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

When creating molecular orbitals via the linear combination of atomic orbitals method I am told that Slater orbitals provide a more accurate representation whilst Gaussian orbitals are computationally more straightforward to integrate (two-electron integrals for instance), since the product of Gaussians is another Gaussian.

However, the Slater-type orbital can itself be approximated by a linear combination of a few Gaussians. This means when we represent the integrals we do so via many more basis functions than if we were to use Slater-type orbitals, however my supervisor assures me that despite there being a higher number of functions when using Gaussian-type orbitals (compared to Slater) the Gaussian-type orbitals' far simpler integration makes it substantially more computationally efficient.

I was wondering if there was anyone who could provide me with a source or some data that quantifies just how much more computationally efficient doing Hartree-Fock calculations with Gaussian-type orbitals is than using Slater-type orbitals.

According to Wikipedia:

The speedup by 4—5 orders of magnitude compared to Slater orbitals more than outweighs the extra cost entailed by the larger number of basis functions generally required in a Gaussian calculation.

However, the Wikipedia article does not give a reference for this and I was just wondering if this is actually an accurate statement and any evidence for this would be really helpful. I could not find any sources that verified this and was wondering if anyone knew of such a source.

In the realm of high-accuracy wavefunction-based ab initio calculations, Michal Lesiuk and co-workers have in recent years put some serious effort into "re-examining" the option of using Slater-type orbitals. At first sight, their paper "Reexamination of the calculation of two-center, two-electron integrals over Slater-type orbitals. III. Case study of the beryllium dimer" seemed to show evidence that Slater-type orbitals gave a better result for the well depth of the $$\ce{Be2}$$ potential than any previous calculation: their calculated energy with Slater-type orbitals was 929.0±1.9 cm$$^{−1}$$ and the empirical result that they compared to was 929.7±2.0 cm$$^{−1}$$. However, that empirical result was based on an older paper that had a major error (the author's didn't know that $$\ce{Be2}$$ has one more vibrational level before the molecule dissociated). A more careful analysis by two different groups, who both used different models to fit the experimental data, arrived at empirical values for this energy to be 934.8±0.3 cm$$^{−1}$$ and 935.0±0.3 cm$$^{−1}$$, and guess what: That's the answer that Gaussian-type orbitals predicted several years earlier when Jacek Koput used an aug-cc-pCV7Z(i) basis set on the same system and got an energy of 935 cm$$^{−1}$$.