Searching....

# B.6.1 Overview

(December 20, 2021)

The fundamental ERI $[ss|ss]^{(0)}\equiv[\bm{0}]^{(0)}$, which is the basis of all ERI algorithms, is usually represented as 369 Gill P. M. W.
(1994), 25, pp. 141.

 $[\bm{0}]^{(0)}=D_{A}D_{B}D_{C}D_{D}\int e^{-\alpha|\mathbf{r}_{1}-\mathbf{A}|^% {2}}e^{-\beta|\mathbf{r}_{1}-\mathbf{B}|^{2}}\left(\frac{1}{r_{12}}\right)e^{-% \gamma|\mathbf{r}_{2}-\mathbf{C}|^{2}}e^{-\delta|\mathbf{r}_{2}-\mathbf{D}|^{2% }}\;d\mathbf{r}_{1}d\mathbf{r}_{2}$ (B.3)

which can be reduced to a one-dimensional integral of the form

 $[\bm{0}]^{(0)}=U(2\,{\vartheta}^{2})^{1/2}\left({\frac{2}{\pi}}\right)^{1/2}\,% \int_{0}^{1}e^{-Tu^{2}}du$ (B.4)

and can be efficiently computed using a modified Chebyshev interpolation scheme. 364 Gill P. M. W., Johnson B. G., Pople J. A.
Int. J. Quantum Chem.
(1991), 40, pp. 745.
Equation (B.4) can also be adapted for the general case $[\mathbf{0}]^{(m)}$ integrals required for most calculations. Following the fundamental ERI, building up to the full bra-ket ERI (or intermediary matrix elements, see later) are the problems of angular momentum and contraction.

Note:  Square brackets denote primitive integrals and parentheses denote fully-contracted integrals.