Two-electron integrals: the N⁴ wall

Electrons repel, and that interaction lives in the two-electron repulsion integrals (ERIs), written (ab|cd): the Coulomb repulsion between the charge cloud ab belonging to one electron and cd belonging to another.

Each ERI carries four basis-function indices, so there are formally N⁴ of them — the cost bottleneck of the whole method. Permutation symmetry (each value is shared by up to 8 index orderings) and distance screening shrink the constant factor, but the N⁴ scaling is exactly what makes big molecules expensive.