Understanding Illustration for Pseudopotential

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.

https://en.wikipedia.org/wiki/Pseudopotential#/media/File:Sketch_Pseudopotentials.png

1. Besides the potential $$V(r) \approx -\frac{Z}{r}$$ there is also the kinetic energy operator in the Kohn-Sham equations. For such spherical problems this equation can be reformulated such that the radial and the angular part of the kinetic energy operator are separated and the angular part can analytically be integrated. As a result one obtains the angular-momentum barrier $$\propto \frac{l(l+1)}{2r^2}$$ as part of the radial Kohn-Sham equation. As the radius enters this term quadratically it is clear that for very small radii and finite angular momenta $$l$$ it is much larger than the potential and has the opposite sign. It implies that a large value of the wave function near the nucleus is also related to a very large energy. The smaller the radius the larger the needed energy. This leads to a vanishing of the wave function near the nucleus. Obviously this consideration is independent of whether one uses a pseudopotential or the real potential. But it depends on the angular momentum $$l$$. Actually the radial behavior of the wave function near the nucleus is $$\propto r^l$$. For $$l=0$$ one can thus have a finite value of the wave function near the nucleus. The illustration you show is a little bit inconsistent as it seems to sketch all-electron and pseudo wave functions for different $$l$$. In general illustrations are never perfect. :D