# Non-collinear spin {#sec:noncollinear}

In non-collinear spin DFT, the fundamental quantity is the spin-density
matrix instead of the density or the spin-density. RESCU can perform
certain computations within the non-collinear spin DFT framework. It is
currently advised to perform only $\Gamma$-point calculations. In this
section, we present the example of a Cr$_3$ cluster in the spiral
configuration, which is the ground state [\[HKH00\]](#HKH00){.citation}.

# Example: chromium trimer

We save the following file under the name cr3.

``` matlab
info.calculationType = 'self-consistent'
info.savepath = './results/cr3';
element(1).species = 'Cr';
element(1).path = './Cr_TM_LDA.mat';
atom.xyz = 4.743 * ...
[-0.5 0.0 0.0
  0.0 0.0 sqrt(3)/2
  0.5 0.0 0.0];
atom.element = [1 1 1];
domain.latvec = diag([20 20 20]);
domain.fourierInit = false;
domain.lowres = 0.25;
functional.libxc = 1;
functional.list = {'XC_GGA_X_PBE','XC_GGA_C_PBE'};
mixing.type = 'density';
mixing.tol = [1e-4,1e-4];
spin.type = 'non-collinear';
atom.magmomd = [5  90 179.9
                5  45   0.1
                5 135   0.1];
```

The meaning of the spin parameters is described below.

-   `spin.type` this is a string that determines the spin description.
    The allowed values are degenerate, collinear and non-collinear.
    Degenerate is the default value;
-   `atom.magmom` this is a list of magnetization vectors. It is used to
    calculate the initial spin-density matrix. If the system contains
    $n$ atoms, then the array is $n\times3$. The vector is express in
    spherical coordinates, the norm is in unit of Bohr magneton and the
    angles in radiant;
-   `atom.magmomd` same as atom.magmom but with angles in degree;
-   `atom.magmomcart` same as atom.magmom but in Cartesian coordinates.

  $|\mathbf{m}|$   $\theta$   $\phi$
  ---------------- ---------- --------
  4.14             -90        -180
  4.14             30         0
  4.14             150        0

  : The atomic magnetic moments of the chromium trimer. The total
  magnetic moment is 0 in the spiral configuration ground state. The
  atomic moments are calculated by projecting the spin density within
  1.3 Bohr radii from the atomic centers and the angle are average
  within the same ball.

The file is run using

``` shell
rescu -i cr3 -i cr3
```

This is a more advanced calculation mode. RESCU first performs an NAO
calculation from the cr3 input file. Then, it project the density and
all the wave-functions in real space and use that as an initial guess
for a real space calculation done from the cr3 input file. In this
example, we use cr3 for both inputs, but these are in fact independent.

The converged spin-density is shown in
`fig:cr3_quiver`{.interpreted-text role="numref"}. The spiral
configuration is readily apparent. We also note that the magnetization
all but vanishes at the mid-point between atoms, which is the so-called
Bloch wall.

<figure>
<img src="figs/cr3_quiver.png" id="fig:cr3_quiver"
alt="figs/cr3_quiver.png" />
<figcaption>Spin-density of the Cr<span
class="math inline"><sub>3</sub></span> trimer</figcaption>
</figure>

::: {#citations}

[HKH00]{#HKH00 .citation-label}

:   D. Hobbs, G. Kresse, and J. Hafner.
    `Fully unconstrained noncollinear magnetism within the projector augmented-wave method. <10.1103/PhysRevB.62.11556>`{.interpreted-text
    role="doi"} Phys. Rev. B 62 (17 Nov. 2000), pp. 11556--11570.
:::
