B.3 AOInts: Calculating ERIs with Q-Chem

The area of molecular integrals with respect to Gaussian basis functions has recently been reviewed³⁰⁵ 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

$$(\mathrm{𝐚𝐛}|\mathrm{𝐜𝐝})=\int {\varphi }_{𝐚}({𝐫}_1){\varphi }_{𝐛}({𝐫}_1)\theta (r_{12}){\varphi }_{𝐜}({𝐫}_2){\varphi }_{𝐝}({𝐫}_2)𝑑{𝐫}_1𝑑{𝐫}_2$$

(B.1)

where the basis functions are contracted Gaussians (CGTF)

$${\varphi }_{𝐚}(𝐫)={(x-A_x)}^{a_x}{(y-A_y)}^{a_y}{(z-A_z)}^{a_z}\sum _{i=1}^{K_a}{D}_{ai}{e}^{-{\alpha }_i{|𝐫-𝐀|}^2}$$

(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 $(\mathrm{𝐚𝐛}|$ and kets $|\mathrm{𝐜𝐝})$.