## Appendix A  
In the PAW method, the all-electron crystal wave function \(|\Psi_{kn}\rangle\) is expressed in terms of a soft pseudo-wave function \(|\widetilde{\Psi}_{kn}\rangle\) usually expanded in a plane-wave basis set and all-electron and pseudo-atomic partial waves \(|\psi_{ilm}\rangle\) and \(|\widetilde{\psi}_{ilm}\rangle\) localized near the nuclei as follows,  
\[|\Psi_{kn}\rangle\ =\ |\widetilde{\Psi}_{kn}\rangle\ +\ \sum_{ilm}\Bigl{(}| \psi_{ilm}\rangle\ -\ |\widetilde{\psi}_{ilm}\rangle\Bigr{)}\langle\overline{p}_{ilm}| \widetilde{\Psi}_{kn}\rangle. \tag{1}\]  
Here, the crystal wave function indices \(k\),\(n\) refer to the Bloch wave vector and band index, while the partial wave expansion indices \(l\),\(m\) refer to the atomic angular momentum quantum numbers, and the index i enumerates the number of l-channels used in the representation of the crystal wave function (Equation (1)). Note that the pseudo-atomic partial wave functions are constructed such that outside the atomic muffin-tin radius \(r_{c}\) they are equal to the all-electron partial wave functions,  
\[\phi_{ilm}(\mathbf{r})\ =\ \overline{\phi}_{ilm}(\mathbf{r})\ \text{for}\ |\mathbf{r}|\ >\ |\mathbf{r}_{c}| \tag{2}\]  
The projector functions \(|\overline{p}_{ilm}\rangle\) in Equation (1) are constructed in such a way that the following relation holds  
\[\langle\overline{p}_{ilm}|\widetilde{\phi}_{l^{\prime}l^{\prime}m^{\prime}} \rangle\ =\ \delta_{l^{\prime}l^{\prime}}\delta_{l^{\prime}l^{\prime}}\delta_{m,m^{\prime}}. \tag{3}\]  
The above integrations over the projector functions and the pseudo-partial wave functions are conducted within \(r_{c}\). The PAW method is exact if the projector functions and the pseudo-atomic partial wave functions constitute a complete basis expansion for the crystal wave function in the muffin-tin radius near each nucleus, i.e.,  
\[\sum_{ilm}|\widetilde{\phi}_{ilm}\rangle\langle\overline{p}_{ilm}|\ =\ 1. \tag{4}\]  
It is easy to see that the above assumption of completeness can become questionable without explicit addition of the p\({}_{1/2}\)-orbitals in the partial-wave expansion of the crystal wave function inside \(r_{c}\), when relativistic corrections are added to the scalar-relativistic equations. Nevertheless, the PAW equations are derived under the assumption of Equation (A4). As a result, the expectation value of any local operator \(\hat{B}\) confined to the muffin-tin sphere around a nucleus can be simply expressed as follows  
\[\langle\vec{\Psi}_{kn}|\hat{B}|\vec{\Psi}_{kn}\rangle\ =\ \sum_{ilm,p^{\prime}l^{ \prime}m^{\prime}}\langle\vec{\Psi}_{klm}|\vec{p}_{ilm}\rangle\langle\phi_{ilm }|\hat{B}|\phi_{p^{\prime}l^{\prime}m^{\prime}}\rangle\langle\vec{p}_{p^{ \prime}l^{\prime}m^{\prime}}|\vec{\Psi}_{kn}\rangle.\] (A5)  
Finally, when dealing with non-collinear magnetism, the crystal wave functions acquire an extra dimension and become spinor-wave functions \(\vec{\Psi}_{kn}^{\tau}\), with \(\tau\) now indexing the two spinor components. We are now in a position to derive the expression for the expectation value of any component (\(\alpha=x,y,z\)) of the local angular-momentum operator \(L_{\alpha}^{f}\) that acts on the \(f\) electrons within an atomic sphere. It can be written as  
\[\langle L_{\alpha}^{f}\rangle=\ \sum_{ij}\langle\phi_{3i}|\phi_{3j}\rangle\sum_{ m,m^{\prime}}\langle 3m|\underline{L}_{\alpha}|3m^{\prime}\rangle\sum_{kn\tau}f_{ k,n}\langle\ \vec{\Psi}_{kn}^{\tau}|\vec{p}_{3 im}\rangle\langle\vec{p}_{3 jm}\ \vec{\Psi}_{kn}^{\tau}\rangle\] (A6)  
Here, \(\langle\phi_{3i}|\phi_{3j}\rangle\) is the radial overlap integral within the muffin-tin radius between two all-electron partial wave f-orbitals, \(\langle 3m|\ \underline{L}_{\alpha}|3m^{\prime}\rangle\) is the spherical-harmonics representation of the angular momentum operator in the subspace of the f-orbitals, and \(f_{kn}\) are the partial occupations of the crystal wave functions.