Operator Formalism for Noncollinear Functionals in Multicollinear Approach
Xiaoyu Zhang, Taoni Bao
TL;DR
This work develops an operator formalism for noncollinear functionals within the multicollinear extension of collinear functionals in a two-component GKS framework. It derives a multiplicative noncollinear potential operator $\hat{V}_{\mathrm{xc}}$ via angular projection and constructs a corresponding xc magnetic field $\bm{B}_{\mathrm{xc}}$ and noncollinear kernel for spin-flip TDDFT, including practical expressions and stability improvements. The NCXC implementation is validated on periodic systems, including noncollinear spin spirals in $\gamma$-Fe, band structures of Bi$_2$Se$_3$ under SOC with six functionals, and band-gap predictions for inorganic semiconductors, demonstrating correct collinear limits and well-defined behavior as $|\bm{m}|\to 0$. The framework is modular and compatible with common DFT packages, enabling straightforward extension to higher derivatives and real-time spin dynamics for spintronics and topological materials research, while highlighting the need for further work on orbital-dependent mGGA via OEP. $V_{\mathrm{xc}}$ is given by $\hat{V}_{\mathrm{xc}} = \frac{1}{4 \pi} \int_0^{2\pi} \int_0^{\pi} \left[ \left.\frac{\delta E_{\mathrm{xc}}^{\mathrm{eff}}}{\delta n}\right|_{s=\mathbf{m}\cdot \mathbf{e}} \hat{I} + \left.\frac{\delta E_{\mathrm{xc}}^{\mathrm{eff}}}{\delta s}\right|_{s=\mathbf{m}\cdot \mathbf{e}} (\boldsymbol{\sigma}\cdot\mathbf{e}) \right] \sin\theta \, d\theta \, d\phi$ and $\bm{B}_{\mathrm{xc}} = \delta E_{\mathrm{xc}}/\delta \bm{m}$, with a spin-flip TDDFT kernel built from $\delta^2 E^{\mathrm{eff}}$.
Abstract
Accurate modeling of spin-orbit coupling and noncollinear magnetism requires noncollinear density functionals within the two-component generalized Kohn-Sham (GKS) framework, yet constructing and implementing noncollinear functionals remains challenging. Recently, a well-defined methodology called the multicollinear approach was proposed to extend collinear functionals into noncollinear ones. While previous research focuses on its matrix representation, the present work derives its operator formalism. We implement these new equations in our noncollinear functional ensemble named NCXC, which is expected to facilitate compatibility with most DFT software packages. Since the multicollinear approach was proposed for solving nonphysical properties and mathematical singularities in noncollinear functionals, we validate its accuracy in practical periodic systems, including noncollinear magnetism in spin spirals, band structures in topological insulators, and band gaps in semiconducting inorganic materials.