Which basis set families were optimized to give the lowest possible variational energy for a given number of orbitals?

Question

Dunning's basis set families (such as cc-pVXZ and aug-cc-pCVXZ) were not optimized to give the lowest variational energy for a given number of orbitals. For example the S-type orbitals are optimized, then held fixed while P-type orbitals are optimized, then the S-type and P-type orbitals are both held fixed while the D-type orbitals are optimized, and so on. Likewise when the "tight" functions of the CV correction are optimized to create an cc-pCVXZ basis set, the exponents of the cc-pVXZ basis set are held fixed. When holding some parameters fixed during the optimization, one will not get the absolute optimal outcome possible for a given number of parameters.

Furthermore, the Dunning basis set sequence is constructed in such a way that when increasing the X in cc-pVXZ by one, a fixed number of new orbitals are added: for example when going from X=3 to X=4 (TZ to QZ) for some second row atom, there will be one new G-type exponent, one new F-type exponent, one new D-type exponent, and so on; but it's possible that a lower variational energy would be obtained with the same number of orbitals if we were to avoid adding the G-type function and instead invest in more P-type functions. This is because the goal of Dunning's basis set sequences, is to construct them in a systematic manner so that there will be a smooth extrapolation to the CBS (complete basis set) limit for some actual property (e.g. energy difference), whether or not a lower total energy for a particular X value could be obtained by adding more exponents for one type of orbital than another.

Which basis set families are constructed simply to get the lowest variational energy for a given number of orbitals, regardless of the ability to extrapolate well or the ability to optimize the exponents in a systematic way? I appreciate that this:

• will be much more costly than holding some exponents fixed while optimizing others, and that
• total energies might be improved at the expense of actual properties (like ionization energies or atomization energies) being worsened, and that
• extrapolations to the CBS limit will not be as smooth,

but I'm curious to know what exists!

For the purposes of this question, I am interested in one-particle basis set families, so not ECGs (explicitly correlated Gaussians) for example. You may choose to answer by giving just one basis set family in one answer, or by giving all examples you know in one answer.

Answer 1

I think there are none. If you look at e.g. the work by Dunning, the problem is that the functions appear in a semi-random order: e.g. the second D function may be energetically more important than the first F function, so you go from 1D -> 2D -> 2D1F -> etc, but it can also be less important, and then you would go 1D -> 1D1F -> 2D1F, etc. This is something I've seen first hand when trying to form property-optimized basis sets automatically; you can get pretty weird-looking results from such a procedure. (By 1D -> 2D -> 2D1F, I mean having one d-orbital, then two d-orbitals, then two d-orbitals and one f-orbital, in the basis set sequence).

The solution of picking the functions in groups like 1D, 2D1F, 3D2F1G, etc is elegant in that you get a more or less consistent basis in any case, since you collect all contributions of the same order. Petersson's nZaP basis sets that are designed for constant error in the energy per electron across the periodic table also use a similar composition, since they're likewise designed for post-HF calculations. In contrast, Jensen's sets have a different composure since the polarization effects behave differently to the correlation effects; for the pc sets the composition is more like 1D -> 2D1F -> 4D2F1G etc.

As to the part that deals with the exponents: Dunning-style sets are designed to be additive, so the exponents are optimized in steps - with the exception of the cc-pwCVXZ sets for transition metals where the exponents would overlap too much and so the exponents are relaxed. In contrast, Petersson's nZaP sets and the polarization-consistent Jensen basis sets use optimized primitives, although uses Hartree-Fock contracted orbitals and adds fresh primitives to describe the polarization and correlation effects, so it's not fully optimized at post-HF level of theory.