## 3 Results
\begin{table}
\begin{tabular}{c c c c} \hline
**Element** & **FPLAPW** & **FPLMTO** & **VASP** \\ \hline Th & 0.12 & 0.02 & 0.02 \\ Pa & 0.53 & 0.50 & 0.57 \\ U & 1.40 & 1.33 & 1.40 \\ Np & N/A & 2.30 & 2.16 \\ Pu & N/A & 3.55 * & 5.50 * \\ \hline \end{tabular} Note: * includes spin–orbit coupling and orbital polarization.  
\end{table} 
Table 2: Density-functional-theory relative expansion (%) of the atomic volume due to inclusion of spin–orbit (SO) coupling, (V\({}_{\text{SO}}\)-V\({}_{\text{NO-SO}}\))/V\({}_{\text{SO}}\). For full-potential linear augmented plane-wave (FPLAPW), SO is included exactly via the Dirac equation, while for full-potential linear muffin-tin orbital (FPLMTO) and Vienna ab initio simulation package (VASP), a scalar-relativistic basis set is used with perturbative SO that is ignored for the p states. For plutonium, orbital polarization is important and therefore included. accurate electronic-structure calculations, recognizing the magnetic interactions and relaxing the crystal structure, to predict the atomic volume accurately. Next, we discuss the theory relative to experimental data. In order to do this, we need to establish the experimental volumes at zero temperature to enable a proper comparison with the zero-temperature modeling. Thankfully, [PERSON] and [PERSON] [61] carefully determined these experimental zero-temperature volumes (V\({}_{0}\)) for Th, Pa, and U (32.55, 24.72, and 20.47 A\({}^{3}\), respectively). For neptunium we use the experimental linear coefficient of thermal expansion (\(27\times 10^{-6}\) K\({}^{-1}\)) [62] combined with the quasi-harmonic Debye-Gruneisen model [4] to back out V\({}_{0}\) (18.9 A\({}^{3}\)). For \(\alpha\)-plutonium the Debye-Gruneisen procedure is inaccurate because the temperature dependence of the volume is highly anomalous at low temperatures. At temperatures close to 50 K, there is negative thermal expansion observed in dilatation experiments [63]. This anomaly was proposed to be due to magnetism [63], and this speculation is plausible because the negative thermal expansion observed in \(\delta\)-plutonium is explained from first-principles theory by magnetic interactions [40]. Here, we extract V\({}_{0}\) (19.53 A\({}^{3}\)) for \(\alpha\)-plutonium from a linear extrapolation, from the three lowest temperatures, to zero temperature of the dilatation data shown in Figure 1. We compare our VASP results, that include spin-orbit interaction and orbital polarization, with the experimental V\({}_{0}\) values in Figure 2. The agreement is very good and almost as good as all-electron (FPLMTO) data that also include spin-orbit coupling and orbital polarization [24]. The small differences are likely due to the PAW method used in VASP. Figure 1: Experimental temperature dependence of the atomic volume for \(\alpha\)-plutonium. Redrawn from [63] and assuming that the room-temperature (300 K) atomic volume is 20.0 Å\({}^{3}\). The 0 K value is obtained from a linear extrapolation from the three lowest temperatures (red line). We assess the pseudopotential approximation in Figure 3 where scalar-relativistic VASP and FPLMTO atomic volumes are compiled. Notice that the all-electron FPLMTO volumes are slightly larger, as are the FPLAPW volumes (not shown), but overall the agreement is quite good. For \(\alpha\)-plutonium the difference is more significant but, in this case, when spin-orbit and orbital polarization are included, the agreement between the methods improves. In our judgement, the VASP pseudopotential is very reasonable for Th to Pu. We thus have a model (DFT + SO + OP), that reproduces the equilibrium volumes for the actinides very accurately. The parabolic behavior of the volumes is duplicated as well as the upturn observed between Np and Pu. The upturn for plutonium is due to several effects: Crystal structure, spin-orbit coupling, orbital polarization, and formation of magnetic moments. The present model does not depend on an ad hoc Hubbard \(U\) parameter.