We present a detailed investigation of the magnetic properties of the spin-$frac{3}{2}$ system Li$_2$Co(WO$_4$)$_2$ by means of magnetic susceptibility and specific heat. Our experimental results show that in Li$_2$Co(WO$_4$)$_2$ a short-range antife
rromagnetic (AFM) correlations appear near $chi$$_{max}$ $sim$ 11 K and two successive long-range AFM phase transitions are observed at T$_{N1}$$sim$ 9 K and T$_{N2}$$sim$ 7 K. The frustration factor, $mid$$Theta$$mid$/T$_{N1}$$sim$3, indicates that the system is moderately frustrated, which is identifiable by the broken triangular symmetry within both $ab$- and $bc$-planes for the triclinic crystal structure. The magnetic isotherm at temperatures below T$_{N2}$ shows a field-induced spin-flop transition, and a complete H-T phase diagram for the two-step AFM system is mapped. $Ab$~$initio$ band structure calculations suggest that the strongest exchange coupling does not correspond to the shortest Co-Co distance along the $a$-axis, but rather along the diagonal direction through a Co-O-W-O-Co super-superexchange path within the $bc$-plane
MCrS2 compounds (M=Li, Na, K, Cu, Ag, and Au) with triangular Cr layers show large variety of magnetic ground states ranging from 120-degree antiferromagnetic order of Cr spins in LiCrS2 to double stripes in AgCrS2, helimagnetic order in NaCrS2, and,
finally, ferromagnetic Cr layers in KCrS2. On the base of ab-initio band structure calculations and an analysis of various contributions to exchange interactions between Cr spins we explain this tendency as originating from a competition between antiferromagnetic direct nearest-neighbor d-d exchange and ferromagnetic superexchange via S p states which leads to the change of the sign of the nearest neighbor interaction depending on the radius of a M ion. It is shown that other important interactions are the third-neighbor interaction in a layer and interlayer exchange. We suggest that strong magneto-elastic coupling is most probably responsible for multiferroic properties of at least one material of this family, namely, AgCrS2.
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.
