## 2 Computational Methods
This formulation of the orbital-polarization Hamiltonian has mainly been implemented in conjunction with spin-orbit coupling in all-electron codes within the second-variation approximation assuming collinear spin polarization. Analogous to the conventional formulation of the orbital polarization (OP) method, while having the computational expediency of plane-wave codes and accurate Hellman-Feynman forces, we make use of a variational total-energy formulation of the OP Hamiltonian in the context of non-collinear magnetism. It is easily constructed within the PAW method following the DFT + \(U\) implementation by [PERSON] et al. [53]. See Appendix A for a brief technical description of the PAW method for the purpose of implementation of the orbital-polarization Hamiltonian. As mentioned, Equation (1) is proposed for collinear magnetism with the spin moment pointing in the \(z\) direction. This formalism can be generalized to non-collinear magnetism with arbitrary spin orientations by modifying the quadratic energy functional within each muffin-tin radius to favor orbital polarization in the direction of the expectation value of the local self-consistent spin vector. It is then straightforward to propose the following total-energy functional, consistent with a non-collinear generalization of Equation (1),  
\[E_{DFT\ +\ OP}\ =\ E_{DFT}\ -\ \frac{1}{2}\sum_{I}E_{I}^{3}\frac{\left[ \langle\overline{L}_{I}^{f}\rangle\cdot\langle\overline{S}_{I}^{f}\rangle \right]^{2}}{\langle\overline{S}_{I}^{f}\rangle\cdot\langle\overline{S}_{I}^{f}\rangle} \tag{2}\]  
where \(\langle\overline{L}_{I}^{f}\rangle\) and \(\langle\overline{S}_{I}^{f}\rangle\) are expectation values of the angular-momentum and the spin-moment vectors of the f states inside the muffin-tin sphere centered on site I. See the Appendix A for the expressions for \(\langle\overline{L}_{I}^{f}\rangle\) and \(\langle\overline{S}_{I}^{f}\rangle\) within the PAW method. In addition to our plane-wave calculations, we apply two all-electron methods to compare our pseudopotential results with. First, a full-potential linear muffin-tin orbital (FPLMTO) method [54] that has been detailed [55]. Here, no approximations are applied for the core electrons, unlike the pseudopotential method. The core-electron approximation made in the latter technique is computationally expedient but will cause inaccuracies that are not present in all-electron calculations. The present implementation does not make any assumptions beyond the GGA. Basis functions, electron densities, and potentials are calculated without any geometrical approximation, and these are expanded in spherical harmonics (with a cut-off L\({}_{\text{max}}\) = 8) inside non-overlapping (muffin-tin, MT) spheres surrounding each atom and in Fourier series in the region between these muffin-tin spheres. One has to define an MT sphere with a radius, sMT, and here it is chosen so that sMT/sWS \(\sim\) 0.8, where sWS is the Wigner-Seitz (atomic sphere) radius. The radial parts of the basis functions inside the MT spheres are calculated from a wave equation for the \(L\) = 0 component of the potential that includes all relativistic corrections including spin-orbit coupling for d and f states, but not for the p states [20]. It should be noted that excluding the p states also removes the strong and artificial dependence of the calculated volumes on the sMT/sWS ratio [2] that is also not present in the exact relativistic treatment (see below). The orbital polarization is only considered for d and f states. In the actinides, however, the OP contribution from the d states is minimal and ignored in the VASP calculations. The formulation is self-consistent and parameter-free and given in [52]. The self-consistent FPLMTO Racah parameters for the 5f states are employed in the VASP calculations described above. All crystal-structure geometries were relaxed and for neptunium and plutonium, that have more complex phases, we apply a scheme [56] that avoids the necessity of Hellman-Feynman forces that are troublesome to compute.