Are primitive GTOs normalised before contraction, or is the resultant basis function normalised afterwards?

Question

I'm quite curious about this:

In a basis set (I'll just use minimal-basis STO-nG basis sets for convenience), the basis functions are written as a linear combination of primitive GTOs. Are the GTOs normalised first, like this:

$$\Psi^\mathrm{STO} = \sum_i N_i c_i \phi_i^\mathrm{GTO}$$

where $\phi_i^\mathrm{GTO}$ are un-normalised GTO functions, and $N_i^{2} \left<\phi_i\left|\phi_i\right.\right> = 1$

or are they normalised like this (GTOs are not normalised):

$$\Psi^\mathrm{STO} = N_\mathrm{STO} \sum_i c_i \phi_i^\mathrm{GTO}$$

where $N_\mathrm{STO}^{2} \left<\Psi^\mathrm{STO}\left|\Psi^\mathrm{STO}\right.\right> = 1$?

I thought it was the former and not the latter, but after poking around the internet I'm not so sure anymore.

Answer 1

As a matter of fact, I believe it's both.

According to "Fundamentals of Molecular Integrals Evaluation", the primitive GTOs are first normalised, and then the entire linear combination of contracted GTOs are then normalised to unity.