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arXiv:2605.31203

Rigorous extension of semilocal collinear functionals to noncollinear DFT using $SU(2)$ rotations

Konstantin Gaul

physics.chem-phcond-mat.mtrl-sciphysics.comp-phquant-phSubmitted May 29, 2026• Announced Jun 1, 2026• Revised Jun 14, 2026 (v2)
v1 · May 29, 2026v2 · Jun 14, 2026
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Abstract

In the presence of spin-orbit coupling and in geometrically frustrated materials, a noncollinear treatment the magnetization density is essential. However, in density functional theory most exchange--correlation functional approximations were originally developed for locally collinear magnetization. Many practical approaches to noncollinear DFT have emerged over the past decade. However, a first-principles connection between widely used semilocal collinear functionals and their noncollinear generalizations remains lacking. In this work, a locally exact relation between collinear and noncollinear exchange--correlation functionals is derived at the level of gradient expansions within a $u(2)$ matrix representation of the energy functional. Within this framework, collinear semilocal variables naturally acquire distinct dependencies on transverse and longitudinal magnetization gradient components. The widely used Scalmani--Frisch scheme emerges as a first-order approximation. The transformation of collinear functional derivatives to noncollinear space is implemented through numerically robust $SU(2)$ rotations. A consistent description of local magnetic torques is demonstrated for the prototypical spin-frustrated Cr$_3$ cluster. The approach further extends to fully nonlocal functionals and provides a direct route towards numerically stable relativistic response calculations. The influence on magnetic properties in presence of spin-orbit coupling is illustrated through calculations of hyperfine couplings in the high-spin ground states of uranium and the uranium ion.

Comments
14 pages, 3 figures, 2 tables; minor revisions to the text

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Density-Functional Theory (DFT), although widely used and very successful in the calculation of several observables, fails to correctly describe strongly correlated materials. In the first part of this work we, therefore, introduce reduced-density-matrix-functional theory (RDMFT) which is one possible way to treat electron correlation beyond DFT. Within this theory the one-body reduced density matrix (1-RDM) is used as the basic variable. Our main interest is the calculation of the fundamental gap which proves very problematic within DFT. In order to calculate the fundamental gap we generalize RDMFT to fractional particle numbers M by describing the system as an ensemble of an N and an N+1 particle system (with N≤ M≤N+1). For each fixed particle number, M, the total energy is minimized with respect to the natural orbitals and their occupation numbers. This leads to the total energy as a function of M. The derivative of this function with respect to the particle number has a discontinuity at integer particle number which is identical to the gap. In addition, we investigate the necessary and sufficient conditions for the 1-RDM of a system with fractional particle number to be N-representable. Numerical results are presented for alkali atoms, small molecules, and periodic systems. Another problem within DFT is the description of non-relativistic many- electron systems in the presence of magnetic fields. It requires the paramagnetic current density and the spin magnetization to be used as basic variables besides the electron density. However, electron-gas-based functionals of current-spin-density-functional Theory (CSDFT) exhibit derivative discontinuities as a function of the magnetic field whenever a new Landau level is occupied, which makes them difficult to use in practice. Since the appearance of Landau levels is, intrinsically, an orbital effect it is appealing to use orbital-dependent functionals. We have developed a CSDFT version of the optimized effective potential (OEP) method which allows for the use of explicitly orbital-dependent functionals in the context of CSDFT. We present the derivation of corresponding equations and show results for a quantum dot in external magnetic fields in the second part of this thesis.
We establish theoretical foundations for embedding a correlated wavefunction in an environment formed by Kohn-Sham orbitals. We show that introducing an approximation that equates two, in principle, distinct kinetic-energy functionals yields an embedding functional identical to the projection-based wavefunction-in-density functional theory (DFT) formulation of Miller and co-workers. We demonstrate that this functional is inherently nonvariational: its minimum is not guaranteed to coincide with the exact ground-state energy and remains bounded from above by it. Building on this formal framework, we analyze the dominant sources of error in projection-based density matrix renormalization group-in-DFT embedding with approximate exchange-correlation (xc) functionals. Using molecules with dissociating covalent bonds as a diagnostic example, we demonstrate that the primary source of error is the nonadditive exchange-correlation energy describing the nonclassical coupling between the active subsystem and its environment. Eliminating the fractional-spin error by employing a pair-density xc functional (pair-density functional theory) instead of a semilocal generalized gradient approximation (GGA) does not remedy this deficiency, because the inaccuracy stems from self-interaction effects at the subsystem-environment interface.
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We establish the theoretical foundations for embedding a correlated wave function in an environment formed by Kohn-Sham orbitals. We show that introducing an approximation which equates two, in principle distinct, kinetic-energy functionals yields an embedding functional identical to the projection-based wavefunction-in-DFT formulation of Miller and co-workers. We demonstrate that this functional is inherently nonvariational: its minimum is not guaranteed to coincide with the exact ground-state energy and remains bounded from above by it. Building on this formal framework, we analyze the dominant sources of error in projection-based DMRG-in-DFT embedding with approximate exchange-correlation (xc) functionals. Using molecules with dissociating covalent bonds as a diagnostic example, we demonstrate that the primary source of error is the nonadditive exchange-correlation energy describing the nonclassical coupling between the active subsystem and its environment. Eliminating the fractional-spin error by employing a pair-density xc functional (PDFT) instead of a semilocal GGA does not remedy this deficiency, because the inaccuracy stems from self-interaction effects at the subsystem-environment interface.
In this article we study the ground-state properties of two square-lattice Heisenberg quantum spin models with competing bonds using a high-order coupled cluster treatment. The first model is a spin-half model with competing nearest-neighbour bonds with and without frustration. We discuss the influence of quantum fluctuations on the ground-state phase diagram and in particular on the nature of the zero-temperature phase transitions from phases with collinear magnetic order at small frustration to phases with noncollinear spiral order at large frustration. The second model is a highly frustrated ferrimagnet, which contains one sublattice (A) entirely populated with spin-one spins and an other sublattice (B) entirely populated with spin-half spins. Sublattice A sites are nearest-neighbours to sublattice B sites and vice versa and frustration is introduced by next-nearest-neighbour bonds. The model shows two collinear ordered phases and a noncollinear phase in which (classically) the spin-one spins are allowed to cant at an angle. Both examples show that the coupled-cluster method is able to describe the zero-temperature transitions well and provides a consistent description of collinear, noncollinear, and disordered phases, for cases in which other standard techniques (e.g. the quantum Monte Carlo technique for spin systems which are frustrated) are not applicable.