^{1}, Daniel S. Lambrecht

^{1,a)}, Denis Flaig

^{1}and Christian Ochsenfeld

^{1,b)}

### Abstract

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 Häser and Ahlrichs published in 1989 [J. Comput. Chem.10, 104 (1989)10.1002/jcc.540100111] and our multipole-based integral estimates (MBIE) introduced in 2005 [D. S. Lambrecht and C. Ochsenfeld, J. Chem. Phys.123, 184101 (2005)10.1063/1.2079967]. 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.

We dedicate this work in honour of Professor Dr. Christoph Bräuchle (LMU Munich) on the occasion of his 65th birthday. C.O. thanks Professor Dr. Peter Pulay (University of Arkansas) for his question during a conference in Bad Herrenalb in 2006 whether the rigorous bound property of integral estimates is essential. While C.O. answered at that time, also by citing the work of Jan Almlöf that a rigorous bound is expected to be crucial, we believe that *reliable tightness* is more important in the present context as exemplified by our present work. The authors also thank Dr. Jörg Kussmann (University of Munich, LMU) for valuable discussions and an anonymous referee for useful comments on our manuscript. Furthermore, C.O. acknowledges financial support by the Volkswagen Stiftung within the funding initiative “New Conceptual Approaches to Modeling and Simulation of Complex Systems.”

I. INTRODUCTION

II. THEORY

III. COMPUTATIONAL DETAILS

A. Error statistics

B. Assessment of QQR on S22 test set

C. Benchmark calculations for recommended testsuite

IV. CONCLUSION

### Key Topics

- Basis sets
- 10.0
- DNA
- 10.0
- Correlation functions
- 5.0
- Carbon nanotubes
- 4.0
- Charge coupled devices
- 4.0

## Figures

Error distribution of (a) combined and (b) pure far-field estimates for the DNA_{2} 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.

Error distribution of (a) combined and (b) pure far-field estimates for the DNA_{2} 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.

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

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

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

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

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 (ϑ_{K} = 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.

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 (ϑ_{K} = 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.

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

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

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: ϑ_{K} = 10^{−8} and left endpoint: ϑ_{K} = 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}

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: ϑ_{K} = 10^{−8} and left endpoint: ϑ_{K} = 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}

Error and speedup (via ratio of integrals) for (a) MBIE and (b) scaled MBIE (scaling factor 0.3) calculations (right endpoint: ϑ_{K} = 10^{−8} and left endpoint: ϑ_{K} = 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}

Error and speedup (via ratio of integrals) for (a) MBIE and (b) scaled MBIE (scaling factor 0.3) calculations (right endpoint: ϑ_{K} = 10^{−8} and left endpoint: ϑ_{K} = 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

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.

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.

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.

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.

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

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

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