Noncollinear

Non colinear magnetism

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 a functional of \(n(\rr)\) and the norm of the magnetization vector \(|m(\rr)|\) only. In the GGA approximation, however, 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

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

where we define \(\widehat {m(\rr)} = {\vec m(\rr) \over |m(\rr)|}\), the unit vector along the magnetization direction. 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 :

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

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:

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

For more information about noncollinear magnetism see [Hobbs2000] and [Perdew1992] for the xc functional.