  We perform a numerical simulation of mapping of charge confined in quantum
dots by the scanning probe technique. We solve the few-electron Schr\"odinger
equation with the exact diagonalization approach and evaluate the energy maps
in function of the probe position. Next, from the energy maps we try to
reproduce the charge density distribution using an integral equation given by
the perturbation theory. The reproduced density maps are confronted with the
original ones. The present study covers two-dimensional quantum dots of various
geometries and profiles with the one-dimensional (1D) quantum dot as a limit
case. We concentrate on large quantum dots for which strong electron-electron
correlations appear. For circular dots the correlations lead to formation of
Wigner molecules that in the presence of the tip appear in the laboratory
frame. The unperturbed rotationally-symmetric charge density is surprisingly
well reproduced by the mapping. We find in general that the size of the
confined droplet as well as the spatial extent of the charge density maxima is
underestimated for repulsive tip potential and overestimated for the attractive
tip. In lower-symmetry quantum dots the Wigner molecules with single-electron
islands nucleate for some electron numbers even in the absence of the tip.
These charge densities are well resolved by the mapping. The single-electron
islands appear in the laboratory frame provided that classical point charge
density distribution is unique, in the 1D limit of confinement in particular.
We demonstrate that for electron systems which possess a few equivalent
classical configurations the repulsive probe switches between the
configurations. In consequence the charge density evades mapping by the
repulsive probe.
