What Are the Physical Contents of Hubbard and Heisenberg Hamiltonian Interactions Extracted from Broken Symmetry DFT Calculations in Magnetic Compounds?

Analytical expressions of the interactions present in the Heisenberg-Dirac van Vleck and Hubbard Hamiltonians have been derived as functions of both the energy of several broken symmetry DFT solutions and their expectation value of the S spin operator. Then, following a strategy of decomposition of the magnetic exchange coupling into its main contributions (direct exchange, kinetic exchange, and spin polarization) and using a recently proposed method of spin decontamination, values of these interactions have been extracted. As already observed, they weakly depend on the correlation functional but strongly depend on the exchange one. In order to distinguish between the effect of the delocalization of the magnetic orbitals and that of the amount of Hartree-Fock exchange (HFX) when hybrid exchange-correlation functionals are used, we have disentangled these two contributions by either freezing the magnetic orbitals and varying the amount of HFX or varying the magnetic orbitals while keeping the same amount of HFX. As expected, increasing the amount of HFX induces a slight relocalization of the magnetic orbitals on the magnetic center which results in a weak increase of the repulsion energy U parameter and a weak decrease of both the direct exchange K and hopping |t| parameters. Conversely, the amount of HFX has a huge effect on all the parameters, even when some of the parameters should be exchange-independent, like U. Indeed, it is analytically demonstrated that the physical content of the U parameter extracted from several broken-symmetry solutions depends on the amount of HFX and that this pathological behavior has the same origin as the self-interaction error. This result is interesting not only to theoretical chemists working in the field of magnetic systems but also to DFT methodologists interested in using this theory for studying either excited states or strongly correlated systems. Finally, the performance of the range-separated ωB97XD functional for both ferromagnetic and antiferromagnetic transition-metal compounds and organic systems must be noted.