# B.8 Contraction Problem

The contraction problem may be described by considering a general contracted ERI of $s$-type functions derived from the STO-3G basis set. Each basis function has degree of contraction $K=3$. Thus, the ERI may be written

 \displaystyle\begin{aligned} \displaystyle(ss|ss)&\displaystyle=\sum_{i=1}^{3}% \sum_{j=1}^{3}\sum_{k=1}^{3}\sum_{\ell=1}^{3}D_{Ai}D_{Bj}D_{Ck}D_{D\ell}\\ &\displaystyle\qquad\times\int e^{-\alpha_{i}|\mathbf{r}_{1}-\mathbf{A}|^{2}}e% ^{-\beta_{j}|\mathbf{r}_{1}-\mathbf{B}|^{2}}\left(\frac{1}{r_{12}}\right)e^{-% \gamma_{k}|\mathbf{r}_{2}-\mathbf{C}|^{2}}e^{-\delta_{\ell}|\mathbf{r}_{2}-% \mathbf{D}|^{2}}d\mathbf{r}_{1}d\mathbf{r}_{2}\\ &\displaystyle=\sum_{i=1}^{3}\sum_{j=1}^{3}\sum_{k=1}^{3}\sum_{\ell=1}^{3}[s_{% i}s_{j}|s_{k}s_{\ell}]\end{aligned} (B.5)

and requires 81 primitive integrals for the single ERI. The problem escalates dramatically for more highly contracted sets (STO-6G, 6-311G) and has been the motivation for the development of techniques for shell-pair modeling,11 in which a second shell-pair is constructed with fewer primitives that the first, but introduces no extra error relative to the integral threshold sought.

The Pople-Hehre axis-switch method776 is excellent for high contraction low angular momentum integral classes.