## 4 Summary and Conclusions  
We conducted DFT and DFT + \(U\) calculations for the purpose of evaluating the potential need for an explicit Hubbard \(U\) intra-atomic Coulomb repulsion for the actinide metals Th to Pu. We limit our study to the \(\alpha\) phases of these metals (except for \(\delta\)-plutonium) and find that one carefully needs to address several key approximations in the theory in order to reach a definite conclusion.  
First, the spin-orbit interaction has to be introduced in an appropriate fashion. Based on comparisons with Dirac-FPLAPW calculations that include spin-orbit interaction exactly, we recommend a way to accurately account for SO interaction without having to solve the spin-polarized Dirac equation. Namely, by excluding this interaction on the p states, one can accurately apply the usual scalar-relativistic Schrodinger-type equation with perturbative spin-orbit coupling [64] as implemented in VASP and other codes.  
Second, one has to make sure that the pseudopotential employed in plane-wave methods is a good representation of the core electrons. We have confirmed that this is the case here by comparisons with results from all-electron codes. The most accurate result is obviously obtained from all-electron approaches, however.  
Figure 5: DFT + \(U\) (VASP) total energies for \(\alpha\)- and \(\delta\)-plutonium. The calculations include a Hubbard \(U\) = 0.4 eV and spin–orbit coupling (not on p states) but no orbital polarization. The dashed lines are guides to the eye only.  
Third, the crystal structures need to be relaxed (i.e., optimized so that the lowest energy is obtained). This is particularly important for \(\alpha\)-plutonium because of the sensitive correlation between its crystal and electronic structure. The plutonium computations we have performed are thus non-trivial because forces are required for structural relaxations and these are troublesome to evaluate in the presence of strong spin-orbit and orbital-orbital coupling.  
The conclusion is that DFT, with the GGA for the electron exchange and correlations, is sufficient for all studied actinide metals (no Hubbard \(U\) needed). Recently, numerous others came to the same conclusion [65; 66; 67; 68] for uranium. For plutonium, however, treating the orbital-orbital interaction is necessary for a good electronic structure, total energy, and magnetic form factor [69]. Importantly, we show that DFT + \(U\) (\(U>0.2\) eV) for plutonium predicts the wrong ground-state phase. Because DFT + \(U\) defines the static limit of dynamical-mean field theory, that incorporates similar electron correlations, it seems unavoidable that it is also inappropriate for plutonium.  
B.S. and P.S. conceived the study. B.S., P.S., and A.K. performed electronic-structure calculations. All authors helped writing the paper.  
We thank [PERSON] and [PERSON] for helpful discussions. This work was performed under the auspices of the U.S. DOE by LLNL under Contract DE-AC52-07 NA27344 and was partly funded by the Laboratory Directed Research and Development Program at LLNL under Project Tracking Code No. 17-ERD-041. The work of [PERSON] was supported by the U.S. DOE, Office of Science, Basic Energy Sciences as a part of the Computational Materials Science Program. The authors declare no conflict of interest.