  Generic low-energy spectral features related to $d$-$d$ excitations in nearly
cubic vanadium perovskites are predicted for the expected $L$-edge resonant
inelastic X-ray scattering (RIXS) measurements. Model Hamiltonian describing
local electronic properties, including crystal-field effects, of vanadium 3d
orbitals in the basic $t_{2g}^2$ configuration is formulated with the help of
complementary {\it ab initio} quantum-chemical cluster calculations. In the
presence of 2p-core hole, the local Hamiltonian includes strong 2p-3d electron
interactions. As a prerequisite for evaluating RIXS transition amplitudes
beyond the fast collision approximation, a symmetry-group approach is applied
to generate a basis set of many-electron wavefunctions of the intermediate
core-hole states accessible in RIXS processes. Although a comprehensive
description of the core-hole multiplets still remains a formidable task and
requires using specially designed numerical codes, for particular resonant
states the analysis is simplified and the calculation of RIXS amplitude can be
carried out analytically to the end.

Strongly Correlated Electrons