By far the largest storage requirement in correlated methods involves quantities with four virtual indices. In the calculation of gradients, there are two such quantities: Γ(ab,cd) density matrix elements and <ab||cd> molecular orbital integrals. Direct evaluation of Γ(ab,cd) is reported here.

<ab||cd> integrals are used in three AMEs in ACESII:

VCC - to create a ladder contribution T(ij,cd)*<cd||ab>
LAMBDA - to create a ladder contribution Λ(cd,ij)*<ab||cd>
DENS - to create increments for I(ab) and X(ia) intermediates:
  I(ab) += Γ(ef,bg)*<ef||ag>
  X(ai) += Γ(ef,gi)*<ef||ga>

Since Γ(ef,bg) = τ(ij,ef)*Λ(bg,ij), it is easy to create this contraction by performing a ladder contraction of Z(μν,ij) = <μν||λσ> * Λ(λσ,ij) as in LAMBDA, and contracting the result with τ(ij,ef) after transforming Z(μν,ij) into MO basis.