  We discuss the quantum dot-ring nanostructure (DRN) as canonical example of a
nanosystem, for which the~interelectronic interactions can be evaluated
exactly. The system has been selected due to its tunability, i.e., its electron
wave functions can be modified much easier than in, e.g., quantum dots. We
determine many-particle states for $N_e=2$ and $3$ electrons and calculate the
3- and 4-state interaction parameters, and discuss their importance. For that
purpose, we combine the first- and second-quantization schemes and hence are
able to single out the component single-particle contributions to the resultant
many-particle state. The method provides both the ground- and the
first-excited-state energies, as the exact diagonalization of the many-particle
Hamiltonian is carried out. DRN provides one of the few examples for which one
can determine theoretically all interaction microscopic parameters to a high
accuracy. Thus the evolution of the single-particle vs. many-particle
contributions to each state and its energy can be determined and tested with
the increasing system size. In this manner, we contribute to the wave-function
engineering with the~interactions included for those few-electron systems.
