A new method of circumscribing and evaluating the effective electronic states of ``atoms-in-compounds'' (AiC)  and properties of molecules and solids described by the operators heavily concentrated in atomic cores or most sensitive to variation of electronic densities in the atomic cores is discussed. Among these properties (AiC properties) are hyperfine structure, time reversal (T) and space parity (P) nonconservation effects , chemical shifts of X-ray emission  and Mössbauer lines, etc. An advantage of the approach is that a good quantitative agreement of predicted and experimental data can be attained and it is correct from the quantum mechanical point of view. The common feature of AiC properties (leading contributions from tails of valence orbitals in atomic cores) can be well-exploited and new concepts such as density matrices reduced on the radial quantum numbers for evaluating AiC properties, effective AiC configuration and partial-wave charges can be introduced. Such reduction utilizes the property of proportionality of valence and low-lying virtual spinors (W-spinors) within an atomic core region with radius Rc. Another problem that can be considered in some sense as ``adjoined’’ one to calculation of the AiC properties is economical evaluation of chemical, spectroscopic etc. properties, which are determined by the wavefunctions in the valence regions localized between atoms. The most efficient (economical) way to study such properties is to apply the pseudopotential (PP) theory  to exclude the atomic core electrons from explicit consideration. Such approach is most applicable to heavy-atom systems, when relativistic effects should be taken into account along with the electron correlation ones. It is shown that the radially-local (semi-local) relativistic PP theory can also be constructed on the basis of the proportionality property discussed above when satisfying the ``hardness’’ condition for the PP components in the core region. However, for the most accurate versions of the radially-local PPs they should satisfy the requirement of their ``physicity’’ in the valence region, out of Rc (or absence of unphysical interaction in the valence region that is closely related to the shape-consistent / norm-conserving PP formulations).
This work is supported by the grant of Russian Science Foundation #14-31-00022.