density matrixes $n^{1}_{\alpha\beta}$ and $\tilde n^{1}_{\alpha\beta}$ are calculated on radical support grid around every atom.

The Kohn-Sham Hamilton and Kohn-Sham equation can be obtained from the variation of total energy to the soft pseudo-density matrix and written as $$\begin{aligned}
      H^{\alpha\beta}\left[n\right]&=&-\frac{1}{2}\Delta \delta^{\alpha\beta}+\tilde{v}^{\alpha\beta}_{eff}
      \\ \nonumber
      &+& \sum_{(ij)}|\tilde{p}_{i}\rangle(\hat{D}^{\alpha\beta}_{ij}+^{1}D^{\alpha\beta}_{ij}
      -^{1}\tilde{D}^{\alpha\beta}_{ij})\langle\tilde{p}_{j}| ,
      \\ \nonumber
      \sum_{\beta}H^{\alpha\beta}|\tilde{\Psi}^{\beta}_{n}\rangle&=&
      \varepsilon_{n}S^{\alpha\alpha}|\tilde{\Psi}_{n}^{\alpha}\rangle .
 \end{aligned}$$ where $S^{\alpha\alpha}$ is the overlapping operator, $\tilde{D}^{\alpha\beta}_{ij}$, $^{1}\hat{D}^{\alpha\beta}_{ij}$ and $^{1}D^{\alpha\beta}_{ij}$ are the nonlocal interaction obtained by the PAW transformation, which are equivalent to the nonlocal pseudo-potential in the ultrasoft-potential methods. In non-collinear DFT formulas, the one-electron effective potential $\tilde{v}^{\alpha\beta}_{eff}=v_{H}(r)\delta_{\alpha\beta}+v_{xc}(r)\delta_{\alpha\beta}
 +\vec{b}(r)\cdot\vec{\sigma}^{\alpha\beta}$, which is the summation of electrostatic potential $v_{H}(r)$, the nonmagnetic part of GGA exchange-correlation correction $v_{xc}(r)$ and the magnetic exchange-correlation potential $\vec{b}(r)\cdot\vec{\sigma}^{\alpha\beta}$, where $\vec{b}=\delta E_{xc}/\delta \vec{m}(r)$. The magnetic exchange-correlation potential will potentially enhance magnetic moments in non-collinear calculations.

The spin-orbit coupling enters into PAW formulas by all-electron part of PAW Hamiltonian and is expressed as $$\begin{aligned}
 H_{SOI}^{\alpha\beta}=\frac{\hbar^{2}}{(2m_{e}c)^{2}}\sum_{ij}
 \langle\phi_{i}|\frac{1}{r}\frac{dV_{sphere}}{dr}|\phi_{j}\rangle
 |\tilde{p}_{i}\rangle\vec{\sigma}_{\alpha\beta}\cdot\vec{L}_{ij}
 \langle\tilde{p}_{j}|.
 \end{aligned}$$ where $\tilde{p}_{i}$ is the projector function and $\phi_{i}$ all-electron wave function. The one-electron effective potential is modified $\tilde{v}^{\alpha\beta}_{eff}\rightarrow\tilde{v}^{\alpha\beta}_{eff}+\tilde{H}_{SOI}^{\alpha\beta}$, and the same time, one part of nonlocal PAW potentials is modified according to $\hat{D}^{\alpha\beta}_{ij}\rightarrow\hat{D}^{\alpha\beta}_{ij}=\sum_{L}
 \int(\tilde{v}^{\alpha\beta}_{eff}(r)+\tilde{H}_{SOI}^{\alpha\beta}(r))\hat{Q}_{ij}^{L}(r)dr$. The roles of spin-orbit interaction in $^{1}\hat{D}^{\alpha\beta}_{ij}$ and $^{1}D^{\alpha\beta}_{ij}$ are similar to 