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There is this paper: Norm-Conserving Pseudopotentials mentioning a procedure to make a pseudopotential. There are many points that I did not get. Let's suppose we already have the results from the all-electron simulation: the potential $V$, $\phi_l$ the valence wavefunction and $u_l(r) = r\phi_l(r)$. The steps that I do not understand are:

  • The form of the potential $V_{1l}^{PS}(r) = (1-f(r/r_{cl}))V(r) + c_lf(r/r_{cl})$, where $r_{cl}$ is some cutoff radius and $f$ the cutoff function
  • The value of the constant $\gamma_l$, involved in the form wavefunction $\omega_{2l}$ and defined as $\gamma_l\omega_{1l} \xrightarrow[r>r_{cl}]{} u_l(r)$

In the first point, $c_l$ is defined as a constant that is "adjustable" so that the nodeless wavefunction $w_{1l}$ has the same energy as the original wavefunction. To me, it feels like trial and error. Am I missing something?

For the second point, how can we get the constant $\gamma_l$?

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We can start from the first term that you are providing, the purpose of this form is to smoothly transition the potential from the true all-electron potential $V(r)$ to a pseudopotential form $V_{1l}^{PS}(r)$ within a cutoff radius $r_{cl}$. Here, the purpose of the cutoff radius is to limit the range of the pseudopotential, we just want to capture the essential physics of the valence electrons and try to avoid unnecessary computational overhead coming from describing the core electrons.

The cutoff function $f(r/r_{cl})$ is usually a smooth function that goes from 1 to 0 as $r$ approaches $r_{cl}$. This transition ensures that the pseudopotential reproduces the correct behavior of the all-electron potential within a certain region.

Regarding the constant $cl$, it is adjustable as you have described and the way to determine it is iterative rather than trial and error cause you are not brute-forcing a solution using different possibilities, rather you are starting with an initial condition and move towards the solution by matching the energy of the nodeless pseudowavefunction $\omega_{1l}$ with the energy of the original all-electron wavefunction $\phi_{1l}$.

Now for the second point 'The constant $\gamma_{l}$' is determined based on the behavior of the all-electron wavefunction $\omega_{1l}$ outside the cutoff radius $r_{cl}$. It represents the ratio of the logarithmic derivatives of the all-electron wavefunction and the pseudowavefunction $\omega_{1l}$ at $r$ = $r_{cl}$. In order to get $\gamma_{l}$, you need to calculate the logarithmic derivative of the all-electron wavefunction $\omega_{1l}$ at $r$ = $r_{cl}$, which is given by: $$\tag{1}u_l(r) = r_{\omega_{1l}}(r),$$ and $\gamma_{l}$ can be computed as the ratio of the logarithmic derivatives of $\omega_{1l}$ and $\omega_{1l}$ at $r$ = $r_{cl}$.

You can have further reads from here which is based on the paper you have mentioned in your question.

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  • $\begingroup$ I wonder why subscripts aren't used, and the larger equations aren't written in equation blocks? $\endgroup$ Jan 16 at 2:59
  • $\begingroup$ @NikeDattani, it is fixed now $\endgroup$ Jan 16 at 5:56
  • $\begingroup$ I think there are some typos in the fourth paragraph: you need to calculate the logarithmic derivative of the all-electron wavefunction ω1l and gamma_l can be computed as the ratio of the logarithmic derivatives of ω1l and ω1l . Since $\omega_{1l}$ is the notation for the pseudo wavefunction $\endgroup$
    – mle
    Jan 25 at 8:39

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