An approach to compute exchange parameters of the Heisenberg model in plane-wave-based methods is presented. This calculation scheme is based on the Greens function method and Wannier function projection technique. It was implemented in the framework
of the pseudopotential method and tested on such materials as NiO, FeO, Li2MnO3, and KCuF3. The obtained exchange constants are in a good agreement with both the total energy calculations and experimental estimations for NiO and KCuF3. In the case of FeO our calculations explain the pressure dependence of the Neel temperature. Li2MnO3 turns out to be a Slater insulator with antiferromagnetic nearest-neighbor exchange defined by the spin splitting. The proposed approach provides a unique way to analyze magnetic interactions, since it allows one to calculate orbital contributions to the total exchange coupling and study the mechanism of the exchange coupling.
The magnetic and electronic properties of Eu2Ru2O7 are discussed in terms of the local ruthenium and europium coordinations, electronic band structure calculations and molecular orbital energy levels. A preliminary electronic structure was calculated
within the LDA and LSDA+U approximations. The molecular orbital energy level diagrams have been used to interpret the Eu-Ru ligand spectrum and the ensuing magnetic properties. The orbital hybridizations and bonds are discussed.
Using ab initio band structure and model calculations we studied magnetic properties of one of the Mn$_4$ molecular magnets (Mn4(hmp)6), where two types of the Mn ions exist: Mn3+ and Mn2+. The direct calculation of the exchange constants in the GGA+
U approximation shows that in contrast to a common belief the strongest exchange coupling is not between two Mn3+ ions (J_{bb}), but along two out of four exchange paths connecting Mn3+ and Mn2+ ions (J_{wb}). The microscopic analysis performed within the perturbation theory allowed to establish the mechanism for this largest ferromagnetic exchange constant. The charge ordering of the Mn ions results in the situation when the energy of the excited state in the exchange process is defined not by the large on-site Coulomb repulsion U, but by much smaller energy V, which stabilizes the charge ordered state. Together with strong Hunds rule coupling and specific orbital order this leads to a large ferromagnetic exchange interaction for two out of four Mn2+ --Mn3+ pairs.
Special features of the crystal field splitting of $d-$levels in the transition metal compounds with the small or negative charge-transfer gap $Delta_{CT}$ are considered. We show that in this case the Coulomb term and the covalent contribution to th
e $t_{2g} - e_g$ splitting have different signs. In order to check the theoretical predictions we carried out the ab-initio band structure calculations for Cs$_2$Au$_2$Cl$_6$, in which the charge-transfer gap is negative, so that the $d-$electrons predominantly occupy low-lying bonding states. For these states the $e_g$-levels lie below $t_{2g}$ ones, which demonstrates that at least in this case the influence of the $p-d$ covalency on the total value of the crystal field splitting is stronger than the Coulomb interaction (which would lead to the opposite level order). We also show that the states in conduction band are made predominantly of $p-$states of ligands (Cl), with small admixture of $d-$states of Au.
The results of the LSDA+U calculations for pyroxenes with diverse magnetic properties (Li,Na)TM(Si,Ge)$_2$O$_6$, where TM is the transition metal ion (Ti,V,Cr,Mn,Fe), are presented. We show that the anisotropic orbital ordering results in the spin-ga
p formation in NaTiSi$_2$O$_6$. The detailed analysis of different contributions to the intrachain exchange interactions for pyroxenes is performed both analytically using perturbation theory and basing on the results of the band structure calculations. The antiferromagnetic $t_{2g}-t_{2g}$ exchange is found to decrease gradually in going from Ti to Fe. It turns out to be nearly compensated by ferromagnetic interaction between half-filled $t_{2g}$ and empty $e_g$ orbitals in Cr-based pyroxenes. The fine-tuning of the interaction parameters by the crystal structure results in the ferromagnetism for NaCrGe$_2$O$_6$. Further increase of the total number of electrons and occupation of $e_g$ sub-shell makes the $t_{2g}-e_g$ contribution and total exchange interaction antiferromagnetic for Mn- and Fe-based pyroxenes. Strong oxygen polarization was found in Fe-based pyroxenes. It is shown that this effect leads to a considerable reduction of antiferromagnetic intrachain exchange. The obtained results may serve as a basis for the analysis of diverse magnetic properties of pyroxenes, including those with recently discovered multiferroic behavior.
