magnetism in transition metal materials.

In non-collinear density functional theory [@Hobbs1; @Hobbs2; @Sandratskii1], the total energy is the functional of a density matrix $n_{\alpha\beta}$ where $\alpha$ and $\beta$ are the spin index along a defined quantization axis. $\alpha=\uparrow$ or $\downarrow$ represent the spin angular momentum point to the positive direction or negative direction of the quantization axis. The sum of the diagonal elements is the total charge density, that is $n_{_{Tr}}=\sum_{\alpha}n_{\alpha\alpha}$. In a non-collinear magnetic system, the off-diagonal elements are non-zero. The 2$\times$2 matrix can be expanded using the complex matrix basis ($I$, $\sigma_{x}$, $\sigma_{y}$, $\sigma_{z}$) and expressed as $$\begin{aligned}
   n_{\alpha\beta}=[n_{_{Tr}}\delta_{\alpha\beta}+\vec m
   \cdot\vec\sigma_{\alpha\beta}]/2 ,
   \end{aligned}$$ where $\vec m=\sum_{\alpha\beta}n_{\beta\alpha}\cdot\vec\sigma_{\alpha\beta}$ is the magnetic electronic density and $\vec\sigma$ is the Pauli matrix. In PAW methods, the pseudo-density matrix can be expressed using the pseudo-wave functions as $$\begin{aligned}
    \tilde n_{\alpha\beta}(\vec{r})=\sum_{n}f_{n}\langle\tilde\Psi_{n}^{\beta}|\vec{r}\rangle
     \langle\vec{r}|\tilde\Psi_{n}^{\alpha}\rangle ,
    \end{aligned}$$ where $f_{n}$ is the occupation number. The total electronic density matrix $n_{\alpha\beta}=\tilde n_{\alpha\beta}+n^{1}_{\alpha\beta}-\tilde n^{1}_{\alpha\beta}$, where $n^{1}_{\alpha\beta}$ and $\tilde n^{1}_{\alpha\beta}$ are the on-site electronic density matrixes. The total energy as the sum of three parts $E=\tilde{E}+E^{1}-\tilde{E}^{1}$, where $\tilde E$ the smooth functional of the pseudo-density matrix $\tilde n_{\alpha\beta}(\vec{r})$ is evaluated on a regular grid, $E^{1}$ and $\tilde{E}^{1}$ the functionals of on-site electronic 