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### STO-1G in Gaussian

I don't believe there is a "STO-1G." Even if there were, it would exhibit poor performance. The STO-nG basis sets were first published by Hehre, Stewart, and Pople in 1969: "Self-...
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One Electron Starting with your trial wavefunction: $$\Phi = \sum^k_{i=1} c_i \chi_i,$$ you seem to be thinking about one-electron orbitals $\chi_i(r)$ which will add up to a one-electron molecular ...