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SAXIS

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SAXIS = [real array]
Default: SAXIS = (0+, 0, 1)

Description: SAXIS specifies the quantisation axis for noncollinear spins.

The spin quantization axis may be specified by means of the SAXIS-tag,

 SAXIS =   sx sy sz    ! global spin quantisation axis


where the default for SAXIS = (0+,0,1) (the notation 0+ implies an infinitesimal small positive number in x-direction). All magnetic moments are now given with respect to the axis (sx,sy,sz), where we have adopted the convention that all magnetic moments and spinor-like quantities written or read by VASP are given with respect to this axis. This includes the MAGMOM line in the INCAR file, the total and local magnetizations in the OUTCAR and PROCAR file, the spinor-like orbitals in the WAVECAR file, and the magnetization density in the CHGCAR file. With respect to the Cartesian coordinates, the components of the magnetization are (internally) given by

{\displaystyle {\begin{aligned}m_{x}&=&\cos(\beta )\cos(\alpha )m_{x}^{\rm {axis}}-\sin(\alpha )m_{y}^{\rm {axis}}+\sin(\beta )\cos(\alpha )m_{z}^{\rm {axis}}\\m_{y}&=&\cos(\beta )\sin(\alpha )m_{x}^{\rm {axis}}+\cos(\alpha )m_{y}^{\rm {axis}}+\sin(\beta )\sin(\alpha )m_{z}^{\rm {axis}}\\m_{z}&=&-\sin(\beta )m_{x}^{\rm {axis}}+\cos(\beta )m_{z}^{\rm {axis}}\end{aligned}}}

Where maxis is the externally visible magnetic moment. Here, ${\displaystyle \alpha }$ is the angle between the SAXIS vector (sx,sy,sz) and the Cartesian vector ${\displaystyle {\hat {x}}}$, and ${\displaystyle \beta }$ is the angle between the vector SAXIS and the Cartesian vector ${\displaystyle {\hat {z}}}$:

{\displaystyle {\begin{aligned}\alpha &=&{\rm {atan}}{\frac {s_{y}}{s_{x}}}\\\beta &=&{\rm {atan}}{\frac {\sqrt {s_{x}^{2}+s_{y}^{2}}}{s_{z}}}\end{aligned}}}

The inverse transformation is given by

{\displaystyle {\begin{aligned}m_{x}^{\rm {axis}}&=&\cos(\beta )\cos(\alpha )m_{x}+\cos(\beta )\sin(\alpha )m_{y}-\sin(\beta )m_{z}\\m_{y}^{\rm {axis}}&=&-\sin(\alpha )m_{z}+\cos(\alpha )m_{y}\\m_{z}^{\rm {axis}}&=&\sin(\beta )\cos(\alpha )m_{x}+\sin(\beta )\sin(\alpha )m_{y}+\cos(\beta )m_{z}\end{aligned}}}

It is easy to see that for the default (sx,sy,sz)=(0+,0,1), both angles are zero, i.e. ${\displaystyle \beta =0}$ and ${\displaystyle \alpha =0}$. In this case, the internal representation is simply equivalent to the external representation:

{\displaystyle {\begin{aligned}m_{x}&=&m_{x}^{\rm {axis}}\\m_{y}&=&m_{y}^{\rm {axis}}\\m_{z}&=&m_{z}^{\rm {axis}}\end{aligned}}}

The second important case, is maxis=(0,0,m). In this case

{\displaystyle {\begin{aligned}m_{x}&=&\sin(\beta )*\cos(\alpha )m=ms_{x}/{\sqrt {s_{x}^{2}+s_{y}^{2}+s_{z}^{2}}}\\m_{y}&=&\sin(\beta )*\sin(\alpha )m=ms_{y}/{\sqrt {s_{x}^{2}+s_{y}^{2}+s_{z}^{2}}}\\m_{z}&=&\cos(\beta )m=ms_{z}/{\sqrt {s_{x}^{2}+s_{y}^{2}+s_{z}^{2}}}\end{aligned}}}

Hence now the magnetic moment is parallel to the vector SAXIS. Thus there are two ways to rotate the spins in an arbitrary direction, either by changing the initial magnetic moments MAGMOM or by changing SAXIS.

To initialize calculations with the magnetic moment parallel to a chosen vector (x,y,z), it is therefore possible to either specify (assuming a single atom in the cell)

MAGMOM = x y z   ! local magnetic moment in x,y,z
SAXIS =  0 0 1   ! quantisation axis parallel to z


or

MAGMOM = 0 0 total_magnetic_moment   ! local magnetic moment parallel to SAXIS
SAXIS =  x y z   ! quantization axis parallel to vector (x,y,z)


Both setups should in principle yield exactly the same energy, but for implementation reasons the second method is usually more precise. The second method also allows to read a preexisting WAVECAR file (from a collinear or non collinear run), and to continue the calculation with a different spin orientation. When a non collinear WAVECAR file is read, the spin is assumed to be parallel to SAXIS (hence VASP will initially report a magnetic moment in the z-direction only).

The recommended procedure for the calculation of magnetic anisotropies is therefore (please check the section on LMAXMIX):

• Start with a collinear calculation and calculate a WAVECAR and CHGCAR file.
  LSORBIT = .TRUE.