# B.4 Shell-Pair Data

It is common to characterize a bra, a ket and a bra-ket by their degree of contraction and angular momentum. In general, it is more convenient to compile data for shell-pairs rather than basis-function pairs. A shell is defined as that sharing common exponents and centers. For example, in the case of a number of Pople derived basis sets, four basis functions, encompassing a range of angular momentum types (i.e., $s$, $p_{x}$, $p_{y}$, $p_{z}$ on the same atomic center sharing the same exponents constitute a single shell.

The shell-pair data set is central to the success of any modern integral program for three main reasons. First, in the formation of shell-pairs, all pairs of shells in the basis set are considered and categorized as either significant or negligible. A shell-pair is considered negligible if the shells involved are so far apart, relative to their diffuseness, that their overlap is negligible. Given the rate of decay of Gaussian basis functions, it is not surprising that most of the shell-pairs in a large molecule are negligible, that is, the number of significant shell-pairs increases linearly with the size of the molecule. Second, a number of useful intermediates which are frequently required within ERI algorithms should be computed once in shell-pair formation and stored as part of the shell-pair information, particularly those which require costly divisions. This prevents re-evaluating simple quantities. Third, it is useful to sort the shell-pair information by type (i.e., angular momentum and degree of contraction). The reasons for this are discussed below.

Q-Chem’s shell-pair formation offers the option of two basic integral shell-pair cutoff criteria; one based on the integral threshold (\$rem variable THRESH) and the other relative to machine precision.

Intelligent construction of shell-pair data scales linearly with the size of the basis set, requires a relative amount of CPU time which is almost entirely negligible for large direct SCF calculations, and for small jobs, constitutes approximately 10% of the job time.