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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

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These are quite a few questions and it is difficult to grab the essence of what you actually want to know, but maybe I can clarify a few aspects that help to understand these things.

  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

  2. The oscillatory behavior of the all-electron wave function stems from the singularity of the real potential at the atomic nucleus. This leads to a high kinetic energy contribution and thus to such an oscillatory behavior. To suppress this oscillatory behavior one has to construct a potential that has no singularity and is also not near this. The absolute values of the potential should not become large. One condition on the pseudopotential is that it agrees with the all-electron potential outside the critical radius and due to its construction principle it is also continuous in value and slope. This implies a large negative slope near the critical radius, leading to large negative values for small radii. As indicated before this is not wanted because it leads to the oscillatory behavior of the wave function. One thus obtains a bump of the potential to keep its absolute values small everywhere in space. The bump in the pseudopotential definitely has an effect on the wavefunction: It keeps it from oscillating.

  3. In general I think one should see the pseudopotential approximation in the opposite direction. One starts with a pseudo-wavefunction for some prototype system and defines this wave function such that it fulfills certain conditions. It should have no nodes, it should nicely converge with respect to a plane-wave expansion, it should coincide with the all-electron wave function outside the critical radius, and its energy eigenvalue should agree with that of the all-electron wave function. There may also be a normalization condition. With this starting point one then constructs a pseudopotential that leads to such a pseudo-wavefunction. So, in the end it is not the wavefunction that one obtains from the pseodopotential, but the pseudopotential is a consequence of what properties you define for your pseudo-wavefunction. The hope then is that this construction also transfers to a good approximation of the physics in chemical environments deviating from your prototype system. For good pseudopotential constructions this often is the case.

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