18.3. Non-collinear spin

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].

18.4. Example: chromium trimer

We save the following file under the name cr3.

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.

Table 18.4.1 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.

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

The file is run using

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. 18.4.1. 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.

Fig. 18.4.1 Spin-density of the Cr\(_3\) trimer

[HKH00]

D. Hobbs, G. Kresse, and J. Hafner. Fully unconstrained noncollinear magnetism within the projector augmented-wave method. Phys. Rev. B 62 (17 Nov. 2000), pp. 11556–11570.