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Distance-dependent Schwarz-based integral estimates for two-electron integrals: Reliable tightness vs. rigorous upper bounds
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10.1063/1.3693908
/content/aip/journal/jcp/136/14/10.1063/1.3693908
http://aip.metastore.ingenta.com/content/aip/journal/jcp/136/14/10.1063/1.3693908

Figures

Image of FIG. 1.
FIG. 1.

Error distribution of (a) combined and (b) pure far-field estimates for the DNA2 test system in the 6-31G* basis. The plot shows the number of integrals with log(F) smaller than (log F)max, where F is defined as F = I estimate /I exact . In (b) only integrals were evaluated for which the MBIE/QQR estimate is lower than the Schwarz estimate, which causes different total integral counts for the two methods.

Image of FIG. 2.
FIG. 2.

Results of the error statistics in the 6-31G* basis corresponding to the values given in Table I.

Image of FIG. 3.
FIG. 3.

Errors in S22 test set calculations with Schwarz and QQR screening with (a) SV(P) and (b) aug-cc-pVTZ basis sets (ϑK = 10−8). Reference for the errors are QQ (ϑK = 10−10) calculations.

Image of FIG. 4.
FIG. 4.

Errors of Schwarz and QQR screening with respect to (a) the distance of hydrogen to the benzene plane in the benzene·HCN complex (#19) of the S22 test set with SV(P) basis and (b) the distance along the h-bonds in the h-bonded uracil dimer (#5) of the S22 test set with aug-cc-pVTZ basis (ϑK = 10−8). Note the big errors in (b) of up to 93 μhartree for both methods that indicate an insufficiently tight threshold for the augmented basis. At a distance of 1.4 Å the threshold is not tight enough to converge the SCF procedure with QQR screening.

Image of FIG. 5.
FIG. 5.

Error vs. number of integrals for Schwarz and QQR screening with different thresholds ϑ K in (a) the h-bonded uracil dimer (#5) and (b) the phenol dimer (#22) of the S22 test set with aug-cc-pVTZ basis set. The points correspond to calculations with ϑ K values of 10−8, 10−9, 10−10, and 10−11. Note the anomalous behavior of the QQ curve in (b) due to fortuitous small errors for the less tight thresholds

Image of FIG. 6.
FIG. 6.

Error and speedup (via ratio of integrals) with the (a) 6-31G*, (b) SV(P), and (c) cc-pVTZ basis set for QQR calculations (right endpoint: ϑK = 10−8 and left endpoint: ϑK = 10−9) of the whole test set relative to the values of a pure Schwarz calculation with ϑ K = 10−8. The Schwarz reference is indicated as a black asterisk. Values to the right of this reference point indicate increased speed, while values below the reference indicate improved accuracy. Data tables can be found in the supplementary information.50

Image of FIG. 7.
FIG. 7.

Error and speedup (via ratio of integrals) for (a) MBIE and (b) scaled MBIE (scaling factor 0.3) calculations (right endpoint: ϑK = 10−8 and left endpoint: ϑK = 10−9) of the whole test set in the SV(P) basis relative to the values of a pure Schwarz calculation with ϑ K = 10−8. The Schwarz reference is indicated as a black asterisk. Values to the right of this reference point indicate increased speed, while values below the reference indicate improved accuracy. Data tables can be found in the supplementary information.50

Tables

Generic image for table
Table I.

Comparison of error statistics for MBIE and QQR integral estimates. Shown are the statistics of the ratio F = I estimate /I exact as the average , its smallest and largest values (F min , F max ), and the standard deviations of F averaged over all iterations.

Generic image for table
Table II.

Comparison of the logarithmic standard deviation σ(log(F)) of the ratio F = I estimate /I exact for MBIE and QQR far-field integral estimates. The standard deviation was determined in each iteration and averaged over all calculations.

Generic image for table
Table III.

QQR benchmark calculations in a 6-31G* basis. Errors are given with respect to the QQ (ϑK = 10−10) reference calculations. Speedups are given as the ratio of the number of integrals.

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/content/aip/journal/jcp/136/14/10.1063/1.3693908
2012-04-11
2014-04-21
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Distance-dependent Schwarz-based integral estimates for two-electron integrals: Reliable tightness vs. rigorous upper bounds
http://aip.metastore.ingenta.com/content/aip/journal/jcp/136/14/10.1063/1.3693908
10.1063/1.3693908
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