Sixteen entries, four numbers
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With two basis functions, the ERI tensor (μν|λσ) has 2⁴ = 16 entries. The 8-fold permutation symmetry crushes that to just four unique values. From the H₂ run at 1.4 bohr:
(11|11) = 0.7746 both electrons on the same atom
(11|22) = 0.5697 one cloud on each atom
(11|12) = 0.4441 one on-atom, one shared
(12|12) = 0.2970 both clouds spread across the bond The ordering is physical, not accidental. Two charge clouds piled on the same atom repel hardest (0.77). Clouds on different atoms repel like point charges at distance R — and indeed 0.5697 is close to 1/1.4 = 0.714 softened by their spread. The exchange-type integral (12|12), built from charge distributions that are themselves smeared across the bond, is smallest. When the Fock matrix later weighs Coulomb against exchange, these four numbers are the entire menu.
Real-orbital ERIs obey an 8-fold permutation symmetry, e.g. (ab|cd) = (ba|cd) = (cd|ab) = …. Practically, this means a code can…
In chemist notation, what does the two-electron integral (ab|cd) represent?