When considering the orbital model of the atom it seems like the shape of each orbital corresponds the shape that contains a volume such that there is a 90% chance of an electron being there. I also assume that in the majority of the literature, the equation that defines the shape of the different orbitals corresponds to the solution of some kind of 3-dimensional version of the Schrödinger Equation the configuration of which corresponds to only one electron. Are we then Supposed to add up all the orbitals? Don't all those orbitals with different shapes intersect? (if they do, i would doubt its a good solution) Would the orbitals of atoms with multiple electrons get distorted instead of cutting trough each other?
Taking all of this into consideration, is there any way to represent the three dimensional probability density of the electrons in a carbon atom for example?