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Discrete variable representation in electronic structure theory: Quadrature grids for least-squares tensor hypercontraction

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10.1063/1.4802773

### Abstract

We investigate the application of molecular quadratures obtained from either standard Becke-type grids or discrete variable representation (DVR) techniques to the recently developed least-squares tensor hypercontraction (LS-THC) representation of the electron repulsion integral (ERI) tensor. LS-THC uses least-squares fitting to renormalize a two-sided pseudospectral decomposition of the ERI, over a physical-space quadrature grid. While this procedure is technically applicable with any choice of grid, the best efficiency is obtained when the quadrature is tuned to accurately reproduce the overlap metric for quadratic products of the primary orbital basis. Properly selected Becke DFT grids can roughly attain this property. Additionally, we provide algorithms for adopting the DVR techniques of the dynamics community to produce two different classes of grids which approximately attain this property. The simplest algorithm is radial discrete variable representation (R-DVR), which diagonalizes the finite auxiliary-basis representation of the radial coordinate for each atom, and then combines Lebedev-Laikov spherical quadratures and Becke atomic partitioning to produce the full molecular quadrature grid. The other algorithm is full discrete variable representation (F-DVR), which uses approximate simultaneous diagonalization of the finite auxiliary-basis representation of the full position operator to produce non-direct-product quadrature grids. The qualitative features of all three grid classes are discussed, and then the relative efficiencies of these grids are compared in the context of LS-THC-DF-MP2. Coarse Becke grids are found to give essentially the same accuracy and efficiency as R-DVR grids; however, the latter are built from explicit knowledge of the basis set and may guide future development of atom-centered grids. F-DVR is found to provide reasonable accuracy with markedly fewer points than either Becke or R-DVR schemes.

© 2013 AIP Publishing LLC

Received 30 January 2013
Accepted 03 April 2013
Published online 21 May 2013

Acknowledgments: R.M.P. is a DOE Computational Science Graduate Fellow (Grant No. DE-FG02-97ER25308). This work was supported by the National Science Foundation through grants to C.D.S. (Grant No. CHE-1011360) and T.J.M. (Grant No. CHE-1047577) and by the Department of Defense (Office of the Director of Defense Research and Engineering) through a National Security Science and Engineering Faculty Fellowship (T.J.M.). T.J.M. thanks David Manolopoulos for early discussions about non-direct product DVR grids for electronic structure theory. The Center for Computational Molecular Science and Technology is funded through a NSF CRIF award (Grant No. CHE-0946869) and by Georgia Tech.

Article outline:

I. INTRODUCTION

II. THEORY

A. 1-Dimensional Gaussian quadrature

B. 1-Dimensional discrete variable representation

C. *N*-dimensional discrete variable representation

D. Radial discrete variable representation

E. Full discrete variable representation

III. RESULTS

A. Computational details

B. Qualitative grid features

C. Quantitative grid fidelity

IV. CONCLUSIONS

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2013-05-21

2014-04-16

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