  We consider electron systems in quantum dots and imaging of the confined
charge density by the Coulomb blockade microscopy (CBM) with the scanning probe
technique. We apply an exact diagonalization method to study the reaction of
the electron system to the potential induced by the model potential of the
probe and calculate the energy maps as functions of the position of the probe.
The charge densities derived from the energy maps are confronted to the exact
charge densities. We focus on the transition of the electron system to the
fractional quantum Hall conditions in external magnetic field. For magnetic
fields corresponding to the integer fillings of the lowest Landau level the
electron system exhibits a liquid-like reaction to the potential of the probe
and the confined charge density can be quite accurately mapped by the CBM. For
fractional fillings of the lowest Landau level the single-electron charge
density islands nucleate in presence of the tip. In circular quantum dots the
single-electron islands evade imaging by CBM. We demonstrate that mapping the
molecular charge densities is possible for confinement potentials of symmetry
that is lower and consistent with the geometry of the lowest-energy charge
distribution of the single-electron islands.
