I am new to solid state calculation. When I was reading about the pseudopotential documents, I always ran into a diagram that shows how pseudopotential will help to smooth the all-electron wavefunction. Figure from Wikipedia
In the diagram, it makes sense that the real potential will goes to negative infinity because the electron will get closer to the core. Then the wavefunctions have lots of nodes.
However, I am confused that how to explain the shape of the pseudopotential (red solid line). For example, why there is a bump for the V-pseudo inside of the cutoff radius? How to use the V-pseudo shape to explain the shape for the pseudo-wavefunction above? Why the V-pseudo starts at a negative value but the pseudo-wavefunction starts from 0? Why the bump in the V-pseudo does not have any effect on the pseudo-wave function?
I really appreciate any comments on this question.