“Know how to solve every problem that has been solved.” “What I cannot create, I do not understand.” — Richard Feynman

Sixteen entries, four numbers

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Lesson 11 of 24 standard ~5 min

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 complete two-electron content of H₂/STO-3G.

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.

standardMultiple choice

Real-orbital ERIs obey an 8-fold permutation symmetry, e.g. (ab|cd) = (ba|cd) = (cd|ab) = …. Practically, this means a code can…

standardMultiple choice

In chemist notation, what does the two-electron integral (ab|cd) represent?