Orbital magnetic moment

The orbital magnetic moment at each atomic site is calculated as default in the non-collinear DFT. Since the orbital magnetic moment appears as a manifestation of spin-orbit coupling (SOC), the calculated values become finite when the SOC is included [63,64]. As an example, a non-collinear LDA+U (U=5eV) calculation of iron monoxide bulk is illustrated using an input file FeO_NC.dat in the directory 'work'. As for the LDA+U calculation, see the Section 'LDA+U'. The calculated orbital and spin magnetic moments at the Fe site are listed in Table 4. Also, you can find the orientation of the (decomposed) orbital moment in *.out, where * means 'System.Name' as follows:

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                     Orbital moments                       
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   Total Orbital Moment (muB)   0.000000070   Angles  (Deg) 113.644105951  -65.722115195

          Orbital moment (muB)   theta (Deg)  phi (Deg)
    1   Fe   1.01127           128.64444   50.80973
    2   Fe   1.01127            51.35556  230.80973
    3    O   0.00000           122.13287    8.40916
    4    O   0.00000            58.29296  151.31925


  Decomposed Orbital Moments

    1   Fe         Orbital Moment(muB)    Angles (Deg)
            multiple
  s           0    0.000000000           90.0000    0.0000
   sum over m      0.000000000           90.0000    0.0000
  s           1    0.000000000           90.0000    0.0000
   sum over m      0.000000000           90.0000    0.0000
  px          0    0.000032282           44.0757   90.0000
  py          0    0.000027194           31.5419   -0.0000
  pz          0    0.000026842           90.0000   57.4970
   sum over m      0.000070741           49.0444   57.5709
  px          1    0.004596036           10.8026  -90.0000
  py          1    0.004533432            5.2237  180.0000
  pz          1    0.000955444           90.0000  244.3929
   sum over m      0.009229130           11.9479  244.3959
  d3z^2-r^2   0    0.045401124           90.0000  224.3492
  dx^2-y^2    0    0.075657665           24.3023  228.5632
  dxy         0    0.453606172           81.2632   50.2745
  dxz         0    0.495766350          143.9475  -10.8730
  dyz         0    0.531382963          138.9632   98.7434
   sum over m      0.997255210          131.7287   51.1391
  d3z^2-r^2   1    0.001075694           90.0000  254.7742
  dx^2-y^2    1    0.012694575           26.6388  225.7504
  dxy         1    0.036086417           71.5849   49.3240
  dxz         1    0.031150186          132.6513  -13.0079
  dyz         1    0.033740724          128.7200   99.3874
   sum over m      0.058459849          109.4476   49.1020
  f5z^2-3r^2  0    0.007365273           90.0000   39.4321
  f5xz^2-xr^2 0    0.005659459           26.2551  124.3549
  f5yz^2-yr^2 0    0.006152658           34.4173  -38.4581
  fzx^2-zy^2  0    0.015290504           34.2465  224.2021
  fxyz        0    0.012904266           11.6263  244.9193
  fx^3-3*xy^2 0    0.004957037           43.3387  -84.7645
  f3yx^2-y^3  0    0.004826463           41.6700  183.4396
   sum over m      0.043385660           10.6323  246.7139
  .....
  ...

As shown in Table 4, OpenMX gives a good agreement for both the spin and orbital magnetic moments of a series of $3d$-transition metal oxides with other calculation results. However, it is noted that the absolute value of orbital magnetic moment seems to be significantly influenced by calculation conditions such as basis functions and on-site 'U' in the LDA+U method, while the spin magnetic moment is relatively insensitive to the calculation conditions, and that a rather rich basis set including polarization functions will be needed for convergent calculations of the orbital magnetic moment.

Table: Spin magnetic moment $M_s (\mu_{\rm B})$ and orbital magnetic moment $M_o (\mu_{\rm B})$ of transition metal oxides, MO (M=Mn,Fe,Co,Ni). In the LDA+U scheme [16], for the first d-orbital of M, the effective U of 4.0 (eV) for Mn, 5.0 (eV) for Fe, Co for 7.0 (eV), and Ni for 7.0 (eV) were used. For the others zero. The local spin moment was calculated by the Voronoi decomposition discussed in the Section 'Voronoi charge' rather than Mulliken charge, since the Mulliken analysis tends to give a larger spin moment in the use of multiple basis functions. The input files are MnO_NC.dat, FeO_NC.dat, CoO_NC.dat, and NiO_NC.dat in the directory 'work'. The other theoretical value [41] and experimental value [41] are also shown for comparison.

	$M_s$		$M_o$
Compound	OpenMX	Other calc.	OpenMX	Other calc.	Expt. in total
MnO	4.560	4.49	0.001	0.00	4.79,4.58
FeO	3.582	3.54	1.010	1.01	3.32
CoO	2.684	2.53	1.088	1.19	3.35,3.8
NiO	1.594	1.53	0.173	0.27	1.77,1.64,1.90