I CONSTRAINED M: Difference between revisions

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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 ''r''={{TAG|RWIGS}}) into a direction given by the {{TAG|M_CONSTR}}-tag.
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 ''r''={{TAG|RWIGS}}) into a direction given by the {{TAG|M_CONSTR}}-tag.


*{{TAG|I_CONSTRAINED_M}}=1
*{{TAG|I_CONSTRAINED_M}}=1: Constrain the ''direction'' of the magnetic moments.
:Constrain the direction of the magnetic moments. The total energy is given by
:The total energy is given by


::<math>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</math>
::<math>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</math>
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:where <math>\vec{\sigma}=(\sigma_x,\sigma_y,\sigma_z)</math> are the Pauli spin-matrices.
:where <math>\vec{\sigma}=(\sigma_x,\sigma_y,\sigma_z)</math> are the Pauli spin-matrices.


*{{TAG|I_CONSTRAINED_M}}=2
*{{TAG|I_CONSTRAINED_M}}=2: Constrain the ''size and direction'' of the magnetic moments.
:The total energy is given by
 
::<math>E=E_0+ \sum_I\lambda \left( \vec{M}_I-\vec{M}^0_I \right)^2</math>
 
:where <math>\vec{M}^0_I</math> is the desired magnetic moment at site ''I'' (as specified using {{TAG|M_CONSTR}}).
:The additional potential that arises from the penalty contribution to the total energy is given by
::<math>V_I (\mathbf{r})=2\lambda \left( \vec{M}_I-\vec{M}^0_I \right)\cdot \vec{\sigma} F_I(|\mathbf{r}|)</math>


== Related Tags and Sections ==
== Related Tags and Sections ==

Revision as of 16:32, 16 February 2011

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 r=RWIGS) into a direction given by the M_CONSTR-tag.

The total energy is given by
where E0 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 ΩI (the radius must be specified by means of RWIGS) around the position of atom I,
where FI(|r|) is a function of norm 1 inside ΩI, that smoothly goes to zero towards the boundary of ΩI.
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

Related Tags and Sections

M_CONSTR, LAMBDA, RWIGS, LNONCOLLINEAR


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