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I would like to know what are the different types of pseudopotentials, the pro and cons, and what properties can/cannot be calculated with them?

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Summary of the "milestone" pseudopotential (PP) papers

Since it wasn't available anywhere & took me a few hours, it's an answer rather than question-edit:

Local pseudo-potentials:

  • 1935: Zusatzpotential / Hellmann (Generally credited as the first pseudopotential).
  • 1936: Fermi pseudopotential (for s-wave scattering of a free neutron by a nucleus).
  • 1958: Harrison (FPPM: First-Principle PP method, fitted to nearly free e- Fermi surface of $\ce{Al}$).
  • 1959: Phillips-Kleinman (core-val. orthogonalization terms replaced by "hard-core" potential).
  • 1968: Weeks-Rice (extended the single valence e- work of Phillips to many valence e-).
  • 1973: Appelbaum-Hamann potential (smooth potential for $\ce{Si^{4+}}$ $\rightarrow$ works for band gaps in $\ce{Si}$).

Non-local pseudo-potentials:

  • 1979: HSC (norm-conserving PP: exact energies & nodeless $\psi$ for $r>r_c$).
  • 1980: Kerker PP (non-singluar PP: exact for $r>r_c$, $\dot\psi$ and $\ddot\psi$ matched with ab initio).
  • 1982: Kleinman-Bylander (highly simple).
  • 1990: RRKJ Optimized PPs (improve plane-wave convergence).
  • 1990: Ultra-soft / Vanderbilt (norm-conservation not needed for $\psi$: allows far more flexibility).
  • 1990: Troullier-Martins (scheme for generating very soft norm-conserving PPs for PWs).

Generalizations of pseudo-potentials:

  • 1994: Blöchl (generalization of PPs and LAPWs based on a linear transformation).
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    $\begingroup$ +1. Hard to know where to draw the line, but you might consider the RRKJ paper. doi.org/10.1103/PhysRevB.41.1227 $\endgroup$ – wcw Jul 16 '20 at 17:17
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    $\begingroup$ @wcw Thanks. You are free to edit my answer to add it in this list. It has 1000+ citations, so certainly can be considered a "milestone". I just don't know which sub-section to put it in, and how to summarize it in one line. You would also be encouraged to write a full answer explaining the RRKJ pseudopotentials! I think the author wants each of the PPs listed in my answer, to be explained in separate answers. $\endgroup$ – Nike Dattani Jul 17 '20 at 22:03
  • $\begingroup$ I suggest that the ultra-soft/vanderbilt method should also be considered a "generalization", since it relaxes the norm-conservation constraint and is (almost) formally equivalent to the Blochl PAW method (in fact it's more general, but the extra freedom isn't actually useful in the scientific context). $\endgroup$ – Phil Hasnip Aug 4 '20 at 0:03
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To add to Nike's list, one should also differentiate between the energy consistent pseudopotentials used in quantum chemistry and the shape consistent pseudopotentials that dominate in the solid state community. Energy consistent pseudopotentials are also shape consistent, while shape consistent pseudopotentials result in much larger errors in the energy than with the energy consistent potentials.

This minireview by Peter Schwerdtfeger might also be useful: ChemPhysChem 12, 3143 (2011).

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