^{1}, Daniel S. Lambrecht

^{1,a)}, Jörg Kussmann

^{1}and Christian Ochsenfeld

^{1,b)}

### Abstract

Efficient estimates for the preselection of two-electron integrals in atomic-orbital based Møller-Plesset perturbation theory (AO-MP2) theory are presented, which allow for evaluating the AO-MP2 energy with computational effort that scales linear with molecular size for systems with a significant HOMO-LUMO gap. The estimates are based on our recently introduced QQR approach [S. A. Maurer, D. S. Lambrecht, D. Flaig, and C. Ochsenfeld, J. Chem. Phys.136, 144107 (Year: 2012)10.1063/1.3693908], which exploits the asympotic decay of the integral values with increasing bra-ket separation as deduced from the multipole expansion and combines this decay behavior with the common Schwarz bound to a tight and simple estimate. We demonstrate on a diverse selection of benchmark systems that our AO-MP2 method in combination with the QQR-type estimates produces reliable results for systems with both localized and delocalized electronic structure, while in the latter case the screening essentially reverts to the common Schwarz screening. For systems with localized electronic structure, our AO-MP2 method shows an early onset of linear scaling as demonstrated on DNA systems. The favorable scaling behavior allows to compute systems with more than 1000 atoms and 10 000 basis functions on a single core that are clearly not accessible with conventional MP2 methods. Furthermore, our AO-MP2 method is particularly suited for parallelization and we present benchmark calculations on a protein-DNA repair complex comprising 2025 atoms and 20 371 basis functions.

C.O. acknowledges financial support by the Volkswagen Stiftung within the funding initiative “New Conceptual Approaches to Modeling and Simulation of Complex Systems” and by the SFB 749 “Dynamik und Intermediate molekularer Transformationen” (DFG).

I. INTRODUCTION

II. THEORY

A. Laplace AO-MP2

B. Integral estimates

III. COMPUTATIONAL DETAILS

IV. RESULTS

A. Robustness and efficiency of QQZZR4 screening

B. Benchmark calculations

C. Scaling behavior and timings

D. Parallelization and application to the MutM-DNA complex

V. CONCLUSION AND OUTLOOK

### Key Topics

- DNA
- 12.0
- Density functional theory
- 10.0
- Integral transforms
- 8.0
- Electronic structure
- 6.0
- Perturbation theory
- 5.0

## Figures

Decay behavior of a (*p* _{ F } *p* _{ F }|*p* _{ H } *p* _{ H }) integral shell block in a HF⋯HF dimer (6-31G** basis).

Decay behavior of a (*p* _{ F } *p* _{ F }|*p* _{ H } *p* _{ H }) integral shell block in a HF⋯HF dimer (6-31G** basis).

Dependence of the error of the Coulomb-type term for the first Laplace point on the number of calculated HTI shell quartets in QQZZR4 and pure Schwarz (QQZZ) screening calculations for a linear alkane C_{40}H_{82} (6-31G* basis) with external thresholds of ϑ = 10^{−6} (left point) and 10^{−7} (right point) as well as results based on a hypothetical exact screening with thresholds of ϑ = 10^{−8}, 10^{−9}, 10^{−10} (from left to right). The reference energy corresponds to a pure Schwarz calculation with ϑ = 10^{−10}. The exact screening is simulated by neglecting all integral shell blocks whose largest actual value is smaller than the chosen threshold. The internal screening threshold was fixed to 10^{−10} in all calculations.

Dependence of the error of the Coulomb-type term for the first Laplace point on the number of calculated HTI shell quartets in QQZZR4 and pure Schwarz (QQZZ) screening calculations for a linear alkane C_{40}H_{82} (6-31G* basis) with external thresholds of ϑ = 10^{−6} (left point) and 10^{−7} (right point) as well as results based on a hypothetical exact screening with thresholds of ϑ = 10^{−8}, 10^{−9}, 10^{−10} (from left to right). The reference energy corresponds to a pure Schwarz calculation with ϑ = 10^{−10}. The exact screening is simulated by neglecting all integral shell blocks whose largest actual value is smaller than the chosen threshold. The internal screening threshold was fixed to 10^{−10} in all calculations.

Error in the opposite-spin MP2 term and speedup (via ratio of integrals) for QQZZR4 AO-MP2 calculations of the whole test set with the 6-31G* basis set (right endpoint: ϑ = 10^{−6}, left endpoint: ϑ = 10^{−7}) relative to the values of a pure Schwarz (QQZZ) calculation with ϑ = 10^{−6}. The Schwarz reference is indicated as a black asterisk at the intersection of the horizontal and vertical line. Values to the right of this reference point indicate increased speed, while values below the reference indicate improved accuracy. The lines that end close to the reference point correspond to very compact or delocalized systems, where QQZZR4 essentially reverts to pure Schwarz screening. The underlying data to the plot can be found in Table II . System sizes in the test set range from 30 to 204 atoms and 250 to 1499 basis functions.

Error in the opposite-spin MP2 term and speedup (via ratio of integrals) for QQZZR4 AO-MP2 calculations of the whole test set with the 6-31G* basis set (right endpoint: ϑ = 10^{−6}, left endpoint: ϑ = 10^{−7}) relative to the values of a pure Schwarz (QQZZ) calculation with ϑ = 10^{−6}. The Schwarz reference is indicated as a black asterisk at the intersection of the horizontal and vertical line. Values to the right of this reference point indicate increased speed, while values below the reference indicate improved accuracy. The lines that end close to the reference point correspond to very compact or delocalized systems, where QQZZR4 essentially reverts to pure Schwarz screening. The underlying data to the plot can be found in Table II . System sizes in the test set range from 30 to 204 atoms and 250 to 1499 basis functions.

CPU times for SOS-MO-MP2, RI-MP2, and SOS-RI-MP2 (both with aux-SVP auxiliary basis) as well as SOS-AO-MP2 (ϑ = 10^{−6}) calculations on DNA systems with the 6-31G* basis. The number in brackets indicates the scaling behavior with respect to the previous point. Neither the common MO-MP2 nor any of the RI-MP2 versions are feasible for the largest systems due to their steep scaling with system size, so the data points (*) were extrapolated conservatively with the scaling behavior of the previous points (MO-MP2 4.41, RI-MP2 4.85, SOS-RI-MP2 3.94).

CPU times for SOS-MO-MP2, RI-MP2, and SOS-RI-MP2 (both with aux-SVP auxiliary basis) as well as SOS-AO-MP2 (ϑ = 10^{−6}) calculations on DNA systems with the 6-31G* basis. The number in brackets indicates the scaling behavior with respect to the previous point. Neither the common MO-MP2 nor any of the RI-MP2 versions are feasible for the largest systems due to their steep scaling with system size, so the data points (*) were extrapolated conservatively with the scaling behavior of the previous points (MO-MP2 4.41, RI-MP2 4.85, SOS-RI-MP2 3.94).

Cutout of the X-ray structure of a DNA double strand with an 8-oxoguanine lesion in complex with the MutM repair protein. Calculations of two conformers were performed at the SOS-AO-MP2 level of theory. The cutout comprises 2025 atoms and 20 371 basis functions in a 6-31G** basis. The SOS-AO-MP2 calculation took 8.5 days on a cluster with 160 cores in parallel.

Cutout of the X-ray structure of a DNA double strand with an 8-oxoguanine lesion in complex with the MutM repair protein. Calculations of two conformers were performed at the SOS-AO-MP2 level of theory. The cutout comprises 2025 atoms and 20 371 basis functions in a 6-31G** basis. The SOS-AO-MP2 calculation took 8.5 days on a cluster with 160 cores in parallel.

## Tables

Errors in the opposite-spin MP2 term for DNA_{2} (128 atoms) and Amylose_{8} (171 atoms) in 6-31G* basis and hydrogen chains with 6-31G** basis. Pure Schwarz screening (QQZZ), a modified—not recommended—QQZZR4 screening using only untransformed extents (QQZZR4 untrf), and the usual QQZZR4 screening are compared for a screening threshold of ϑ = 10^{−7}.

Errors in the opposite-spin MP2 term for DNA_{2} (128 atoms) and Amylose_{8} (171 atoms) in 6-31G* basis and hydrogen chains with 6-31G** basis. Pure Schwarz screening (QQZZ), a modified—not recommended—QQZZR4 screening using only untransformed extents (QQZZR4 untrf), and the usual QQZZR4 screening are compared for a screening threshold of ϑ = 10^{−7}.

Total number of half-transformed integrals (HTIs) and error of the opposite-spin AO-MP2 energy with respect to MO-MP2 for calculations in the 6-31G* basis. The absolute energies as well as the canonical reference values are provided in the supplementary material. ^{ 76 }

Total number of half-transformed integrals (HTIs) and error of the opposite-spin AO-MP2 energy with respect to MO-MP2 for calculations in the 6-31G* basis. The absolute energies as well as the canonical reference values are provided in the supplementary material. ^{ 76 }

Total number of half-transformed integrals (HTIs) and error of the opposite-spin AO-MP2/QQZZR4 energy with respect to MO-MP2 for calculations in the SV(P) basis. The absolute energies as well as the canonical reference values are provided in the supplementary material. ^{ 76 }

Total number of half-transformed integrals (HTIs) and error of the opposite-spin AO-MP2/QQZZR4 energy with respect to MO-MP2 for calculations in the SV(P) basis. The absolute energies as well as the canonical reference values are provided in the supplementary material. ^{ 76 }

The number of basis functions, the total number of half-transformed integrals (HTIs), the scaling with system size, and the error of the opposite-spin AO-MP2/QQZZR4 energy with respect to MO-MP2 for calculations on linear alkanes.

The number of basis functions, the total number of half-transformed integrals (HTIs), the scaling with system size, and the error of the opposite-spin AO-MP2/QQZZR4 energy with respect to MO-MP2 for calculations on linear alkanes.

Counterpoise-corrected SOS-AO-MP2/QQZZR4 results for the S22 test set with a SVP basis. The root-mean-square deviation (RMSD) of the errors in the interaction energy with respect to full SOS-MO-MP2 are given.

Counterpoise-corrected SOS-AO-MP2/QQZZR4 results for the S22 test set with a SVP basis. The root-mean-square deviation (RMSD) of the errors in the interaction energy with respect to full SOS-MO-MP2 are given.

Number of basis functions and HTI products as well as the scaling behavior of the number of HTIs with respect to the next smaller system for DNA double-strands with a 6-31G* basis set (ϑ = 10^{−6}).

Number of basis functions and HTI products as well as the scaling behavior of the number of HTIs with respect to the next smaller system for DNA double-strands with a 6-31G* basis set (ϑ = 10^{−6}).

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