### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 1732-1754 |

Number of pages | 23 |

Journal | Journal of Mathematical Chemistry |

Volume | 57 |

Issue number | 7 |

DOIs | |

State | Published - Aug 15 2019 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Chemistry(all)
- Applied Mathematics

### Cite this

*Journal of Mathematical Chemistry*,

*57*(7), 1732-1754. https://doi.org/10.1007/s10910-019-01034-z

**Sparse low rank approximation of potential energy surfaces with applications in estimation of anharmonic zero point energies and frequencies.** / Rai, Prashant; Sargsyan, Khachik; Najm, Habib; Hirata, So.

Research output: Contribution to journal › Article

*Journal of Mathematical Chemistry*, vol. 57, no. 7, pp. 1732-1754. https://doi.org/10.1007/s10910-019-01034-z

}

TY - JOUR

T1 - Sparse low rank approximation of potential energy surfaces with applications in estimation of anharmonic zero point energies and frequencies

AU - Rai, Prashant

AU - Sargsyan, Khachik

AU - Najm, Habib

AU - Hirata, So

PY - 2019/8/15

Y1 - 2019/8/15

N2 - 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.

AB - 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.

KW - Anharmonic vibrations

KW - Compressed sensing

KW - Green’s function theory

KW - High dimensional integration

KW - Potential energy surfaces

KW - Tensor decomposition

UR - http://www.scopus.com/inward/record.url?scp=85067250745&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85067250745&partnerID=8YFLogxK

U2 - 10.1007/s10910-019-01034-z

DO - 10.1007/s10910-019-01034-z

M3 - Article

AN - SCOPUS:85067250745

VL - 57

SP - 1732

EP - 1754

JO - Journal of Mathematical Chemistry

JF - Journal of Mathematical Chemistry

SN - 0259-9791

IS - 7

ER -