  The magnetic field dependence of the electronic properties of \textit{real}
single vertical quantum dots in circular and rectangular mesas is investigated
within a full three-dimensional multiscale self-consistent approach without any
{\it \'a priori} assumptions about the shape and strength of the confinement
potential. The calculated zero field electron addition energies are in good
agreement with available experimental data for both mesa geometries. Charging
diagrams in a magnetic field for number of electrons up to five are also
computed. Consistent with the experimental data, we found that the charging
curves for the rectangular mesa dot in a magnetic field are flatter and exhibit
less features than for a circular mesa dot. Evolution of the singlet-triplet
energy separation in the two electron system for both dot geometries in
magnetic field was also investigated. In the limit of large field, beyond the
singlet-triplet transition, the singlet-triplet energy difference continues to
become more negative in a circular mesa dot without any saturation within the
range of considered magnetic fields whilst it is predicted to asymptotically
approach zero for the rectangular mesa dot. This different behavior is
attributed to the symmetry "breaking" that occurs in the singlet wave-functions
in the rectangular mesa dot but not in the circular one.
