Highly Efficient and Scalable Compound Decomposition of TwoElectron Integral Tensor and Its Application in Coupled Cluster Calculations
Abstract
The representation and storage of twoelectron integral tensors are vital in large scale applications of accurate electronic structure methods. Lowrank representation and efficient storage strategy of integral tensors can significantly reduce the numerical overhead and consequently timetosolution of these methods. In this paper, by combining pivoted incomplete Cholesky decomposition (CD) with a followup truncated singular vector decomposition (SVD), we develop a decomposition strategy to approximately represent the twoelectron integral tensor in terms of lowrank vectors. A systematic benchmark test on a series of 1D, 2D, and 3D carbonhydrogen systems demonstrates high efficiency and scalability of the compound twostep decomposition of the twoelectron integral tensor in our implementation. For the size of atomic basis set N_b ranging from ~ 100 up to ~ 2, 000, the observed numerical scaling of our implementation shows O(N_b^{2.5~3}) versus O(N_b^{3~4}) of single CD in most of other implementations. More importantly, this decomposition strategy can significantly reduce the storage requirement of the atomicorbital (AO) twoelectron integral tensor from O(N_b^4) to O(N_b^2 log_{10}(N_b)) with moderate decomposition thresholds. The accuracy tests have been performed using ground and excitedstate formulations of coupled cluster formalism employing single and double excitations (CCSD) on several bench mark systems including the C_{60} moleculemore »
 Authors:
 William R. Wiley Environmental Molecular Sciences Laboratory, Battelle, Pacific Northwest National Laboratory, K891, P. O. Box 999, Richland, Washington 99352, United States
 Publication Date:
 Research Org.:
 Pacific Northwest National Laboratory (PNNL), Richland, WA (US), Environmental Molecular Sciences Laboratory (EMSL)
 Sponsoring Org.:
 USDOE
 OSTI Identifier:
 1414535
 Report Number(s):
 PNNLSA126829
Journal ID: ISSN 15499618; 49220
 DOE Contract Number:
 AC0576RL01830
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Chemical Theory and Computation; Journal Volume: 13; Journal Issue: 9
 Country of Publication:
 United States
 Language:
 English
 Subject:
 54 ENVIRONMENTAL SCIENCES; Environmental Molecular Sciences Laboratory
Citation Formats
Peng, Bo, and Kowalski, Karol. Highly Efficient and Scalable Compound Decomposition of TwoElectron Integral Tensor and Its Application in Coupled Cluster Calculations. United States: N. p., 2017.
Web. doi:10.1021/acs.jctc.7b00605.
Peng, Bo, & Kowalski, Karol. Highly Efficient and Scalable Compound Decomposition of TwoElectron Integral Tensor and Its Application in Coupled Cluster Calculations. United States. doi:10.1021/acs.jctc.7b00605.
Peng, Bo, and Kowalski, Karol. 2017.
"Highly Efficient and Scalable Compound Decomposition of TwoElectron Integral Tensor and Its Application in Coupled Cluster Calculations". United States.
doi:10.1021/acs.jctc.7b00605.
@article{osti_1414535,
title = {Highly Efficient and Scalable Compound Decomposition of TwoElectron Integral Tensor and Its Application in Coupled Cluster Calculations},
author = {Peng, Bo and Kowalski, Karol},
abstractNote = {The representation and storage of twoelectron integral tensors are vital in large scale applications of accurate electronic structure methods. Lowrank representation and efficient storage strategy of integral tensors can significantly reduce the numerical overhead and consequently timetosolution of these methods. In this paper, by combining pivoted incomplete Cholesky decomposition (CD) with a followup truncated singular vector decomposition (SVD), we develop a decomposition strategy to approximately represent the twoelectron integral tensor in terms of lowrank vectors. A systematic benchmark test on a series of 1D, 2D, and 3D carbonhydrogen systems demonstrates high efficiency and scalability of the compound twostep decomposition of the twoelectron integral tensor in our implementation. For the size of atomic basis set N_b ranging from ~ 100 up to ~ 2, 000, the observed numerical scaling of our implementation shows O(N_b^{2.5~3}) versus O(N_b^{3~4}) of single CD in most of other implementations. More importantly, this decomposition strategy can significantly reduce the storage requirement of the atomicorbital (AO) twoelectron integral tensor from O(N_b^4) to O(N_b^2 log_{10}(N_b)) with moderate decomposition thresholds. The accuracy tests have been performed using ground and excitedstate formulations of coupled cluster formalism employing single and double excitations (CCSD) on several bench mark systems including the C_{60} molecule described by nearly 1,400 basis functions. The results show that the decomposition thresholds can be generally set to 10^{4} to 10^{3} to give acceptable compromise between efficiency and accuracy.},
doi = {10.1021/acs.jctc.7b00605},
journal = {Journal of Chemical Theory and Computation},
number = 9,
volume = 13,
place = {United States},
year = 2017,
month = 8
}

Coupledcluster methods provide highly accurate models of molecular structure through explicit numerical calculation of tensors representing the correlation between electrons. These calculations are dominated by a sequence of tensor contractions, motivating the development of numerical libraries for such operations. While based on matrix–matrix multiplication, these libraries are specialized to exploit symmetries in the molecular structure and in electronic interactions, and thus reduce the size of the tensor representation and the complexity of contractions. The resulting algorithms are irregular and their parallelization has been previously achieved via the use of dynamic scheduling or specialized data decompositions. We introduce our efforts tomore »

Relativistic generalorder coupledcluster method for highprecision calculations: Application to the Al{sup +} atomic clock
We report the implementation of a generalorder relativistic coupledcluster method for performing highprecision calculations of atomic and molecular properties. As a first application, the blackbody radiation shift of the Al{sup +} clock has been estimated precisely. The computed shift relative to the frequency of the 3s{sup 2} {sup 1}S{sub 0}{sup e}{yields}3s3p {sup 3}P{sub 0}{sup o} clock transition given by (3.66{+}0.60)x10{sup 18} calls for an improvement over the recent measurement with a reported result of (9{+}3)x10{sup 18}[Phys. Rev. Lett. 104, 070802 (2010)].