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# I CONSTRAINED M

I_CONSTRAINED_M = 1 | 2

Default: **I_CONSTRAINED_M** = none

Description: I_CONSTRAINED_M switches on the constrained local moments approach.

VASP offers the possibility to add a penalty contribution to the total energy expression (and consequently a penalty functional to the Hamiltonian) that drives the local magnetic moment (integral of the magnetization in a site centered sphere of radius RWIGS) into a direction given by the M_CONSTR-tag.

- I_CONSTRAINED_M=1: Constrain the
*direction*of the magnetic moments.

- The total energy is given by

- 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*, is the desired direction of the magnetic moment at site*I*(as specified using M_CONSTR), and is the integrated magnetic moment inside a sphere Ω_{I}(the radius**must**be specified by means of RWIGS) around the position of atom*I*,

- where
*F*_{I}(|**r**|) is a function of norm 1 inside Ω_{I}, that smoothly goes to zero towards the boundary of Ω_{I}.

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

- where are the Pauli spin-matrices.

- I_CONSTRAINED_M=2: Constrain the
*size and direction*of the magnetic moments.

- The total energy is given by

- where is the desired magnetic moment at site
*I*(as specified using M_CONSTR). - The additional potential that arises from the penalty contribution to the total energy is given by

The weight λ, with which the penalty terms enter into the total energy expression and the Hamiltonian in the above is specified through the LAMBDA tag.

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 decreasing LAMBDA and can in principle be made vanishingly small. Decreasing 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.

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. 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 density 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.

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.

## Related Tags and Sections

M_CONSTR, LAMBDA, RWIGS, LNONCOLLINEAR