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(June 30, 2021)

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
^{
124
}
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
^{
123
}
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.
^{
885
}
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
^{
1176
}
Chem. Phys. Lett.

(1994),
230,
pp. 8.
Link
^{,}
^{
24
}
Chem. Phys. Lett.

(1996),
254,
pp. 329.
Link
^{,}
^{
284
}
J. Phys. Chem.

(1996),
100,
pp. 6272.
Link
^{,}
^{
189
}
J. Chem. Phys.

(1997),
106,
pp. 5526.
Link
^{,}
^{
977
}
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 $\mathcal{O}(N)$ 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.

1950 | Boys | 124 | ERIs with low $L$ and low $K$ |

1969 | Pople | 885 | ERIs with low $L$ and high $K$ |

1976 | Dupuis | 290 | Integrals with any $L$ and low $K$ |

1978 | McMurchie | 751 | Integrals with any $L$ and low $K$ |

1982 | Almlöf | 36 | Introduction of the direct SCF approach |

1986 | Obara | 803 | Integrals with any $L$ and low $K$ |

1988 | Head-Gordon | 428 | Integrals with any $L$ and low $K$ |

1991 | Gill | 354, 359 | Integrals with any $L$ and any $K$ |

1994 | White | 1176 | J matrix in linear work |

1996 | Schwegler | 977, 978 | HF exchange matrix in linear work |

1997 | Challacombe | 189 | Fock matrix in linear work |