Distance-dependent Schwarz-based integral estimates for two-electron integrals: reliable tightness vs. rigorous upper bounds

A new integral estimate for four-center two-electron integrals is introduced that accounts for distance information between the bra- and ket-charge distributions describing the two electrons. The screening is denoted as QQR and combines the most important features of the conventional Schwarz screening by Ha?ser and Ahlrichs published in 1989 J. Comput. Chem. 10, 104 (1989)] and our multipole-based integral estimates (MBIE) introduced in 2005 D. S. Lambrecht and C. Ochsenfeld, J. Chem. Phys. 123, 184101 (2005)]. At the same time the estimates are not only tighter but also much easier to implement, so that we recommend them instead of our MBIE bounds introduced first for accounting for charge-distance information. The inclusion of distance dependence between charge distributions is not only useful at the SCF level but is particularly important for describing electron-correlation effects, e.g., within AO-MP2 theory, where the decay behavior is at least 1/R(4) or even 1/R(6). In our present work, we focus on studying the efficiency of our QQR estimates within SCF theory and demonstrate the performance for a benchmark set of 44 medium to large molecules, where savings of up to a factor of 2 for exchange integrals are observed for larger systems. Based on the results of the benchmark set we show that reliable tightness of integral estimates is more important for the screening performance than rigorous upper bound properties.