# Critical difference and roles of Slater- and Gaussian-type orbitals

I was very interested in the following question and answer about the Slater- and Gaussian-type orbitals (STO and GTO) used in quantum chemistry and a practical calculation.

Gaussian-type orbitals vs Slater-type orbitals in terms of computational efficiency

After reading this discussion, I am further interested in the critical difference and roles of these STO and GTO orbitals, because I have considered (or misunderstood!) that the STO is actually hard to calculate but always outperforms the GTO in computational accuracy.

My question is that, does the GTO simply represent the STO by using many Gaussians, or not?

Intuitively, it would be better to use many GTOs to represent the molecular orbitals in a many electron system in a flexible way, because it is not bound by the form of the hydrogen atom wavefunction. However, the GTO does not have a cusp and so... I'm very wondering how to understand this problem.

• +1 & welcome to our new community! Thank you very much for contributing your question here and we hope to see much more of you in the future!! I commented out your second question because we really prefer each question here to go in a separate post. Also, in this community we really like to vote a lot, as you can see here and here and here. Since you found that discussion so fascinating, perhaps give some votes! Jun 19 at 1:50

"My question is that, does the GTO simply represent the STO by using many Gaussians, or not?"

There does exist the STO-nG basis set family in which $$n$$ Gaussian orbitals are fitting to represent a single STO. These are never used for serious applications to predict spectroscopic, thermodynamical, or kinetic properties of molecules (or atoms) though. They can be seen for testing new software, and recently they've started to get used for testing quantum computing hardware, but they are simply not accurate enough for serious applications.

For more commonly used basis sets like the cc-pVXZ series, I don't like to think of GTOs the above way. The parameters of the GTOs are not optimized to reproduce a single STO, but instead for something else such as minimizing the energy given some constraints.

Perhaps for the sole case of a hydrogen atom, where the wavefunction obtained from solving the Schroedinger equation (with a non-relativistic Hamiltonian, not containing any fine, hyperfine, or other terms which would make it more accurate) would have the STO shape, you can think of a large number of GTOs "modeling" the STO shape. But for many-electron systems, the true wavefunction is not necessarily going to have a shape descried by STOs, so I would just say the GTOs are modeling the shape of the wavefunction, not the shape of the STO.

• Thank you very much for your detailed answer! I realized that the role of many GTOs is not only to approximate an STO of the hydrogen atom wavefunction with the cusp (I studied this with Donald A. McQuarrie's textbook) but also to describe the molecular wavefunction of many-electron systems. I found that the latter is more important in practice.
– neco
Jun 19 at 12:04

It's important to also note that the STO-nG family is a Gaussian fit to an old, minimal STO basis. Larger STO basis sets ranging up to quadruple-zeta quality are included in e.g. the ADF program that is based on Slater-type orbital basis sets.

As Nike already said, more reasonable GTO basis sets don't even try to imitate STO basis sets, but rather just aim for the best possible energies and transferability across systems. Once the basis set is big enough, the form of the individual functions matters less than the span of the set of basis functions. A good example of this are fully numerical basis sets, where the basis functions may not look anything like atomic orbitals; however, you are able to describe the exact solution to arbitrary accuracy using such functions.

STOs are also not exact solutions to anything except the hydrogen atom. Numerical atomic orbitals (NAOs) form a much better basis in this respect; they are almost immune to basis set superposition error because already the minimal NAO basis is exact for non-interacting atoms.

You can read my review article in Int. J. Quantum Chem. 119, e25968 for more discussion on GTO vs STO vs NAO and discussion on various fully numerical methods.

• Thank you for the answer. It was very helpful. In particular, I agreed that "Once the basis set is big enough, the form of the individual functions matters less than the span of the set of basis functions." I think that you are familiar with this kind of topic, do you have any references or research papers on this topic? I would like to study more details about the trade off problem about the STO with cusp and many GTOs without cusp and NAO in a practical quantum-chemical calculation for many-electron systems.
– neco
Jun 25 at 10:45
• @neco such studies are not without problems as many technical aspects differ significantly between GTO vs STO vs NAO calculations. Note that while STOs have better cusps and asymptotic decay properties, the latter also causes more linear dependencies in the basis set, so in this respect one could argue GTOs are better! Jun 26 at 21:04