# Thermodynamics Modeling for Actinide Monocarbides and Mononitrides from First Principles
## 1 Introduction
For the present actinide monocarbides and mononitrides, the +U approach has not been shown to be effective either. For example, DFT + U (U-2 eV) generates imaginary phonon frequencies for UN, resulting in a dynamical instability of the compound, and the method was consequently not recommended [18]. A far less practical methodology is that of the dynamical mean-field theory (DMFT) that also relies on the Hubbard U and other parameters. For DMFT, the computational cost accelerates dramatically while the calculated total energy becomes ambiguous because the double-counting energy contribution is not clearly defined. Therefore, energy-derived properties, crucial for thermodynamics, are not accessible from the DMFT method [22]. On a more fundamental level, all +U (and hybrid-DFT) methods suffer from having one or more parameters that are impossible to know for certain and therefore assumptions must be made when deciding their values. These approaches further rely on using fixed parameters, while in reality they must depend on both temperature and pressure. Unfortunately, the calculated properties often depend rather sensitively on these uncertain parameters [23]. As a better alternative, we have employed an approach that includes orbital polarization (OP) in a DFT + OP method [24], involving self-consistent Racah parameters, where OP accounts for magnetic orbital-moment couplings. This orbital-polarization interaction only exists when electrons are moving (orbiting) and is therefore not explicitly addressed in conventional DFT that is free of charge currents. Results from the present DFT + OP scheme, that is thus an extension of conventional DFT, agree well with those obtained from a method where OP is rigorously derived from current-density-functional theory [25]. It has been argued that there is a connection between the +U methods and orbital polarization, namely, that DFT + OP can be mimicked by DFT + U in the limit of very small U [26]. This may be approximately true for very small U, but the two methods are certainly not equivalent. The OP is driven by an orbital-moment interaction that only exists when the spin-orbit interaction and orbital moments are present while DFT + U does not depend on spin-orbit coupling. Another more fundamental difference is that DFT + OP is a self-consistent approach that adapts to changing conditions while DFT + U is not. We have shown that OP is essential for plutonium [27] while less important for thorium and uranium [28], but we have nonetheless chosen to include OP in most of our present calculations because of the small additional computational cost (see discussion below). Aside from the general difficulty of the 5f electronic structure, vacancies (of nitrogen or carbon) tend to form in these materials thus further complicating the modeling of a realistic material. These effects can be modeled at low temperatures [29] but for the present high-temperature study, we do not address any defect mechanisms or dependencies on defects, but rather assume perfect, stochiometric crystals. Our present results compliment those we have published for uranium mononitride [8] and plutonium monocarbide [15] so that we now can present thermodynamical modeling for a complete set of thorium, uranium, and plutonium monocarbides and mononitrides. The uranium and plutonium compounds (and mixes of them) are considered potential advanced nuclear fuels [1; 2]. The thorium compounds we include because they are of interest for very high-temperature reactors [1] but also because they provide a broader fundamental-science context to the results for the other compounds.