Fast linear scaling second-order Møller-Plesset perturbation theory (MP2) using local and density fitting approximations

We apply density fitting approximations to generate the 2-electron integrals in local MP2 (LMP2) to produce a method denoted DF-LMP2. The method can equally be seen as a local version of the well-known RI-MP2 method, which in this work is referred to as DF-MP2. Local approximations reduce the asymptotic scaling of computational resources to $\mathcal{O}\mathrm{(N),}$ and the most expensive step of DF-MP2 [the $\mathcal{O}{\mathrm{(N}}^5]$ assembly) is rendered negligible in DF-LMP2. It is demonstrated that for large molecules DF-LMP2 is much faster (1–2 orders of magnitude) than either LMP2 or DF-MP2. The availablility of LMP2, DF-MP2 and DF-LMP2 has for the first time made it possible to assess the accuracy of local and density fitting approximations for extended molecules using cc-pVDZ and cc-pVTZ basis sets. The density fitting errors are found to be consistently small, but the errors arising from local approximations are somewhat larger than expected from calculations on smaller systems. It is proposed to apply local density fitting approximations also for the Fock matrix construction in Hartree-Fock calculations. Preliminary results demonstrate that this can lead to significant savings in the Hartree-Fock calculation.