I CONSTRAINED M

I_CONSTRAINED_M = 1 | 2
Default: I_CONSTRAINED_M = none 

Description: I_CONSTRAINED_M switches on the constrained local moments approach.

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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=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

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

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, $\hat{M}^0_I$ is the desired direction of the magnetic moment at site I (as specified using M_CONSTR), and $\vec{M}_I$ 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,

$\vec{M}_I=\int_{\Omega_I} \vec{m}(\mathbf{r}) F_I(|\mathbf{r}|) d\mathbf{r}$

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

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

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

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

where $\vec{M}^0_I$ 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

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

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.

Related Tags and Sections

M_CONSTR, LAMBDA, RWIGS, LNONCOLLINEAR

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