# What is the meaning of energy-consistent and shape-consistent in the context of pseudopotentials?

I am trying to use pseudopotentials to model scattering from heavy molecules.

However, I am confused by some of the terminologies of pseudopotentials/ECPs:

Norm-conserving according to Wikipedia and other sources are the potentials which obey a few mathematical conditions and seem to find a compromise between transferrability and softness.

However, I do not understand what is meant by shape consistent and energy consistent pseudopotentials and would like some sort of basic description about them.

• +1 and welcome to our new community! Thank you for contributing your question here and we hope to see much more of you in the future !!! I see you asked a lot of questions on Physics.SE, what brought you here? I would guess that energy-consistent pseudopotentials aim to help users get accurate energies and shape-consistent pseudopotentials aim to help users get accurate wavefunction shapes, but I personally haven't seen those terms before. – Nike Dattani Feb 20 at 5:08
• I had to remove the second part of the question because we have a policy to ask only one question per post, and asking for a list of things, ending in "etc." makes the question very open-ended and hard to answer. My reading of the subject suggests that energy-consistent PPs are superior to solely shape-consistent ones, as energy-consistent PPs are also shape-consistent but the converse is not necessarily true, and the review paper cited at the end of my answer suggests that shape-consistent PPs are fast to optimize, so that's some of the pros and cons, but if you want most, please ask a new Q! – Nike Dattani Feb 20 at 5:37
• Hi @NikeDattani! I only just discovered that this STACK community existed and so joined as my MSc project in the Summer that I am currently in the process of doing research for revolves around the modelling of electron-molecule scattering using the R-matrix method, particularly for heavy molecules. We will use pseudopotentials as they allow us to easily account for relativistic effects and also make the code more computationally efficient since we can reduce the number of electrons in the problem by removing the core electrons and nucleus by replacing them with an effective core potential. – DJA Feb 20 at 12:39

This 2017 paper titled "Shape and Energy Consistent Pseudopotentials for Correlated Electron systems" defines energy-consistent psuedopotentials this way:

"A combined reproduction of core scattering, core polarisation, and atomic excitation energies allows the generation of a new pseudopotential from correlated electron calculations, referred to as an energy consistent correlated electron pseudopotential (eCEPP)."

where by "reproduction", the abstract hints that they compare results using the pseudopotential to all-electron results. This means that they do calculations for core scattering, core polarization, and atomic excitation, all with every electron correlated, and then they do the same calculations but with the pseudopotential taking care of most of the electron correlation, and after comparing the two, if the psuedopotential method gives the same results as the all-electron calculation, then it is labelled as "energy-consistent".

Unfortunately that article does in my opinion, not as good of a job at defining what a shape-consistent pseudopotential is, but thanks to that, I had to look for a better resource to show you, and arrived at this 1999 paper ("On the accuracy of averaged relativistic shape-consistent pseudopotentials") which defines both shape-consistent pseudopotentials and "energy-adjusted" pseudopotentials which gives some perspective on how the terminology can vary slightly from paper to paper. The "energy-adjusted pseudopotentials" or EAPP are described in the following way:

"The EAPP are based on quantum mechanical observables (atomic excitations, ionization energies, etc. and the starting point is either a quasi-relativistic or a 4-component all-electron calculation."

So again, they are comparing energies from relativistic all-electron calculations to energies obtained with the pseudopotential, but the specific energies that are being compared are not necessarily exactly the same from paper to paper.

They also describe shape-consistent pseudopotentials or SCPPs, and as suspected in my initial comment, this has to do with the accuracy of the shape of the wavefunction:

"In contrast to the previous method, the SCPP are based on orbital properties"

I'll also mention that this answer by Susi Lehtola points out that energy-consistent PPs are shape-consistent, but shape-consistent PPs are not necessarily energy-consistent, and refers the reader to this excellent review paper: "The Pseudopotential Approximation in Electronic Structure Theory." That paper describes shape-consistent potentials in the following way:

"If the pseudopotential parameters are fitted in such a way that the valence orbitals of different symmetries reproduce to a high accuracy the corresponding all-electron orbitals from a certain cut-off (valence) radius $$r_c$$ onwards with matching orbital energies, we obtain shape-consistent pseudopotentials."