Shell charging and spin-filling sequences in realistic vertical quantum dots

For nominally high-symmetry cylindrical-shaped vertical quantum dots, we show that Hund's first rule is not violated if maxima appear in the 0-T addition energy spectra at positions other than when the shell is half filled. For the third shell, a minimum at $N=7$ electrons followed by a maximum at $N=8$ or 9 electrons is the only clear signature of sequential spin alignment up to half shell occupation. We demonstrate that the filling of this shell and the consequent spectral shape for realistic dots are quite rich and varied.