The ground-state magnetic properties of clusters with N = 3 atoms are studied by sohing lhe single band Hubbard Hamiltonian in the unrestricted Hartree-Fock approximation. Three-dimensional arrangements oi spins are considcrcd in the solution. Results for the total energy and magnetic arder, as a function of the Coulomb interaction strength Ujt and num- ber of eleetrons v, are prcsented. For half-band filling an antiferromagnetic nOll-collinear spin arrangement is the most stable for the trianglc structurc. For v = 2 ami Uji ~ 16.2, v = 4 and U/t ~ 2.3 the linear chain is the most stable stcucture. In particular, it is found that the inclusion oi non-collinear spins improves in the Hartrce-Fock ground-state cncrgy as compared with collinear spins. Our rcsults are compared and discusscd with exact calculations. RESU1.IEN. Las propiedades magneticas del estado base de cumulos con N = 3 atomos son estu- diadas resolviendo el hamiltoniano de Hubbard para una banda en la aproximacion no restringida de Hartree-Fock. Se consideran soluciones magneticas con arreglos de espines no colineales. Se presentan los resultados de la energia total y el orden magnetico en funcion de la intensidad de la interaccion de Coulomb y el numero de electrones. Para la banda semillena la solucion mas estable para la estructura triangular, corresponde a un arreglo antiferromagnetico no colineal de espines. Para v = 2 Y U/t 2: lG.2, v = 4 Y U/t ~ 2.3 la cadena lineal es la estructura mas estable. En particular 1 se encuentra que considerar espines no colineales mejora la energia del estado base de Hartree-Fock comparada con espines colineales. Se discuten y comparan nuestros resultados con soluciones exactas.The ground-state magnetic properties of clusters with N = 3 atoms are studied by sohing lhe single band Hubbard Hamiltonian in the unrestricted Hartree-Fock approximation. Three-dimensional arrangements oi spins are considcrcd in the solution. Results for the total energy and magnetic arder, as a function of the Coulomb interaction strength Ujt and num- ber of eleetrons v, are prcsented. For half-band filling an antiferromagnetic nOll-collinear spin arrangement is the most stable for the trianglc structurc. For v = 2 ami Uji ~ 16.2, v = 4 and U/t ~ 2.3 the linear chain is the most stable stcucture. In particular, it is found that the inclusion oi non-collinear spins improves in the Hartrce-Fock ground-state cncrgy as compared with collinear spins. Our rcsults are compared and discusscd with exact calculations. RESU1.IEN. Las propiedades magneticas del estado base de cumulos con N = 3 atomos son estu- diadas resolviendo el hamiltoniano de Hubbard para una banda en la aproximacion no restringida de Hartree-Fock. Se consideran soluciones magneticas con arreglos de espines no colineales. Se presentan los resultados de la energia total y el orden magnetico en funcion de la intensidad de la interaccion de Coulomb y el numero de electrones. Para la banda semillena la solucion mas estable para la estructura triangular, corresponde a un arreglo antiferromagnetico no colineal de espines. Para v = 2 Y U/t 2: lG.2, v = 4 Y U/t ~ 2.3 la cadena lineal es la estructura mas estable. En particular 1 se encuentra que considerar espines no colineales mejora la energia del estado base de Hartree-Fock comparada con espines colineales. Se discuten y comparan nuestros resultados con soluciones exactas.

Jan 1, 19970 citations

Magnetic properties of the three-band Hubbard model

Thomas Maier, M. Zölfl, Thomas Pruschke +1 more

The European Physical Journal BFeb 1, 199916 citations

Electron correlations, magnetism, and structure of small clusters

G. M. Pastor, R. Hirsch, Bernhard Mühlschlegel

Many-body properties of N-atom clusters having N\ensuremath{\le}8 are calculated exactly in the framework of the Hubbard model for all possible nonequivalent cluster structures. The most stable structure and the corresponding total spin S are obtained rigorously as a function of the Coulomb-interaction strength U/t and the number of electrons. Results for the stability of cluster ferromagnetism against electronic excitations and structural changes are obtained for the first time. The resulting interplay between electron correlations, magnetism, and cluster structure is determined.Many-body properties of N-atom clusters having N\ensuremath{\le}8 are calculated exactly in the framework of the Hubbard model for all possible nonequivalent cluster structures. The most stable structure and the corresponding total spin S are obtained rigorously as a function of the Coulomb-interaction strength U/t and the number of electrons. Results for the stability of cluster ferromagnetism against electronic excitations and structural changes are obtained for the first time. The resulting interplay between electron correlations, magnetism, and cluster structure is determined.

Physical Review LettersJun 13, 199472 citations

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

Konstantin Gaul

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.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.

physics.chem-phCross-listedarXiv:2605.31203Jun 1, 2026

Rigorous extension of semilocal collinear functionals to noncollinear DFT using

SU(2)

rotations

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.