Noncollinear

Notations and theoretical considerations¶

We will denote the spinor by $\Psi^{\alpha\beta}$, ${\alpha, \beta}$ being the two spin indexes. The magnetic properties are well represented by introducing the spin density matrix:

\rho^{\alpha\beta}(\rr) = \sum_n f_n \la \rr|\Psi_n^\alpha\ra \la\Psi_n^\beta|\rr\ra

where the sum runs over all states and $f_n$ is the occupation of state $n$.

With $\rho^{\alpha\beta}(\rr)$, we can express the scalar density by

\rho(\rr)=\sum_{\alpha} \rho^{\alpha\alpha}(\rr)

and the magnetization density $\vec m(\rr)$ (in units of $\hbar /2$) whose components are:

m_i(\rr) = \sum_{\alpha\beta} \rho^{\alpha\beta}(\rr) \sigma_i^{\alpha\beta},

where the $\sigma_i$ are the Pauli matrices.

In general, $E_{xc}$ is a functional of $\rho^{\alpha\beta}(\rr)$, or equivalently of $\vec m(\rr)$ and $\rho(\rr)$. It is therefore denoted as $E_{xc}[n(\rr), \vec m(\rr)]$.

The expression of $V_{xc}$ taking into account the above expression of $E_{xc}$ is:

V_{xc}^{\alpha\beta}(\rr)={\delta E_{xc} \over \delta \rho (\rr)} \delta_{\alpha\beta} + \sum_{i=1}^3 {\delta E_{xc} \over \delta m_i (\rr) }\sigma_i^{\alpha\beta}

In the LDA approximation, due to its rotational invariance, $E_{xc}$ is indeed a functional of $n(\rr)$ and $|m(\rr)|$ only. In the GGA approximation, on the contrary, we assume that it is a functional of $n(\rr)$ and $|m(\rr)|$ and their gradients. (This is not the most general functional of $\vec m(\rr)$ dependent upon first order derivatives, and rotationally invariant.) We therefore use exactly the same functional as in the spin polarized situation, using the local direction of $\vec m(\rr)$ as polarization direction.

We then have

{\delta E_{xc} \over \delta m_i (\rr) }={\delta E_{xc} \over \delta |m_i (\rr)| } \widehat {m(\rr)},

where $\widehat {m(\rr)} = {m(\rr) \over |m(\rr)|}$. Now, in the LDA-GGA formulations, $n_\uparrow + n_\downarrow =n$ and $|n_\uparrow-n_\downarrow|=|m|$ and therefore, if we set $n_\uparrow = (n+m)/2$ and $n_\downarrow=(n-n_\uparrow)$, we have:

{\delta E_{xc} \over \delta \rho (\rr)} = {1 \over 2} \Bigl( {\delta E_{xc} \over \delta n_\uparrow(\rr)}+ {\delta E_{xc} \over \delta n_\downarrow(\rr)} \Bigr )

and

{\delta E_{xc} \over \delta |m (\rr)| }={1 \over 2} \Bigl ( {\delta E_{xc} \over \delta n_\uparrow(\rr)} - {\delta E_{xc} \over \delta n_\downarrow(\rr)} \Bigr )

This makes the connection with the more usual spin polarized case.

Expression of $V_{xc}$ in LDA-GGA

V_{xc}(\rr) = {\delta E_{xc} \over \delta \rho (\rr)} \delta_{\alpha\beta}+ {\delta E_{xc} \over \delta |m (\rr)| } {\widehat m(\rr)}.\sigma

Implementation¶

Computation of $\rho^{\alpha\beta}(\rr) = \sum_n f_n \la \rr|\Psi^\alpha\ra \la\Psi^\beta|\rr\ra$

One would like to use the routine mkrho which does precisely this but this routine transforms only real quantities, whereas $\rho^{\alpha\beta}(\rr)$ is hermitian and can have complex elements. The trick is to use only the real quantities:

\begin{eqnarray*} \rho^{11}(\rr)& = &\sum_n f_n \la \rr|\Psi^1\ra \la\Psi^1\ra \\ \rho^{22}(\rr)&=&\sum_n f_n \la \rr|\Psi^2\ra \la\Psi^2\ra \\ \rho(\rr)+m_x(\rr)&=&\sum_{n} f_n (\Psi^{1}+\Psi^{2})^*_n (\Psi^{1}+\Psi^{2})_n \\ \rho(\rr)+m_y(\rr)&=&\sum_{n} f_n (\Psi^{1}-i \Psi^{2})^*_n (\Psi^{1}-i \Psi^{2})_n \end{eqnarray*}

and compute $\rho(\rr)$ and $\vec m(\rr)$ with the help of:

\begin{eqnarray*} \rho(\rr)&=&\rho^{11}(\rr)+\rho^{22}(\rr) \\ m_z(\rr)&=&\rho^{11}(\rr) - \rho^{22}(\rr) \end{eqnarray*}