Density functional perturbation theory within noncollinear magnetism

We extend the density functional perturbation theory formalism to the case of noncollinear magnetism. The main problem comes with the exchange-correlation (XC) potential derivatives, which are the only ones that are affected by the noncollinearity of the system. Most of the present XC functionals are constructed at the collinear level, such that the off-diagonal (containing magnetization densities along $x$ and $y$ directions) derivatives cannot be calculated simply in the noncollinear framework. To solve this problem, we consider here possibilities to transform the noncollinear XC derivatives to a local collinear basis, where the $z$ axis is aligned with the local magnetization at each point. The two methods we explore are (i) expanding the spin rotation matrix as a Taylor series and (ii) evaluating explicitly the XC for the local density approximation through an analytical expression of the expansion terms. We compare the two methods and describe their practical implementation. We show their application for atomic displacement and electric field perturbations at the second order, within the norm-conserving pseudopotential methods.