mirror of https://github.com/abinit/abinit.git
104 lines
3.8 KiB
Markdown
104 lines
3.8 KiB
Markdown
---
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authors: G. Zerah
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---
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## Non colinear magnetism
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## Notations and theoretical considerations
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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:
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$$ \rho^{\alpha\beta}(\rr) = \sum_n f_n \la \rr|\Psi_n^\alpha\ra \la\Psi_n^\beta|\rr\ra ,$$
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where the sum runs over all states and $f_n$ is the occupation of state $n$.
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With $\rho^{\alpha\beta}(\rr)$, we can express the scalar density by
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$$ \rho(\rr)=\sum_{\alpha} \rho^{\alpha\alpha}(\rr) $$
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and the magnetization density $\vec m(\rr)$ (in units of $\hbar /2$) whose components are:
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$$ m_i(\rr) = \sum_{\alpha\beta} \rho^{\alpha\beta}(\rr) \sigma_i^{\alpha\beta}, $$
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where the $\sigma_i$ are the Pauli matrices.
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In general, $E_{xc}$ is a functional of $\rho^{\alpha\beta}(\rr)$, or equivalently of $\vec m(\rr)$ and $\rho(\rr)$.
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It is therefore denoted as $E_{xc}[n(\rr), \vec m(\rr)]$.
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The expression of $V_{xc}$ taking into account the above expression of $E_{xc}$ is:
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$$
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V_{xc}^{\alpha\beta}(\rr)={\delta E_{xc} \over \delta \rho (\rr)} \delta_{\alpha\beta} +
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\sum_{i=1}^3 {\delta E_{xc} \over \delta m_i (\rr) }\sigma_i^{\alpha\beta}.
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$$
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In the LDA approximation, due to its rotational invariance, $E_{xc}$ is a functional of $n(\rr)$ and
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the norm of the magnetization vector $|m(\rr)|$ only.
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In the GGA approximation, however, we **assume** that it is a functional of $n(\rr)$ and $|m(\rr)|$ and their gradients.
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(This is not the most general functional of $\vec m(\rr)$ dependent upon first order derivatives, and rotationally invariant.)
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We therefore use exactly the same functional as in the spin polarized situation, using the local direction
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of $\vec m(\rr)$ as polarization direction.
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We then have
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$$
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{\delta E_{xc} \over \delta m_i (\rr) }={\delta E_{xc} \over \delta |m (\rr)| } \Big( \widehat {m(\rr)} \Big)_i,
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$$
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where we define $\widehat {m(\rr)} = {\vec m(\rr) \over |m(\rr)|}$, the unit vector along the magnetization direction.
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Now, in the LDA-GGA formulations, $n_\uparrow + n_\downarrow =n$ and $|n_\uparrow-n_\downarrow|=|m|$
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and therefore, if we set $n_\uparrow = (n+m)/2$ and $n_\downarrow=(n-n_\uparrow)$, we have:
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$$
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{\delta E_{xc} \over \delta \rho (\rr)} = {1 \over 2} \Bigl(
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{\delta E_{xc} \over \delta n_\uparrow(\rr)}+
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{\delta E_{xc} \over \delta n_\downarrow(\rr)}
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\Bigr )
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$$
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and
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$$
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{\delta E_{xc} \over \delta |m (\rr)| }={1 \over 2} \Bigl (
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{\delta E_{xc} \over \delta n_\uparrow(\rr)} -
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{\delta E_{xc} \over \delta n_\downarrow(\rr)}
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\Bigr ) .
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$$
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This makes the connection with the more usual spin polarized case.
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Expression of $V_{xc}$ in LDA-GGA :
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$$
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V_{xc}(\rr) = {\delta E_{xc} \over \delta \rho (\rr)} \delta_{\alpha\beta}+ {\delta E_{xc} \over \delta |m (\rr)| }
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\big( {\widehat m(\rr)}.\sigma \big) .
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$$
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## Implementation
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Computation of $\rho^{\alpha\beta}(\rr) = \sum_n f_n \la \rr|\Psi^\alpha\ra \la\Psi^\beta|\rr\ra$
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One would like to use the routine *mkrho* which does precisely this
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but this routine transforms only real quantities, whereas
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$\rho^{\alpha\beta}(\rr)$ is hermitian and can have complex elements.
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The *trick* is to use only the real quantities:
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\begin{eqnarray*}
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\rho^{11}(\rr)& = &\sum_n f_n \la \rr|\Psi^1\ra \la\Psi^1\ra \\
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\rho^{22}(\rr)&=&\sum_n f_n \la \rr|\Psi^2\ra \la\Psi^2\ra \\
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\rho(\rr)+m_x(\rr)&=&\sum_{n} f_n (\Psi^{1}+\Psi^{2})^*_n (\Psi^{1}+\Psi^{2})_n \\
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\rho(\rr)+m_y(\rr)&=&\sum_{n} f_n (\Psi^{1}-i \Psi^{2})^*_n (\Psi^{1}-i \Psi^{2})_n
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\end{eqnarray*}
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and compute $\rho(\rr)$ and $\vec m(\rr)$ with the help of:
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\begin{eqnarray*}
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\rho(\rr)&=&\rho^{11}(\rr)+\rho^{22}(\rr) \\
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m_z(\rr)&=&\rho^{11}(\rr) - \rho^{22}(\rr)
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\end{eqnarray*}
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For more information about noncollinear magnetism see [[cite:Hobbs2000]]
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and [[cite:Perdew1992]] for the xc functional.
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