Convergence Test

The resulting integral will only be accurate if the binomial expansion converges fast enough. For the concentric case we have a rigorous error bound for the kth term. The error introduced is of the order of the kth-plus-one term, where k is the number of terms in the expansion. Increasing k will obviously increase the accuracy of an integral, allowing for more of the shell-pair interactions to be calculated via this new method (provided that the two shell-pairs are far enough apart that the point-charge approximation still holds). But this will also increase the cost of a calculation. Using all terms up to, and including, ninth order has been found to be a good compromise. This will give an error of
$$\begin{align}\frac{\vert DC\vert}{\vert DB\vert} = \sqrt[10]{\mathrm{Error Bound}}. \end{align}$$

For the non-concentric case, the rigorous error bound is too complex to use in a computation, so an error bound is estimated by
$$\begin{align}\mathrm{Error Bound} = \left(\frac{\vert AB\vert+\vert CD\vert}{\vert DB\vert}\right)^{10}. \end{align}$$
Even if the shell-pair passes the above error-bound, |DB| must still be large enough for the point charge approximation to be valid. This is solved with a final restriction, that |DB| must be greater than some distance, chosen with the most diffuse functions and the desired accuracy in mind.