Spin-Dependent Nonorthogonal Generalized Wannier Functions and their Integration with PAW and Hubbard Corrections in Linear-Scaling DFT

TL;DR

This work extends linear-scaling DFT to spin-polarized systems by introducing spin-dependent nonorthogonal generalized Wannier functions (NGWFs) within ONETEP, enabling independent optimization of spin-up and spin-down orbitals. The authors integrate spin-dependent NGWFs with the projector-augmented-wave (PAW) formalism and DFT+$U$/$J$ corrections, deriving the necessary gradient and force expressions in the PAW framework and implementing a minimum-tracking linear response for in situ U and J. Across defects in hBN, transition-metal complexes, 2D vdW magnets like CrI$_3$, and bulk/nano cobalt, the method yields improved total energies, sharper spin localization, and better spin-resolved spectra, often matching plane-wave and hybrid-functional benchmarks more closely than spin-independent baselines. The approach offers a scalable and accurate route to study spin-polarized materials within LS-DFT, with future prospects including spin–orbit coupling, noncollinear magnetism, and machine-learned Hubbard projectors to further enhance correlation treatments.

Abstract

We present a spin-dependent extension of the non-orthogonal generalized Wannier function (NGWF) formalism within the framework of linear-scaling density functional theory (LS-DFT) as implemented in the ONETEP code. In traditional LS-DFT representations, both spin channels are constrained to share a common variational basis, which limits the accuracy for systems that are spin-polarized or exhibit magnetic order. Our approach allows NGWFs to vary independently for each spin channel, enabling a more accurate representation of spin-polarization in the electronic density. We demonstrate the efficacy of this method through a series of test cases, including localized magnetic defects in two-dimensional hBN, transition metal complexes, two-dimensional van der Waals magnetic materials, and both bulk and nanocluster ferromagnetic Co. In each scenario, the incorporation of spin-dependent NGWFs results in enhanced accuracy for total energy calculations, improved localization of spin density, and accurate predictions of magnetic ground states. This improvement is particularly notable when combined with DFT+U and DFT+U+J corrections. In this work, we take the opportunity to describe the combination of DFT+U+J and the projector-augmented wave (PAW) formalism within the LS-DFT framework, including how PAW participates in the ionic Pulay force, and in the minimum-tracking linear response approach for computing parameters in situ. Our findings demonstrate that spin-dependent NGWFs are a crucial and computationally efficient advancement in the linear-scaling DFT simulation of spin-polarized materials.

Figures (10)

• Figure 1: A test for the consistency between DFT+$U$+$J$ total energy gradients, with respect to ionic displacement, and DFT+$U$+$J$ forces including PAW terms, as defined by Eqns. \ref{['eq:finalU_force']},\ref{['eq:finalJ_force']}, and \ref{['eq:tildeX']}. The curve is the analytical derivative of a least-squares fit of a two-parameter (quadratic and cubic) polynomial, including energy data points outside of the range shown. Shown are data points for single force components (in the direction of displacement) on the Fe atom in the high-spin [Fe(NCH)$_6$]$^{2+}$ complex, both with (PAW augmented) and without (unagumented) the second term of Eq. \ref{['eq:tildeX']}.
• Figure 2: Defect levels (in eV) for substitutional carbon defects in a $6 \times 6$ hBN supercell. Horizontal lines indicate the positions of spin-up and spin-down defect levels computed using plane-wave DFT in Quantum ESPRESSO (green), traditional non-spin-depenent NGWFs (blue), and spin-dependent NGWFs (red). The $\Delta$ values quantify deviations of ONETEP results from the plane-wave reference, and are quoted in meV.
• Figure 3: Structures of the optimized iron(III) clusters in the high-spin state, based on geometries optimized using the PBE0 functional.
• Figure 4: Atomic structure of bilayer CrI3 in the low-temperature (top row) and high-temperature (bottom row) phases. Panels (a) and (d) show top views of the bilayer supercells. Panels (b) and (e) present side views in the $XZ$ plane, while panels (c) and (f) show side views in the orthogonal $Y$Z plane.
• Figure 5: AFM–FM energy differences (in meV/Cr) for CrI$_3$ bilayers in both high-temperature (HT) and low-temperature (LT) stacking configurations. Results are shown for DFT+$U$ and DFT+$U$+$J$, comparing ONETEP with non-spin-dependent NGWFs (blue), ONETEP with spin-dependent NGWFs (red), Quantum Espresso with ortho-atomic projectors (black) and Quantum Espresso with atomic projectors (green).