# Density functional perturbation theory within non-collinear magnetism

## TL;DR

This paper extends density functional perturbation theory to non-collinear magnetism by developing methods to compute exchange-correlation derivatives in a non-collinear framework, enabling accurate simulations of magnetic materials.

## Contribution

The paper introduces two approaches to transform non-collinear exchange-correlation derivatives into a local collinear basis for perturbation theory calculations.

## Key findings

- Methods successfully applied to atomic displacement perturbations.
- Methods enable second-order electric field perturbation calculations.
- Comparison shows trade-offs between Taylor expansion and explicit evaluation.

## Abstract

We extend the density functional perturbation theory formalism to the case of non-collinear magnetism. The main problem comes with the exchange-correlation (XC) potential derivatives, which are the only ones that are affected by the non-collinearity 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 non-collinear framework. To solve this problem, we consider here possibilities to transform the non-collinear 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, 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.