A.1.1 Overview

Within the Q-Chem program, an Atomic Orbital integrals (AOInts) package has been developed which, while relatively invisible to the user, is one of the keys to the overall speed and efficiency of the Q-Chem program. The key challenge was summarized by Gill: ⁴¹³ Gill P. M. W., Head-Gordon M., Pople J. A.
J. Phys. Chem.
(1990), 94, pp. 5564. Link

Ever since Boys’ introduction of Gaussian basis sets to quantum chemistry in 1950, the calculation and handling of the notorious two-electron repulsion integrals (ERIs) over Gaussian functions has been an important avenue of research for practicing computational chemists. Indeed, the emergence of practically useful computer programs has been fueled in no small part by the development of sophisticated algorithms to compute the very large number of ERIs that are involved in calculations on molecular systems of even modest size.

The ERI engine of any competitive quantum chemistry software package will be one of the most complicated aspects of the package as whole. Coupled with the importance of such an engine’s efficiency, a useful yardstick of a program’s anticipated performance can be quickly measured by considering the components of its ERI engine. In recent times, developers at Q-Chem, Inc. have made significant contributions to the advancement of ERI algorithm technology (for example, see Refs.  413 Gill P. M. W., Head-Gordon M., Pople J. A.
J. Phys. Chem.
(1990), 94, pp. 5564. Link , 420 Gill P. M. W.
Adv. Quantum Chem.
(1994), 25, pp. 141. Link , 22 Adams T. R., Adamson R. D., Gill P. M. W.
J. Chem. Phys.
(1997), 107, pp. 124. Link , 380 Frisch M. J. et al.
Chem. Phys. Lett.
(1993), 206, pp. 225. Link , 415 Gill P. M. W., Johnson B. G., Pople J. A.
Int. J. Quantum Chem.
(1991), 40, pp. 745. Link , 419 Gill P. M. W., Pople J. A.
Int. J. Quantum Chem.
(1991), 40, pp. 753. Link , 417 Gill P. M. W., Johnson B. G., Pople J. A.
Chem. Phys. Lett.
(1994), 217, pp. 65. Link , 504 Head-Gordon M., Pople J. A.
J. Chem. Phys.
(1988), 89, pp. 5777. Link , 608 Johnson B. G., Gill P. M. W., Pople J. A.
Chem. Phys. Lett.
(1993), 206, pp. 229. Link , 609 Johnson B. G., Gill P. M. W., Pople J. A.
Chem. Phys. Lett.
(1993), 206, pp. 239. Link ), and it is not surprising that Q-Chem’s AOInts package is considered the most advanced of its kind.

Prior to the 1950s, the most difficult step in the systematic application of Schrödinger wave mechanics to chemistry was the calculation of the notorious two-electron integrals that measure the repulsion between electrons. Boys ¹⁴¹ Boys S. F.
Proc. Roy. Soc. Ser. A
(1950), 200, pp. 542. Link showed that this step can be made easier (although still time consuming) if Gaussian, rather than Slater, orbitals are used in the basis set. Following the landmark paper of computational chemistry ¹⁴⁰ Boys S. F. et al.
Nature
(1956), 178, pp. 1207. Link (again due to Boys) programs were constructed that could calculate all the ERIs that arise in the treatment of a general polyatomic molecule with $s$ and $p$ orbitals. However, the programs were painfully slow and could only be applied to the smallest of molecular systems.

In 1969, Pople constructed a breakthrough ERI algorithm, a hundred time faster than its predecessors. The algorithm remains the fastest available for its associated integral classes and is now referred to as the Pople-Hehre axis-switch method. ¹⁰³⁹ Pople J. A., Hehre W. J.
J. Comput. Phys.
(1978), 27, pp. 161. Link

Over the two decades following Pople’s initial development, an enormous amount of research effort into the construction of ERIs was documented, which built on Pople’s original success. Essentially, the advances of the newer algorithms could be identified as either better coping with angular momentum ($L$) or, contraction ($K$); each new method increasing the speed and application of quantum mechanics to solving real chemical problems.

By 1990, another barrier had been reached. The contemporary programs had become sophisticated and both academia and industry had begun to recognize and use the power of ab initio quantum chemistry, but the software was struggling with “dusty deck syndrome” and it had become increasingly difficult for it to keep up with the rapid advances in hardware development. Vector processors, parallel architectures and the advent of the graphical user interface were all demanding radically different approaches to programming and it had become clear that a fresh start, with a clean slate, was both inevitable and desirable. Furthermore, the integral bottleneck had re-emerged in a new guise and the standard programs were now hitting the $N^2$ wall. Irrespective of the speed at which ERIs could be computed, the unforgiving fact remained that the number of ERIs required scaled quadratically with the size of the system.

The Q-Chem project was established to tackle this problem and to seek new methods that circumvent the $N^2$ wall. Fundamentally new approaches to integral theory were sought and the ongoing advances that have resulted ¹³⁶⁰ White C. A. et al.
Chem. Phys. Lett.
(1994), 230, pp. 8. Link ^{, 23} Adamson R. D., Dombroski J. P., Gill P. M. W.
Chem. Phys. Lett.
(1996), 254, pp. 329. Link ^{, 328} Dombroski J. P., Taylor S. W., Gill P. M. W.
J. Phys. Chem.
(1996), 100, pp. 6272. Link ^{, 218} Challacombe M., Schwegler E.
J. Chem. Phys.
(1997), 106, pp. 5526. Link ^{, 1139} Schwegler E., Challacombe M.
J. Chem. Phys.
(1996), 105, pp. 2726. Link have now placed Q-Chem firmly at the vanguard of the field. It should be emphasized, however, that the $𝒪(N)$ methods that we have developed still require short-range ERIs to treat interactions between nearby electrons, thus the importance of contemporary ERI code remains.

The chronological development and evolution of integral methods can be summarized by considering a time line showing the years in which important new algorithms were first introduced. These are best discussed in terms of the type of ERI or matrix elements that the algorithm can compute efficiently.