# GTOs of higher quantum numbers?

I recently realised that the STO-nG basis functions were fit to 1s/2s/3d STOs of various exponents, instead of being fit to STOs of higher principal quantum numbers (such as 2s/3p orbitals etc.).

Has a Gaussian expansion of STOs of higher quantum numbers (for example, 2s or 3p STOs) been performed, and are there any GTO integration schemes for Cartesian GTOs of higher principal quantum numbers?

• Good question. I gave you a +1. If you have interest in basis sets, you might be interested in committing to the Materials Modeling Stack Exchange: area51.stackexchange.com/proposals/122958/…. Some questions so far: "Is there any hybrid of Gaussians and plane waves that is accepted as mainstream for materials modeling?" "How does a complete basis set remove Pulay forces?" "Is it practical to model [next comment] – user1271772 Jan 11 '20 at 2:50
• "Is it practical to model materials using Gaussians?", "Are plane wave basis sets re-optimized for each material?", "Why are there no correlation consistent basis sets for potassium that are generally contracted?" – user1271772 Jan 11 '20 at 2:51

I recently realised that the STO-nG basis functions were fit to 1s/20/3d STOs of various exponents, instead of being fit to STOs of higher principal quantum numbers (such as 2s/3p orbitals etc.).

If I understand what you are saying, this is incorrect (although feel free to clarify if I am not understanding the question).

Here is the paper documenting STO-nG for Na-Ar: https://doi.org/10.1063/1.1673374. The first sentence of the abstract:

"Least‐squares representations of the 3s and 3p Slater‐type atomic orbitals by a small number of Gaussian functions are presented."

However, it is true that they are expanded in terms of 1s and 2p Gaussians, which leads to your next question

Are there any GTO integration schemes for Cartesian GTOs of higher principal quantum numbers?

I am not aware of any codes that take advantage of higher principal quantum numbers. There likely isn't any need to.

The Molecular Electronic Structure Theory book (Helgaker et al), has this to say about STO-3G (page 290-291):

"Note the ... absence of a radial node in the STO-3G $$2s$$ orbital. In calculations using this basis, the $$2s$$ radial node will arise from a linear combination of the STO-3G 1s and 2s orbitals. The absence of a node in the STO-3G orbital is therefore not a problem."

These nodes arise from higher principal quantum numbers. Note that (almost) all basis set formats do not include principal quantum numbers. (A few do, but they are materials-oriented).

• Thanks for your answer! I was simply wondering if any basis set out there fit 2s basis functions to the 2s STO with radial component r e^-ζr, or the 3p basis function to the 3p STO with radial component r^2 e^-ζr. – ANZGC FlyingFalcon Jan 11 '20 at 0:54
• @Benjamin Pritchard: I gave this answer a +1. If you have interest in basis sets, you might be interested in committing to the Materials Modeling Stack Exchange: area51.stackexchange.com/proposals/122958/… Some questions so far: "Is there any hybrid of Gaussians and plane waves that is accepted as mainstream for materials modeling?" "How does a complete basis set remove Pulay forces?" "Is it practical to model [next comment] – user1271772 Jan 11 '20 at 2:53
• "Is it practical to model materials using Gaussians?", "Are plane wave basis sets re-optimized for each material?", "Why are there no correlation consistent basis sets for potassium that are generally contracted?" – user1271772 Jan 11 '20 at 2:53

Has a Gaussian expansion of STOs of higher quantum numbers (for example, 2s or 3p STOs) been performed

I am unaware of such fitting. Typically, basis sets using higher angular momenta (cc-pV$$n$$Z, def2-$$n$$ZVP, pc-$$n$$) are fitted on total energies or correlation energies. See https://www.basissetexchange.org/ for examples and references.

Are there any GTO integration schemes for Cartesian GTOs of higher principal quantum numbers?

Yes, there are. I have seen the following approach in action before:

• hand-crafted algorithms are used for common, low-angular-momentum cases
• more general schemes, but still optimized schemes for higher cases
• a catch-all, not very efficient, unspecific algorithm above a certain threshold
• You have quite good knowledge of basis sets. If you have interest in basis sets, you might be interested in committing to the Materials Modeling Stack Exchange: area51.stackexchange.com/proposals/122958/… Some questions so far: "Is there any hybrid of Gaussians and plane waves that is accepted as mainstream for materials modeling?" "How does a complete basis set remove Pulay forces?" "Is it practical to model [next comment] – user1271772 Jan 11 '20 at 2:53
• "Is it practical to model materials using Gaussians?", "Are plane wave basis sets re-optimized for each material?", "Why are there no correlation consistent basis sets for potassium that are generally contracted?" – user1271772 Jan 11 '20 at 2:53