The cost of HF Theory

The bottleneck for HF calculations is the generation of all the two-electron integrals in the atomic basis, $(\mu\nu\vert\lambda\sigma)$. The number of these grows as O(N⁴), where N is the total number of basis functions in the system. However, this scaling can be drastically reduced by viewing $(\mu\nu\vert\lambda\sigma)$ as the repulsion between two shell-pairs,
$$\begin{align}(\mu\nu\vert = \phi^{*}_{\mu}({\bf r}_{1})\phi_{\nu}({\bf r}_{1}) \end{align}$$
and
$$\begin{align}\vert\lambda\sigma) = \phi^{*}_{\lambda}({\bf r}_{2})\phi_{\sigma}({\bf r}_{2}). \end{align}$$

If either $(\mu\nu\vert$ or $\vert\lambda\sigma)$ is so small that it is negligible, the integral $(\mu\nu\vert\lambda\sigma)$ will be negligible. If the two shells of a shell-pair are very far apart (relative to their diffuseness) then their overlap will produce a negligible shell-pair. The number of non-negligible shell-pairs grows only linearly with the size of the system. So, forming only the significant shell-pairs will generate a number of integrals which grows only quadratically with molecular size. Hence the cost of HF theory is O(N²).

There is a separate O(N³) cost involved in diagonalising the Fock matrix, yet this scaling does not become noticeable until the system is extremely large, and is not expected to be a problem in the medium term [29]. Recent techniques have made HF theory scale only linearly with molecular size for certain types of systems; these are dealt with in later chapters.