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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.

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    $\begingroup$ They don't necessarily have to be normalised at all, as long as the trial wave function you are using them in is later normalised. Since GTO are supposed to mimic (approximate) STO, the first way you have proposed makes not much sense to me. $\endgroup$ Commented Aug 5, 2019 at 16:50

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