin potential spheres [28]. The EMTO-CPA [29] is well suited to explore compositional disorder and sub-stoichiometric conditions.