The orthoperovskites TbCoO$_3$ and DyCoO$_3$ with Co$^{3+}$ in a non-magnetic low-spin state have been investigated by neutron diffraction down to 0.25 K. Magnetic ordering is evidenced below $T_N=3.3$ K and 3.6 K, respectively, and the ordered arran
gements are of canted type, A$_x$G$_y$ for TbCoO$_3$ and G$_x$A$_y$ for DyCoO$_3$ in Bertauts notation. The experiments are confronted with the first-principle calculations of the crystal field and magnetism of Tb$^{3+}$ and Dy$^{3+}$ ions, located in the $Pbnm$ structure on sites of $C_s$ point symmetry. Both these ions exhibit an Ising behavior, which originates in the lowest energy levels, in particular in accidental doublet of non-Kramers Tb$^{3+}$ ($4f^8$ configuration) and in ground Kramers doublet of Dy$^{3+}$ ($4f^9$) and it is the actual reason for the non-collinear AFM structures. Very good agreement between the experiment and theory is found. For comparison, calculations of the crystal field and magnetism for other systems with Kramers ions, NdCoO$_3$ and SmCoO$_3$, are also included.
Fifteen parameters characterizing the crystal field of rare-earth ions in the RMO$_3$ perovskites (R = Pr, Nd, M = Ga, Co) are calculated by expanding the local Hamiltonian expressed in the basis of Wannier functions into a series of spherical tensor
operators. The method contains a single adjustable parameter that characterizes the hybridization of R($4f$) states with the states of oxygen ligands. Subsequently the energy levels and magnetic moments of trivalent R ion are determined by diagonalization of an effective Hamiltonian which, besides the crystal field, contains the $4f$ electron-electron repulsion, spin-orbit coupling and interaction with magnetic field. In the Ga compounds the energy levels of the ground multiplet agree within few meV with those determined experimentally by other authors. For all four compounds in question the temperature dependence of magnetic susceptibility is measured on polycrystalline samples and compared with the results of calculation. For NdGaO$_3$ theory is also compared with the magnetic measurements on a single crystal presented by Luis {it et al.} Phys. Rev. B {bf 58}, 798 (1998). A good agreement between the experiment and theory is found.
A method to calculate the crystal field parameters {it ab initio} is proposed and applied to trivalent rare earth impurities in yttrium aluminate and to Tb$^{3+}$ ion in TbAlO$_3$. To determine crystal field parameters local Hamiltonian expressed in
basis of Wannier functions is expanded in a series of spherical tensor operators. Wannier functions are obtained by transforming the Bloch functions calculated using the density functional theory based program. The results show that the crystal field is continuously decreasing as the number of $4f$ electrons increases and that the hybridization of $4f$ states with the states of oxygen ligands is important. Theory is confronted with experiment for Nd$^{3+}$ and Er$^{3+}$ ions in YAlO$_3$ and for Tb$^{3+}$ ion in TbAlO$_3$ and a fair agreement is found.
