# B.6 Fundamental ERI

The fundamental ERI $[ss|ss]^{(0)}\equiv[\bm{0}]^{(0)}$, which is the basis of all ERI algorithms, is usually represented as303

 $[\bm{0}]^{(0)}=D_{A}D_{B}D_{C}D_{D}\int e^{-\alpha|\mathbf{r}_{1}-\mathbf{A}|^% {2}}e^{-\beta|\mathbf{r}_{1}-\mathbf{B}|^{2}}\left(\frac{1}{r_{12}}\right)e^{-% \gamma|\mathbf{r}_{2}-\mathbf{C}|^{2}}e^{-\delta|\mathbf{r}_{2}-\mathbf{D}|^{2% }}\;d\mathbf{r}_{1}d\mathbf{r}_{2}$ (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.298 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.