Sparse low rank approximation of potential energy surfaces with applications in estimation of anharmonic zero point energies and frequencies

  • Prashant RaiEmail author
  • Khachik Sargsyan
  • Habib Najm
  • So Hirata
Original Paper


We propose a method that exploits sparse representation of potential energy surfaces (PES) on a polynomial basis set selected by compressed sensing. The method is useful for studies involving large numbers of PES evaluations, such as the search for local minima, transition states, or integration. We apply this method for estimating zero point energies and frequencies of molecules using a three step approach. In the first step, we interpret the PES as a sparse tensor on polynomial basis and determine its entries by a compressed sensing based algorithm using only a few PES evaluations. Then, we implement a rank reduction strategy to compress this tensor in a suitable low-rank canonical tensor format using standard tensor compression tools. This allows representing a high dimensional PES as a small sum of products of one dimensional functions. Finally, a low dimensional Gauss–Hermite quadrature rule is used to integrate the product of sparse canonical low-rank representation of PES and Green’s function in the second-order diagrammatic vibrational many-body Green’s function theory (XVH2) for estimation of zero-point energies and frequencies. Numerical tests on molecules considered in this work suggest a more efficient scaling of computational cost with molecular size as compared to other methods.


Potential energy surfaces Tensor decomposition Anharmonic vibrations Green’s function theory Compressed sensing High dimensional integration 



The authors thank Judit Zador at Sandia National Laboratories, Livermore for fruitful discussions. Support for this work was provided through the Scientific Discovery through Advanced Computing (SciDAC) program funded by the U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research and Basic Energy Sciences under Award No. DE-FG02-12ER46875. Sandia National Laboratories is a multimission laboratory operated by National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. The views expressed in the article do not necessarily represent the views of the U.S. Department of Energy or the United States Government. Sandia has major research and development responsibilities in nuclear deterrence, global security, defense, energy technologies and economic competitiveness, with main facilities in Albuquerque, New Mexico, and Livermore, California. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.


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Copyright information

© This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2019

Authors and Affiliations

  • Prashant Rai
    • 1
    Email author
  • Khachik Sargsyan
    • 1
  • Habib Najm
    • 1
  • So Hirata
    • 2
  1. 1.Sandia National LaboratoriesLivermoreUSA
  2. 2.Department of ChemistryUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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