# MagMom

This page gives hints on how to perform calculation with constrained atomic magnetic moments with the ABINIT package.

## Introduction¶

A complementary magnetic constraint method has been implemented in the ABINIT code, wherein the magnetization around each atom $I$ is pushed to a desired (vectorial) value. The constraint can either be on the full vector quantity, $\vec{M}_I$, or only on the direction $\vec{e}_I$. This is mainly useful for non collinear systems, where the direction and amplitude of the magnetic moment can change. The method follows that used in VASP [Ma2015]: a Lagrangian constraint is applied to the energy, and works through a resulting term in the potential, which acts on the different spin components. The magnetization in a sphere $\Omega_I$ around atom $I$ at position $\vec{R}_I$ is calculated as:

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

and the corresponding potential for spin component $\alpha$ is written as:

V_{\alpha}(\vec{r}) = -2 \lambda F_I(|\vec{r}-\vec{R_I}|) \vec{c} \vec{\sigma}_{\alpha}.

The function $F_I$ is zero outside of a sphere of radius ratsph for atom $I$, and inside the sphere, the function $f(x) = x^2(3+x(1+x(-6+3x)))$ (see Eq. (B4) in [Laflamme2016]), is applied to smooth the transition in a region of thickness $r_s$ (fixed to 0.05 bohr), otherwise it is 1. This minimizes discontinuous variations of the potential from iteration to iteration.

The constraint is managed by the keyword magconon. Value 1 gives a constraint on the direction:

\vec{c} = (\vec{M}_I/|\vec{M}_I|)-(\vec{M}^{spinat}_I / |\vec{M}^{spinat}_I|),

while value 2 gives a full constraint on the vector

\vec{c} = \vec{M}_I - \vec{M}^{spinat}_I,

where in both cases $\vec{M}^{spinat}_I$ defined by spinat, giving a 3-vector magnetic potential for each atom. The latter is quite a stringent constraint, and often may not converge. The former value usually works, provided sufficient precision is given for the calculation of the magnetic moment (kinetic energy cutoff in particular).

The strength of the constraint is given by the keyword magcon_lambda ($\lambda$ above - real valued). Typical values are $10^{-2}$ but vary strongly with system type: this value should be started small (here the constraint may not be enforced fully) and increased. A too large value leads to oscillations of the magnetization (the equivalent of charge sloshing) which do not converge. A corresponding Lagrange penalty term is added to the total energy, and is printed to the log file, along with the effective magnetic field being applied. In an ideal case the energy penalty term should go to 0 (the constraint is fully satisfied).

useful:

• magcon_lambda MAGnetization CONstraint LAMBDA parameter
• magconon turn MAGnetization CONstraint ON
• ratsm Radii of the ATomic spheres SMearing
• ratsph Radii of the ATomic SPHere(s)
• spinat SPIN for AToms

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