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Norm-Conserving Pseudopotentials

Authors: Dou Du, Taylor James Baird and Giovanni Pizzi

Original notebook: pseudopotential.ipynb

The pseudopotential method is a technique employed to simplify the description of a system of interacting electrons and nuclei. It is used to construct an effective potential that includes both the effects of a nucleus and of its core electrons, allowing one to consider explicitly only the valence electrons. This notebook illustrates a method of constructing norm-conserving pseudopotentials and to display them interactively, together with the resulting pseudowavefunctions.

Goals

• Understand why pseudopotentials are needed.
• Learn how to construct pseudopotentials using Kerker’s method.
• Examine the results for various values of the principal quantum number n and of the angular quantum number l.
• Examine the effect of changing the cutoff radius.

Background theory

More on the background theory.

Tasks and exercises

1. Investigate the role of the cutoff radius by varying the $R_c$ slider.

   Solution

   Move the slider for $R_c$ and press the button “Compute pseudopotential” to obtain the results. Check if there are values for which no solutions can be found. Inspect how different the pseudopotential is from the Coulomb potential.

2. Investigate how the pseudopotential changes for different values of the quantum numbers

   Solution

   Try to construct the pseudopotential for various values of n and l. Check what happens when constructing a pseudopotential for a nodeless wavefunction (e.g. $n=1$ and $l=0$, or $n=2$ and $l=1$).

3. Why do we need pseudopotentials?

   Solution

   Wavefunction oscillates rapidly in the core region. In a plane-wave approach, this would require a huge basis set (i.e., a huge number of plane waves) to be described accurately. What is most relevant, however, is that while the largest part of the contribution to the total energy of the system comes from the core electrons, these electrons are essentially frozen and do not participate in chemistry and the creation of bonds; the electronic structure is instead determined by the valence electrons. Avoiding to treat explicitly core electrons avoids that small relative errors on the core electrons completely spoil the calculation of the energy of the full system and, in particular, of the (small, but crucial) energy differences between different atomic configurations or crystalline phases.

4. What is the meaning of the norm-conservation condition?

   Solution

   The condition ensures that the total charge inside the cutoff radius $R_c$ is correct. However, there are more profound consequences that are implied by this condition: it turns out that imposing norm conservation implies also that the first energy derivative of the logarithmic derivatives of the all-electron wavefunction and the pseduowafevuntion agree at $R_c$. This is a very important condition for the transferability of the pseudopotential. It can be shown that this means that the true atom with its electrons and the pseudopotential generate the same phase shift when a plane wave is scattered into a spherical wave. A detailed discussion can be found in Section 11.4 of the book “Electronic Structure: Basic Theory and Practical Methods” by Richard M. Martin, Cambridge University Press (2004).

5. Are the pseudopotentials constructed with this method local?

   Solution

   As discussed earlier, we obtain a different pseudopotential for different values of the quantum numbers n and l. Therefore, the pseudopotential is not local (its action is not not just the product of the same a single function $V^{PS}(r)$ times the wavefunction).

Interactive visualization

Legend

(How to use the interactive visualization)

Controls

Here, we consider the wavefunctions from the solution of the Schrödinger equation of one hydrogen atom. One can choose the state with quantum numbers n and l via the sliders. The position of the cutoff distance $R_c$ can be tuned with the slider. Click the “Compute pseudopotentials” button to calculate the pseudowavefunction and pseudopotential.

Interactive figure

The top left panel shows the calculated radial pseudowavefunction times $r$: $r R^{PS}_{nl}(r)$ (blue line) and the radial wavefunction times r: $rR_{nl}(r)$ (red line). The top right subplot shows their respective square moduli. To fulfill the norm-conserving condition, the green and yellow areas illustrated in this subplot should be equal. The square of the area should be 1 when $r$ goes to $+\infty$. The bottom left panel shows plots of the logarithmic derivatives of the all-electron (in red) and pseudowavefunctions (in blue) respectively. The bottom right panel shows the calculated pseudopotential (in blue).