Given a sorted list of shell-pair data, it is possible to construct all potentially important shell-quartets by pairing of the shell-pairs with one another. Because the shell-pairs have been sorted, it is possible to deal with batches of integrals of the same type or class (e.g., $(ss|ss)$, $(sp|sp)$, $(dd|dd)$, etc.) where an integral class is characterized by both angular momentum ($L$) and degree of contraction ($K$). Such an approach is advantageous for vector processors and for semi-direct integral algorithms where the most expensive (high $K$ or $L$ integral classes can be computed once, stored in memory (or disk) and only less expensive classes rebuilt on each iteration.
While the shell-pairs may have been carefully screened, it is possible for a pair of significant shell-pairs to form a shell-quartet which need not be computed directly. Three cases are:
The quartet is equivalent, by point group symmetry, to another quartet already treated.
The quartet can be ignored on the basis of cheaply computed ERI bounds^{Gill:1994b} on the largest quartet bra-ket.
On the basis of an incremental Fock matrix build, the largest density matrix element which will multiply any of the bra-kets associated with the quartet may be negligibly small.
Note: Significance and negligibility is always based on the level of integral threshold set by the $rem variable THRESH.