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?
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