Theoretical calculations of effective exchange integrals by spin projected and unprojected broken-symmetry methods. I. Cluster models of $K2NiF4$-type solids

Previously, various symmetry-adapted (SA) and broken-symmetry (BS) computations have been performed for strongly correlated transition metal species so as to examine magnetic properties in simple cluster models such as binuclear transition metal oxides. Though SA computations such as the complete active space configuration interaction and CASSCF are desirable for estimating physical constants, these computations are heavy for larger cluster models with strongly correlated electrons. $K2NiF4$ is known to be the two-dimensional perovskite-type antiferromagnet and to have the electronic configuration similar to that of $La2CuO4.$ Here, we have examined the utility and applicability of the BS spin-polarized hybrid-density functional theory (HUDFT) for cluster models of $K2NiF4.$ As the result, HUDFT calculation such as UB2LYP has provided the reasonable effective exchange integral $(Jab)$ followed by our approximately spin projected scheme, in comparison to the experimental one. It was also found that the square planar tetranuclear model has provided the most reasonable $Jab$ value by HUDFT. In addition, external effects such as putting point charges around cluster models and changing distances between nickel and fluorine have been also examined. The natural orbital analysis by HUDFT has been carried out to obtain natural orbitals and their occupation numbers. Charge density, spin density, and chemical indices expressed by the occupation numbers have been also obtained to elucidate the nature of the chemical bonds in the $K2NiF4$-type solids.