Schwarz integral screening

In practice, integral screening is employed before the integrals are computed. The screening is performed using the Schwarz inequality... 

Saddle optimization, 329 Scalar relativistic corrections, 209 Scaled External Correlation (SEC), Scaled All Correlation (SAC) models, 169 Scaling of different methods with basis set, 145 Schrodinger equation, 2, 53 Schwarz inequality, for integral screening, 78 Second quantization, 411 Second-order corrections, in perturbation methods, 126... 

The Schwarz estimates introduced by Haser and Ahlrichs are now used in almost every quantum chemical code for two-electron integral screening. However, they are not optimal in a certain sense They do not describe the 1/R decay behavior between the charge distributions of the two-electron integrals, as we will explain shortly. 

Figure 19 Comparison of the number of significant Coulomb-type integral products (CnHin+i/b-dlG basis in units of 10 ) as estimated over shells by Schwarz-type screening (QQZZ 10 ) and MBIE (10 ) with the exact number of products selected via basis functions. For the latter, a threshold of has been selected to provide comparable accuracy in the absolute energies of 0.1 mH (only data for the first Laplace coefficient in computing the MP2 energy is listed).

Almlof pointed out in 1972 that the missing 1/R dependence in the Schwarz screening might be approximated by the following equation via overlap integrals (Sa and Sb) ... 

Note that this bound is independent of the separation between the distributions pq. Since the integral decreases in magnitude with 7 pq, the Schwarz inequality provides a best estimate for small PQ. However, since Rpq is usually considerably smaller than say 100oo> this overestimate does not reduce the usefulness of the Schwarz screening severely. 

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