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A.2 Calculating Electron Repulsion Integrals (ERIs)

A.2.2 Data Structures: Shell Pairs and Quartets

(May 7, 2024)

A.2.2.1 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, px, py, pz 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.

A.2.2.2 Shell-Quartets and Integral Classes

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 417 Gill P. M. W., Johnson B. G., Pople J. A.
    Chem. Phys. Lett.
    (1994), 217, pp. 65.
    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.