Constraining the direction of magnetic moments

Supported as of VASP.4.6.

VASP offers the possibility to add a penalty contribution to the total energy expression (and consequently a penalty functional to the Hamiltonian) which drives the local moment (integral of the magnetization in a site centered sphere) into a direction specified by the user. This feature is controlled using the following tags:

• I_CONSTRAINED_M=1
  Constrain the direction of the magnetic moments. The total energy is given by

  $$E=E_0+ \sum_I\lambda \left[ \vec{M}_I-\hat{M}^0_I \left( \hat{M}^0_I \cdot \vec{M_I}\right)\right]^2$$ (6.52)

  
  
  where $E_0$ is the usual DFT energy, and the second term on the right-hand-side represents the penalty. The sum is taken over all atomic sites $I$, $\hat{M}^0_I$ is the desired direction of the magnetic moment at site $I$, and $\vec{M}_I$ is the integrated magnetic moment inside a sphere $\Omega_I$ (the radius must be specified through the RWIGS-tag, see below) around the position of atom $I$,

  $$\vec{M}_I=\int_{\Omega_I} \vec{m}({\bf r}) F_I(\vert{\bf r}\vert) d{\bf r}$$ (6.53)

  
  
  where $F_I(\vert{\bf r}\vert)$ is a function of norm 1 inside $\Omega_I$, that smoothly goes to zero towards the boundary of $\Omega_I$.

  The penalty term in the total energy introduces an additional potential inside the aforementioned spheres centered at the atomic sites $I$, given by

  $$V_I ({\bf r})=2\lambda \left[ \vec{M}_I-\hat{M}^0_I \left( \hat{M}^0_I \cdot \vec{M_I}\right)\right] \cdot \vec{\sigma} F_I(\vert{\bf r}\vert)$$ (6.54)

  
  
  where $\vec{\sigma}=(\sigma_x,\sigma_y,\sigma_z)$ are the Pauli spin-matrices.

• I_CONSTRAINED_M=2
  Constrain the size and direction of the magnetic moments. The total energy is given by

  $$E=E_0+ \sum_I\lambda \left( \vec{M}_I-\vec{M}^0_I \right)^2$$ (6.55)

  
  
  where $\vec{M}^0_I$ is the desired magnetic moment at site $I$. The additional potential that arises from the penalty contribution to the total energy is given by

  $$V_I ({\bf r})=2\lambda \left( \vec{M}_I-\vec{M}^0_I \right)\cdot \vec{\sigma} F_I(\vert{\bf r}\vert)$$ (6.56)

  
  

• LAMBDA= [real]
  Specifies the weight $\lambda$, with which the penalty terms enter into the total energy expression and the Hamiltonian (see equations above).
• M_CONSTR= $M^0_{1x}$ $M^0_{1y}$ $M^0_{1z}$ ...
  The desired direction(s) of the integrated local moment(s) with respect to cartesian coordinates (3 coordinates must be specified for each ion). For I_CONSTRAINED_M=1 the norm of this vector is meaningless since only the direction will be constrained. Setting M_CONSTR= 0 0 0 for an ion is equivalent to imposing no constraints.

In addition one must set the RWIGS-tag to specify the radius of integration around the atomic sites which determines the local moments.

When one uses the constrained moment approach, additional information pertaining to the effect of the constraints is written into the OSZICAR file:

 E_p =  0.36856E-07  lambda =  0.500E+02
<lVp>=  0.30680E-02
 DBL = -0.30680E-02
 ion        MW_int                 M_int
  1 -0.565  0.000  0.000   -0.770  0.000  0.000
  2  0.565  0.000  0.000    0.770  0.000  0.000
  3 -0.565  0.000  0.000   -0.770  0.000  0.000
  4  0.565  0.000  0.000    0.770  0.000  0.000
DAV:   8    -0.133293620177E+03    0.15284E-05   -0.29410E-08  4188   0.144E-03    0.119E-04

E_p is the contribution to the total energy arising from the penalty functional. Under M_int VASP lists the integrated magnetic moment at each atomic site. The column labeled MW_int shows the result of the integration of magnetization density which has been smoothed towards the boundary of the sphere (see Eq. 6.53). It is actually the smoothed integrated moment which enters in the penalty terms (the smoothing ensures that the total local potential remains continuous at the sphere boundary). One should look at the latter numbers to check whether enough of the magnetization denstity around each atomic site is contained within the integration sphere and increase RWIGS accordingly. What exactly constitutes ``enough" in this context is hard to say. It is best to set RWIGS in such a manner that the integration spheres do not overlap and are otherwise as large as possible.

At the end of the run the OSZICAR file contains some extra information:

DAV:   9    -0.133293621087E+03   -0.91037E-06   -0.18419E-08  4188   0.104E-03
   1 F= -.13329362E+03 E0= -.13329362E+03  d E =0.000000E+00  mag=     0.0000     0.0000     0.0000

 E_p =  0.36600E-07  lambda =  0.500E+02
 ion             lambda*MW_perp
  1  -0.67580E-03  -0.12424E-22  -0.88276E-23
  2   0.67580E-03   0.14700E-22  -0.24744E-22
  3  -0.67790E-03  -0.82481E-23  -0.19834E-22
  4   0.67790E-03   0.15710E-23   0.34505E-22

Under lambda*MW_perp the constraining ``magnetic field" at each atomic site is listed. It shows which magnetic field is added to the DFT Hamiltonian to stabilize the magnetic configuration.

As is probably clear from the above, applying constraints by means of a penalty functional contributes to the total energy. This contribution, however, decreases with increasing LAMBA and can in principle be made vanishingly small. Increasing LAMBDA stepwise, from one run to another (slowly so the solution remains stable) one thus converges towards the DFT total energy for a given magnetic configuration.