  Starting from the multiorbital Hubbard model for the t2g-bands of RTiO3 (R=
Y, Gd, Sm, and La), where all parameters have been derived from the
first-principles calculations, we construct an effective superexchange (SE)
spin model, by treating transfer integrals as a perturbation. We consider four
approximations for the SE interactions: (i) the canonical crystal-field (CF)
theory, where the form of the the occupied t2g-orbitals is dictated by the CF
splitting, and three extensions, namely (ii) the relativistic one, where
occupied orbitals are confined within the lowest Kramers doublet obtained from
the diagonalization of the crystal field and relativistic spin-orbit (SO)
interactions; (iii) the finite-temperature extension, which consider the effect
of thermal orbital fluctuations near the CF configuration; (iv) the
many-electron extension, which is based on the diagonalization of the full
Hamiltonian constructed in the basis of two-electron states separately for each
bond of the system. The main results are summarized as follows. (i) Thermal
fluctuations of the orbital degrees of freedom can substantially reduce the
value of the magnetic transition temperature. (ii) The anisotropic and
antisymmetric Dzyaloshinsky-Moriya interactions are rigorously derived and
their implications to the magnetic properties are discussed. (iii) The CF
theory, although applicable for YTiO3 and high-temperature structures of GdTiO3
and SmTiO3, breaks down in LaTiO3. Instead, the combination of the
many-electron effects and SO interaction can be responsible for the AFM
character of interatomic correlations in LaTiO3. (iv) The SE interactions in
YTiO3 strongly depend on the details of the crystal structure. Distortions in
the low-temperature structure tend to weaken the ferromagnetic interactions.
