STO-3G, by the numbers

Here is what STO-3G literally is for hydrogen — six numbers, fitted once in 1969 to mimic a ζ = 1 Slater orbital, tabulated, and used unchanged in every calculation since:

alpha = [3.42525091, 0.62391373, 0.16885540]   # exponents
d     = [0.15432897, 0.53532814, 0.44463454]   # contraction coefficients

Read the exponents as job assignments. The tight Gaussian (α = 3.43) patches the region near the nucleus where the cusp should be; the diffuse one (α = 0.17) fakes the exponential tail; the middle one carries the bulk. Their sum is a committee impersonating a Slater orbital — wrong in the first meter and the last mile, decent in between.

The coefficients d never change. What the SCF will optimize later are the molecular-orbital coefficients — how basis functions on different atoms combine — not the recipe inside each basis function.

The price of the impersonation, measured: a hydrogen atom in STO-3G comes out at −0.4666 hartree against the exact −0.5. Thirty-three milli-hartree for analytic integrals everywhere. Quantum chemistry took that deal and never looked back.