%\title{ABINIT data structures and their theoretical justifications}
\documentclass{article}
\title{Non colinear magnetism\\
       {\normalsize{(G. Z\'erah)}}}
\begin{document}
\maketitle

\section{Notations and theoretical considerations}
\label{sec:A1}
\hspace*{\parindent}
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* We will denote the spinor by $\Psi^{\alpha \beta}$, ${\alpha, \beta}$ being the two spin indexes.
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* The magnetic properties are well represented by introducing the spin density matrix:
$\rho^{\alpha,\beta}(r)=\sum_n f_n <r|\Psi_n^\alpha>
<\Psi_n^\beta|r>$ where the sum runs over all states and $f_n$ is the occupation of state $n$.
\vskip 1em
* With  $\rho^{\alpha,\beta}(r)$, we can express the scalar density by
$\rho(r)=\sum_{\alpha,\alpha} \rho^{\alpha,\alpha}(r)$
and the magnetisation density $\vec m(r)$ (in units of $\hbar /2$) whose components are
$m_i(r)=\sum_{\alpha,\beta} \rho^{\alpha,\beta}(r) \sigma_i^{\alpha,\beta}$, where the $\sigma_i$ are
the Pauli matrices.
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* In general, $E_{xc}$ is a functional of $\rho^{\alpha,\beta}(r)$, or equivalently  of $\vec m(r)$
and $\rho(r)$. It is therefore denoted as $E_{xc}(n(r), \vec m(r))$
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* The expression of $V_{xc}$
taking into account the above expression of $E_{xc}$ is:
$$
V_{xc}^{\alpha,\beta}(r)={\delta E_{xc} \over \delta \rho (r)} delta_{\alpha,\beta}+
\sum_{i=1}^3 {\delta E_{xc} \over \delta m_i (r) }\sigma_i^{\alpha,\beta}
$$
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* In the LDA approximation, due to its rotational invariance, $E_{xc}$ is indeed a functional of $n(r)$ and $|m(r)|$ only.
In the GGA approximation, we {\it assume} that it is a functional of $n(r)$ and $|m(r)|$ and their gradients.(This is not the
most general functional of $\vec m(r)$ 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 $\vec m(r)$ as polarization direction.
We the have $ {\delta E_{xc} \over \delta m_i (r) }={\delta E_{xc} \over \delta |m_i (r)| }
\widehat {m(r)}$, where $\widehat {m(r)}={m(r) \over |m(r)|}$.
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:
$$
{\delta E_{xc} \over \delta \rho (r)}={1 \over 2}({\delta E_{xc} \over \delta n\uparrow(r)}+
{\delta E_{xc} \over \delta n\downarrow(r)})
$$
and
$$
{\delta E_{xc} \over \delta |m (r)| }={1 \over 2}({\delta E_{xc} \over \delta n\uparrow(r)}-
{\delta E_{xc} \over \delta n\downarrow(r)})
$$
This makes the connection with the more usual spin polarized case.
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* Expression of $V_{xc}$ in LDA-GGA
$$
V_{xc}(r)={\delta E_{xc} \over \delta \rho (r)} \delta_{\alpha,\beta}+ {\delta E_{xc} \over \delta |m (r)| }
 {\widehat m(r)}.\sigma
$$
* Implementation
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* Computation of $\rho^{\alpha,\beta}(r)=\sum_n f_n <r|\Psi^\alpha>
<\Psi^\beta|r>$
One would like to use the routine {\tt mkrho.f} which does precisely this. But this routine transforms only real quantities, whereas
$\rho^{\alpha,\beta}(r)$ is hermitian and can have complex elements.
The ``trick'' is to use only the real quantities:
\begin{eqnarray*}
\rho^{1,1}(r)&=&\sum_n f_n <r|\Psi^1>
<\Psi^1> \\
\rho^{2,2}(r)&=&\sum_n f_n <r|\Psi^2>
<\Psi^2> \\
\rho(r)+m_x(r)&=&\sum_{n} f_n (\Psi^{1}+\Psi^{2})^*_n (\Psi^{1}+\Psi^{2})_n \\
\rho(r)+m_y(r)&=&\sum_{n} f_n (\Psi^{1}-i \Psi^{2})^*_n (\Psi^{1}-i \Psi^{2})_n
\end{eqnarray*}
and compute  $(\rho(r), \vec m(r))$ with the help of the aditionnal:
\begin{eqnarray*}
\rho(r)&=&\rho^{1,1}(r)+\rho^{2,2}(r) \\
m_z(r)&=&\rho^{1,1}(r) - \rho^{2,2}(r)
\end{eqnarray*}

Note that only the forurier transform are performed in {\tt mkrho.f} the final transformation to
$(\rho(r), \vec m(r))$ is performed in {\tt symrhg.f}.
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* The computation of $V_{xc}$ is performed in {\tt rhohxc.f}. The only transformation to this routine, is
to compute $|\vec m(r)|$ as input of the usual (i.e spin polarized)  {\tt rhohxc.f} and yield back
the four component $V_{xc}$, from the expression of ${\delta E_{xc} \over \delta |m (r)| }$.

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* For more information, see: Hobbs et al., PRB, 62, 11556 ; Perdew et al. PRB, 46, 6671
(for the xc functional)
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