What is the difference, if any, between an effective core potential (ECP) and a pseudopotential?

Question

I've used these terms interchangeably for a while, but have noticed that in PySCF they are separate attributes.

Are there any crucial differences between an effective core potential (ECP) and a pseudopotential?

My initial thought was that ECPs are for gaussian-type orbitals (GTOs) and pseudopotentials are for plane-wave calculations, as I usually hear ECP in the context of molecules with heavy elements and pseudopotentials in the context of exploring extended (periodic) systems.

Answer 1

They are often used as synonyms

Schwerdtfeger's review paper "The Pseudopotential Approximation in Electronic Structure Theory" states that the two terms are used as synonyms in the electronic structure literature:

"As we are mostly interested in chemical properties coming from the valence space, we can try to replace the effect of the core electrons on a valence system in an atom, molecule or infinite system by a pseudopotential, which for this special case we call an effective core potential (ECP) (unfortunately, in electronic structure theory the term effective core potential and the more general term pseudopotential are used synonymously, the latter favoured by European theoretical chemists and the solid-state community, which we will adopt in the following."

But "Pseudopotential" can be more general

The above paragraph also says what my initial comment said, which is that "pseudopotential" can be a more general term (he says that it's "unfortunate" that electornic structure theorists tend to use the terms as synonyms). I would say that "pseudopotential" is a more general term because the name doesn't imply that it's the effects of core electrons that are replaced. The Fermi pseudopotential for neutron scattering by a nucleus might be a good example of a psuedopotential that doesn't replace the effects of core electrons.

PySCF attributes

You provided a link to the PySCF documentation, which indicates that ECPs and PPs are different attributes in PySCF. Indeed that page contains a script that contains the following comment:

 Input ECP with .ecp attribute and PP with .pseudo attribute.

However, immediately before that comment it says:

It is allowed to mix the Quantum chemistry effective core potentail (ECP) with crystal pseudo potential (PP).

and immediately after that same comment it says:

See also
pyscf/pbc/gto/pseudo/GTH_POTENTIALS for the GTH-potential format
pyscf/examples/gto/05-input_ecp.py for quantum chemistry ECP format

These suggest that the attributes have more to do with the format than anything else.

When we look at pyscf/examples/gto/05-input_ecp.py we see:

Na nelec 10
Na ul
0      2.0000000              6.0000000        
1    175.5502590            -10.0000000        
2      2.3365719             -6.0637782        
2      0.7799867             -0.7299393        
Na S
0    243.3605846              3.0000000        
1     41.5764759             36.2847626        
2     13.2649167             72.9304880        
2      0.9764209              6.0123861        
Na P
0   1257.2650682              5.0000000        
1    189.6248810            117.4495683        
2     54.5247759            423.3986704        
2      0.9461106              7.1241813        

But when we look at pyscf/pbc/gto/pseudo/GTH_POTENTIALS we see:

Na GTH-BLYP-q9 GTH-BLYP
    3    6
     0.23396502    2    -2.68948346    -0.50947770
    2
     0.14977690    1    32.85715860
     0.12319901    1   -13.99900802

Not only are these different potentials, but they are in completely different formats.

The PySCF authors may have just stuck with the "convention" that a lot of chemists tend to call them ECPs and a lot of physicists tend to call them PPs, but even in the same script the comments later admit that ECPs can also be called PPs, as in the case of the BFD (Burkatzki-Filippi-Dolg) ECP/PP:

Allow mixing quantum chemistry ECP (or BFD PP) and crystal PP in the same calculation.

The original BFD paper uses all three of the terms (pseudopotential, effective potential, effective core potential) in the same paper, and I think it also puts to rest your initial thought:

"My initial thought was that ECPs are for gaussian-type orbitals (GTOs) and pseudopotentials are for plane-wave calculations"

because the BFD PPs were designed for GTOs and molecules rather than for plane-waves and periodic systems.

Funny conclusion

• Effective core potentials are a specific type of psuedopotential
• Psuedopotentials are a specific type of effective potential

Longer conclusion

• "Effective core potentials" are pseudopotentials that are used to treat the effects of core electrons
• Pseudopotentials are a type of "effective potential", but effective potentials don't have to be pseudo (e.g. if we're just adding five non-psuedo potentials together to give us one combined "effective" potential).

PySCF has different attributes for:

• ECPs/PPs that are in a format used mostly in the study of periodic stystems. They call this the "GTH (Goedecker-Teter-Hutter) format" and they call these "crystal PPs"
• ECPs/PPs that are in a format used mostly in the study of molecules. They call this the "quantum chemistry ECP format" and they call these "quantum chemistry effective core potentials (ECP), but they admit that what they call the "BFD PPs" are actually ECPs.

The last point possibly reflects Schwerdtfeger's observation that the difference in terms used is not a matter of chemists vs physicists, but a matter of "non-European chemists", vs "physicists + European chemists", meaning that some chemists like Burkatzki, Filippi, and Dolg (from Germany, Netherlands, and Germany, at the time of the above-linked paper) use the term PP even when they're using GTOs for molecules/non-periodic-systems.