## 3 Results  
The formally correct way to include spin-orbit interaction in an electronic-structure calculation is to solve the four-spinor Dirac equation. Here, we have applied a FPLAPW method that indeed does this (FlapwMBPT [57]), and we regard the corresponding results as the gold standard for our comparisons. We mentioned [20] that utilizing a scalar-relativistic basis set causes inaccuracies for the p states, but it has been suggested that this problem can be circumvented for the actinides by ignoring spin-orbit interaction for these states [20]. This realization has not been widely recognized in studies using plane-wave pseudopotential techniques, and we therefore reiterate this fact by showing our results in Table 2.  
Table 2 presents the effect of spin-orbit coupling on the volumes of the actinide elements. For Th, Pa, and U, FPLAW-Dirac calculations provide the reference, while for Np and Pu, that have substantially more complex and large crystal structures, we show results from VASP and FPLMTO. For VASP and FPLMTO the perturbative form of SO is included but for the p states. All three methods give essentially identical results for Th, Pa, and U, suggesting that ignoring spin-orbit interaction for the p states is a very good approximation. The same conclusion was reached in previous studies [59; 60] where comparisons were made with the exact muffin-tin orbital method that also solves the four-spinor Dirac equation. The trend is that spin-orbit coupling is of increasing importance proceeding from thorium to neptunium, causing an expansion of the volumes (Table 2). These metals are all non-magnetic and the explanation for this behavior is straightforward. As mentioned above, the atomic volumes for Th to Np decrease with atomic number in roughly a parabolic fashion that is due to a gradual population of bonding 5f states. No anti-bonding 5f electrons exist because Th to Np all have less than half-filled 5f band. Inclusion of spin-orbit interaction, however, causes some anti-bonding 5f states to be occupied, and these tend to expand the atomic volumes. The reason is that anti-bonding states mix with bonding states as the 5f band is divided into overlapping 5f\({}_{5/2}\) and 5f\({}_{7/2}\) sub-bands that comprise both bonding and anti-bonding states. The result of this is a reduction of the effective number of bonding electrons and an increase in the atomic volumes (Table 2). For plutonium, on the other hand, the situation becomes more complicated because here both spin and orbital polarization play important roles as well. In addition, for \(\alpha\)-plutonium the crystal and electronic structure correlate sensitively so that the atomic volume cannot be obtained simply from extrapolating the trend established by Th to Np. For plutonium, one must thus carry out \begin{table}
\begin{tabular}{c c c c} \hline
**Element** & **Th** & **Pa** & **U** \\ \hline
**Structure** & **Face-Centered Cubic** & **Body-Centered Tetragonal** & **Orthorhombic** \\ \hline b/a & N/A & N/A & 2.068 \\ c/a & N/A & 0.82 & 1.74 \\ y & N/A & N/A & 0.1017 \\ Basis set for \(\
abla\) & 596–642 & 404–460 & 1000–1048 \\ Plane waves in the interstitial for & 965 & 1263 & 1541 \\ \(\rho/V_{\text{KS}}\) & 965 & 10 & 10 \\ L\({}_{\text{max}}\) in MT for \(\Psi\) & 10 & 10 & 10 \\ L\({}_{\text{max}}\) in MT for \(\rho\)/V\({}_{\text{KS}}\) & 6 & 6 & 6 \\ \(k\)-point mesh & \(12\times 12\times 12\) & \(12\times 12\times 12\) & \(12\times 12\times 8\) \\ \hline \end{tabular}
\end{table} 
Table 1: Structural parameters and cut-offs for the calculations with FlapwMBPT. \(\Psi\) represents the wave function, \(\rho\) the electronic density, and V\({}_{\text{KS}}\) the Kohn–Sham potential. Basis-set size corresponds to the fully relativistic calculations. For the scalar-relativistic approach, the size is half.