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(May 16, 2021)

The area of molecular integrals with respect to Gaussian basis functions has
recently been reviewed
^{
360
}
Adv. Quantum Chem.

(1994),
25,
pp. 141.
Link
and the user is referred to this review
for deeper discussions and further references to the general area. The purpose
of this short account is to present the basic approach, and in particular, the
implementation of ERI algorithms and aspects of interest to the user in the
AOInts package which underlies the Q-Chem program.

We begin by observing that all of the integrals encountered in an *ab
initio* calculation, of which overlap, kinetic energy, multipole moment,
internuclear repulsion, nuclear-electron attraction and inter electron
repulsion are the best known, can be written in the general form

$$(\mathrm{\mathbf{a}\mathbf{b}}|\mathrm{\mathbf{c}\mathbf{d}})=\int {\varphi}_{\mathbf{a}}({\mathbf{r}}_{1}){\varphi}_{\mathbf{b}}({\mathbf{r}}_{1})\theta ({r}_{12}){\varphi}_{\mathbf{c}}({\mathbf{r}}_{2}){\varphi}_{\mathbf{d}}({\mathbf{r}}_{2})\mathit{d}{\mathbf{r}}_{1}\mathit{d}{\mathbf{r}}_{2}$$ | (B.1) |

where the basis functions are contracted Gaussians (CGTF)

$${\varphi}_{\mathbf{a}}(\mathbf{r})={(x-{A}_{x})}^{{a}_{x}}{(y-{A}_{y})}^{{a}_{y}}{(z-{A}_{z})}^{{a}_{z}}\sum _{i=1}^{{K}_{a}}{D}_{ai}{e}^{-{\alpha}_{i}{|\mathbf{r}-\mathbf{A}|}^{2}}$$ | (B.2) |

and the operator $\theta $ is a two-electron operator. Of the two-electron operators (Coulomb, CASE, anti-Coulomb and delta-function) used in the Q-Chem program, the most significant is the Coulomb, which leads us to the ERIs.

An ERI is the classical Coulomb interaction, $\theta (x)=1/x$ in Eq. (B.1), between two charge distributions referred to as bras $(\mathrm{\mathbf{a}\mathbf{b}}|$ and kets $|\mathrm{\mathbf{c}\mathbf{d}})$.