H₂'s matrices, with real numbers
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Formulas earn their keep when you run them. For H₂ at R = 1.4 bohr in STO-3G — two basis functions, one per atom — every one-electron matrix is 2×2. Here they are from an actual run (the same one the in-browser SCF executes):
Now read them like a chemist. The off-diagonal overlap S₁₂ = 0.66 is enormous — these two 'atomic' orbitals are two-thirds the same function, because Gaussians at 1.4 bohr separation still have plenty of mass between the nuclei. The kinetic matrix is positive (confinement always costs). The attraction matrix is the deepest thing on the page, and even its off-diagonal element is −1.19: an electron shared between the atoms still feels both nuclei strongly. That large negative H₁₂ is, numerically, the chemical bond waiting to happen.
Two Gaussian primitives have exponents α (large) and β (small) on centres A and B. Their product Gaussian sits at P. Where is P?
Every entry came from a closed-form expression — the Gaussian product theorem collapsed each two-center product into one Gaussian, and the only special function needed was the Boys function inside V.
Why do the overlap and kinetic-energy integrals between two Gaussians on different centres have simple closed forms?