Noncollinear density functional theory having proper invariance and local torque properties

Noncollinear spins are among the most interesting features of magnetic materials, and their accurate description is a central goal of density functional theory applied to periodic solids. However, these calculations typically yield a magnetization vector that is everywhere parallel to the exchange-correlation magnetic field. No meaningful description of spin dynamics can emerge from a functional constrained to have vanishing local magnetic torque. In this contribution we present a generalization to periodic systems of the extension of exchange-correlation functionals to the noncollinear regime, proposed by Scalmani and Frisch [J. Chem. Theory Comput. 8, 2193 (2012)]. This extension does afford a nonvanishing local magnetic torque and is free of numerical instabilities. As illustrative examples, we discuss frustrated triangular and kagome lattices evaluated with various density functionals, including screened hybrid functionals.

Figure 1

The structure of the triangular Cr lattice. The blue points represent the magnetic unit cell employed in the calculations. The dark blue arrows present the converged magnetization obtained with the HSE functional, partitioned according to the Hirshfeld scheme. Additionally, the translation vectors are presented (green arrows).

Figure 2

The directions of the magnetization (red arrows) and the exchange-correlation magnetic field (blue arrows) obtained with the PBE functional. The color map depicts the magnitude (in atomic units) of $\vec{m}\times {\vec{B}}_{\mathrm{xc}}$ in the direction perpendicular to the Cr surface. The black points indicate the actual magnetic unit cell used in the calculations.

Figure 3

The structure of the kagome $\vec{q}=0$ lattices of the (left) positive and (right) negative chiralities. The blue points represent the actual magnetic unit cell used in the calculations, with arrows representing the converged magnetization obtained with the HISS functional partitioned according to the Hirshfeld scheme. For the clarity of the picture, the magnetization for the negative chirality has been rotated to stress the difference to the left panel. Additionally, the translation vectors are presented (green arrows).

Figure 4

The directions of the magnetization (red arrows) and the exchange-correlation magnetic field (blue arrows) obtained with the PBE functional for the kagome lattice of (left) positive and (right) negative chirality. The color map depicts the magnitude (in atomic units) of $\vec{m}\times {\vec{B}}_{\mathrm{xc}}$ in the direction perpendicular to the Cr surface. The black points indicate the actual magnetic unit cell used in the calculations.

Figure 5

The structure of the $\sqrt{3}\times \sqrt{3}$ kagome lattice. The blue points represent the magnetic unit cell employed in the calculations, with arrows representing the converged magnetization obtained with the TPSS functional, partitioned according to the Hirshfeld scheme. Additionally, the translation vectors are presented (green arrows).