SCF Methods and Two-Electron Integral Screening

In nondirect SCF methods, all two-electron integrals are calculated once, stored on disk, and later reused in the subsequent SCF iterations. Because the number of integrals scales formally as M, storing and retrieving two-electron integrals is an extremely expensive step as far as disk space and input/ output (I/O) time are concerned. For large molecules the required disk space (and calculation time, see discussion below) easily exceeds all available capacities. 

Almlof et al. observed in a seminal paper that recomputing integrals whenever needed, rather than storing them to disk, could not only be 

Direct schemes have two advantages First, storage requirements are greatly decreased, and second, calculations for large molecules can actually be made much faster than for nondirect methods, because within the SCF iterations, information on the locality of the molecular system (via the one-particle density matrix see discussion below) can be exploited. In this way, the introduction of the direct SCF approach constitutes an important step toward the applicability of quantum chemical methods to large molecules. 

The formation of Fock-type matrices can be schematically divided into two steps  

In the following section, we focus first on the selection of numerically significant integrals, and later we discuss the different contractions of the two-electron integrals. 

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